diff --git a/scripts/generate_synthetic_provers.jl b/scripts/generate_synthetic_provers.jl index f27185c5..aa14f914 100644 --- a/scripts/generate_synthetic_provers.jl +++ b/scripts/generate_synthetic_provers.jl @@ -99,6 +99,150 @@ function generate_nuprl()::Vector{Dict{String,Any}} ("choice_nat", "all R:Nat->Nat->Prop. (all n:Nat. exists m:Nat. R(n,m)) => exists f:Nat->Nat. all n:Nat. R(n, f(n))", "Intro; CountableChoice; Auto"), ("well_ordering_nat", "all P:Nat->Prop. (exists n:Nat. P(n)) => exists m:Nat. P(m) & (all k:Nat. k < m => ~P(k))", "StrongInduction; Auto"), ("brouwer_continuity", "all f:(Nat->Nat)->Nat. exists n:Nat. all a,b:Nat->Nat. (all i:Nat. i < n => a(i) = b(i)) => f(a) = f(b)", "Continuity; Auto"), + ("church_rosser", "all M,N,P:Term. reduces(M, N) => reduces(M, P) => exists Q:Term. reduces(N, Q) & reduces(P, Q)", "Induction; Diamond; Auto"), + ("fan_theorem", "all T:Tree. (all b:Branch. bar(T, b)) => exists n:Nat. uniform_bar(T, n)", "BarInduction; Auto"), + ("kripke_schema", "all A:Prop. (A \\/ ~A) => decidable(A)", "Intro; OrElim; Auto"), + ] + + sets = [ + ("set_ext", "all A:Type, s1,s2:Set(A). (all x:A. x in s1 <=> x in s2) => s1 = s2", "Intro; SetExt; Auto"), + ("union_comm", "all A:Type, s1,s2:Set(A). union(s1, s2) = union(s2, s1)", "SetExt; Auto"), + ("union_assoc", "all A:Type, s1,s2,s3:Set(A). union(union(s1,s2),s3) = union(s1,union(s2,s3))", "SetExt; Auto"), + ("inter_comm", "all A:Type, s1,s2:Set(A). intersect(s1, s2) = intersect(s2, s1)", "SetExt; Auto"), + ("distrib_union_inter", "all A:Type, s1,s2,s3:Set(A). union(s1, intersect(s2, s3)) = intersect(union(s1,s2), union(s1,s3))", "SetExt; Auto"), + ("empty_subset", "all A:Type, s:Set(A). subset(empty, s)", "Intro; ElimEmpty; Auto"), + ("union_superset", "all A:Type, s1,s2:Set(A). subset(s1, union(s1, s2))", "Intro; UnionElim; Auto"), + ("power_set_monotone", "all A:Type, s1,s2:Set(A). subset(s1, s2) => subset(pow(s1), pow(s2))", "Intro; Intro; Apply; Auto"), + ("finite_union", "all A:Type, s1,s2:FiniteSet(A). finite(union(s1, s2))", "Intro; FiniteUnion; Auto"), + ("cardinal_union", "all A:Type, s1,s2:FiniteSet(A). card(union(s1,s2)) = card(s1) + card(s2) - card(intersect(s1,s2))", "InclusionExclusion; Auto"), + ] + + relations = [ + ("refl_closure", "all A:Type, R:A->A->Prop, x:A. refl_trans_closure(R)(x, x)", "Intro; RTC_Refl; Auto"), + ("trans_composition", "all A:Type, R:A->A->Prop, x,y,z:A. trans_closure(R)(x,y) => trans_closure(R)(y,z) => trans_closure(R)(x,z)", "Intro; Induction; Auto"), + ("equiv_sym", "all A:Type, R:A->A->Prop, x,y:A. equivalence(R) => R(x, y) => R(y, x)", "Intro; Intro; ApplySym; Auto"), + ("well_founded_induction", "all A:Type, R:A->A->Prop. well_founded(R) => all P:A->Prop. (all x:A. (all y:A. R(y,x) => P(y)) => P(x)) => all x:A. P(x)", "Intro; WFInduction; Auto"), + ("partial_order_antisym", "all A:Type, R:A->A->Prop, x,y:A. partial_order(R) => R(x,y) => R(y,x) => x = y", "Intro; ApplyAntisym; Auto"), + ("total_order_trichotomy", "all A:Type, R:A->A->Prop, x,y:A. total_order(R) => R(x,y) \\/ x = y \\/ R(y,x)", "Intro; Trichotomy; Auto"), + ("graph_path_compose", "all V:Type, E:V->V->Prop, x,y,z:V. path(E)(x,y) => path(E)(y,z) => path(E)(x,z)", "Induction; Auto"), + ("converse_relation", "all A,B:Type, R:A->B->Prop, x:A, y:B. R(x,y) <=> converse(R)(y,x)", "Intro; Split; Auto"), + ] + + number_theory = [ + ("euclid_lemma", "all p:Prime, a,b:Nat. divides(p, a*b) => divides(p, a) or divides(p, b)", "Intro; CaseGCD; Auto"), + ("fundamental_arith", "all n:Nat. n >= 2 => exists! ps:List(Prime). product(ps) = n", "StrongInduction; Auto"), + ("bezout", "all a,b:Nat. a > 0 \\/ b > 0 => exists x,y:Int. a*x + b*y = gcd(a,b)", "Intro; ExtendedEuclid; Auto"), + ("infinitude_primes", "all n:Nat. exists p:Prime. p > n", "Intro; Witness `factorial(n)+1`; Auto"), + ("wilson_theorem", "all p:Prime. (factorial(p-1) + 1) mod p = 0", "Intro; ModArith; Auto"), + ("fermat_little", "all p:Prime, a:Nat. coprime(a, p) => (a^(p-1)) mod p = 1", "Intro; GroupOrder; Auto"), + ("chinese_remainder", "all m,n:Nat. coprime(m,n) => all a,b:Int. exists x:Int. (x mod m = a) & (x mod n = b)", "Intro; Intro; CRT; Auto"), + ("sum_of_squares", "all n:Nat. (exists a,b:Nat. n = a*a + b*b) <=> all p:Prime. p `divides` n & p mod 4 = 3 => even(mult(p, n))", "Classify; Auto"), + ("pigeon_hole", "all n,m:Nat, f:Fin(n)->Fin(m). n > m => exists i,j:Fin(n). i <> j & f(i) = f(j)", "Intro; PHP; Auto"), + ] + + real_analysis = [ + ("lim_sum", "all f,g:Nat->Real, a,b:Real. lim(f) = a => lim(g) = b => lim(fun n. f(n)+g(n)) = a+b", "Intro; LimitArith; Auto"), + ("lim_prod", "all f,g:Nat->Real, a,b:Real. lim(f) = a => lim(g) = b => lim(fun n. f(n)*g(n)) = a*b", "Intro; LimitArith; Auto"), + ("squeeze", "all f,g,h:Nat->Real. (all n:Nat. f(n) <= g(n) <= h(n)) => lim(f) = lim(h) => lim(g) = lim(f)", "Intro; Squeeze; Auto"), + ("continuous_compose", "all f,g:Real->Real. continuous(f) => continuous(g) => continuous(compose(f, g))", "Intro; EpsilonDelta; Auto"), + ("ivt", "all f:Real->Real, a,b:Real, y:Real. continuous(f) => f(a) <= y <= f(b) => exists c:Real. a <= c <= b & f(c) = y", "Intro; Bisection; Auto"), + ("mvt", "all f:Real->Real, a,b:Real. differentiable(f) => a < b => exists c:Real. a < c < b & derivative(f)(c) = (f(b)-f(a))/(b-a)", "Intro; Rolle; Auto"), + ("cauchy_complete", "all s:Nat->Real. cauchy(s) => exists L:Real. lim(s) = L", "Intro; CompletenessAxiom; Auto"), + ("compact_attains", "all f:Real->Real, K:Set(Real). compact(K) => continuous(f) => exists m,M:Real. m in f(K) & M in f(K) & all y in f(K). m <= y <= M", "Intro; Intro; ExtremeValue; Auto"), + ] + + computability = [ + ("halting_undecidable", "~exists H:(Prog->Input->Bool). all p:Prog, x:Input. H(p, x) = true <=> halts(p, x)", "Diagonalization; Auto"), + ("rice_theorem", "all P:(Prog->Prop). nontrivial(P) => ~decidable(P)", "Intro; Reduction; Auto"), + ("recursion_theorem", "all f:Prog->Prog. exists e:Prog. equal_programs(e, f(e))", "KleeneRecursion; Auto"), + ("smn_theorem", "all n,m:Nat, f:Prog. exists s:Prog. all x,y:Input. run(f, pair(x,y)) = run(run(s, x), y)", "Intro; Intro; SMN; Auto"), + ("kleene_normal", "all f:PartialFn. exists e,U:Prog. all x:Input. f(x) = run(U, pair(e, x))", "Intro; NormalForm; Auto"), + ("church_turing_lambda", "all f:ComputableFn. exists t:Term. reduces_to_numeral(t(numeral(x)), numeral(f(x)))", "Intro; Encode; Auto"), + ] + + category_theory = [ + ("cat_comp_assoc", "all C:Cat, f:Hom(a,b), g:Hom(b,c), h:Hom(c,d). comp(comp(h,g),f) = comp(h,comp(g,f))", "Intro; CatAssoc; Auto"), + ("cat_id_left", "all C:Cat, f:Hom(a,b). comp(id(b), f) = f", "Intro; CatIdLeft; Auto"), + ("cat_id_right", "all C:Cat, f:Hom(a,b). comp(f, id(a)) = f", "Intro; CatIdRight; Auto"), + ("functor_comp", "all F:Functor, f:Hom(a,b), g:Hom(b,c). Fmap(comp(g,f)) = comp(Fmap(g), Fmap(f))", "Intro; FunctorLaw; Auto"), + ("functor_id", "all F:Functor, a:Obj. Fmap(id(a)) = id(Fobj(a))", "Intro; FunctorIdLaw; Auto"), + ("natural_trans", "all F,G:Functor, eta:NatTrans(F,G), f:Hom(a,b). comp(eta(b), Fmap_F(f)) = comp(Fmap_G(f), eta(a))", "Intro; Naturality; Auto"), + ("iso_compose", "all C:Cat, f:Hom(a,b), g:Hom(b,a). comp(g, f) = id(a) => comp(f, g) = id(b) => isomorphism(f)", "Intro; Isomorphism; Auto"), + ("product_universal", "all C:Cat, p:Obj, f:Hom(p,a), g:Hom(p,b). exists! h:Hom(p, prod(a,b)). comp(pi1, h) = f & comp(pi2, h) = g", "Intro; Universal; Auto"), + ] + + topology = [ + ("open_union", "all X:TopSpace, U,V:Open(X). open(union(U, V))", "Intro; OpenAxiom; Auto"), + ("open_finite_inter", "all X:TopSpace, U,V:Open(X). open(intersect(U, V))", "Intro; OpenAxiom; Auto"), + ("continuous_compose", "all X,Y,Z:TopSpace, f:X->Y, g:Y->Z. continuous(f) => continuous(g) => continuous(compose(g, f))", "Intro; Continuous; Auto"), + ("compact_closed", "all X:TopSpace, K:Subspace(X). compact(K) => hausdorff(X) => closed(K)", "Intro; Compact; Auto"), + ("connected_image", "all X,Y:TopSpace, f:X->Y. continuous(f) => connected(X) => connected(image(f))", "Intro; Connected; Auto"), + ("hausdorff_limit", "all X:TopSpace, s:Nat->X, L1,L2:X. hausdorff(X) => lim(s) = L1 => lim(s) = L2 => L1 = L2", "Intro; Hausdorff; Auto"), + ("homeomorphism", "all X,Y:TopSpace, f:X->Y. continuous(f) => continuous(inverse(f)) => homeomorphism(f)", "Intro; Homeomorphism; Auto"), + ] + + algebra = [ + ("group_inv_unique", "all G:Group, a,b,c:G. op(a,b) = e => op(a,c) = e => b = c", "Intro; Cancel; Auto"), + ("subgroup_closed", "all G:Group, H:Subgroup(G), a,b:H. op(a, b) in H", "Intro; SubgroupAxiom; Auto"), + ("lagrange", "all G:FinGroup, H:Subgroup(G). divides(order(H), order(G))", "Intro; Lagrange; Auto"), + ("homomorphism_kernel", "all G,H:Group, f:Homomorphism(G, H). subgroup(kernel(f), G)", "Intro; KernelAxiom; Auto"), + ("ring_add_group", "all R:Ring. group(R, +)", "Intro; RingAxiom; Auto"), + ("ideal_closed", "all R:Ring, I:Ideal(R), a:R, b:I. op(a, b) in I", "Intro; IdealAxiom; Auto"), + ("field_no_zero_div", "all F:Field, a,b:F. a*b = 0 => a = 0 or b = 0", "Intro; FieldAxiom; Auto"), + ("vector_space_sum", "all V:VectorSpace, v,w:V. v + w = w + v", "Intro; VectorAxiom; Auto"), + ] + + probability = [ + ("prob_nonneg", "all P:ProbMeasure, A:Event. P(A) >= 0", "Intro; ProbAxiom; Auto"), + ("prob_total", "all P:ProbMeasure. P(universe) = 1", "Intro; ProbAxiom; Auto"), + ("prob_additivity", "all P:ProbMeasure, A,B:Event. disjoint(A, B) => P(union(A, B)) = P(A) + P(B)", "Intro; Additivity; Auto"), + ("bayes_rule", "all P:ProbMeasure, A,B:Event. P(B) > 0 => P(A | B) = P(A & B) / P(B)", "Intro; BayesDef; Auto"), + ("conditional_indep", "all P:ProbMeasure, A,B,C:Event. indep(A, B | C) <=> P(A & B | C) = P(A | C) * P(B | C)", "Intro; Split; Auto"), + ("markov_ineq", "all X:RandomVar, a:PosReal. P(X >= a) <= E(X) / a", "Intro; Markov; Auto"), + ("expectation_linear", "all X,Y:RandomVar, a,b:Real. E(a*X + b*Y) = a*E(X) + b*E(Y)", "Intro; Linearity; Auto"), + ("variance_nonneg", "all X:RandomVar. var(X) >= 0", "Intro; Variance; Auto"), + ] + + bishop_analysis = [ + ("bishop_real_approx", "all x:Real, n:PosInt. exists q:Rational. abs(x - q) < 1/n", "Intro; ConstructReal; Auto"), + ("bishop_seq_modulus", "all s:Nat->Real, eps:PosReal. cauchy(s) => exists M:Nat. all m,n:Nat. m >= M => n >= M => abs(s(m) - s(n)) < eps", "Intro; Intro; ModulusCauchy; Auto"), + ("bishop_cont_uniform", "all f:Real->Real, K:CompactInterval. continuous(f) => uniformly_continuous_on(f, K)", "Intro; Intro; HeineCantor; Auto"), + ("bishop_rational_dense", "all a,b:Real. a < b => exists q:Rational. a < q < b", "Intro; RationalDensity; Auto"), + ("bishop_sqrt_exists", "all x:NonNegReal. exists y:NonNegReal. y*y = x", "Intro; Newton; Auto"), + ] + + proof_theory = [ + ("cut_elim_nuprl", "all A,B. |- A -> (A |- B) -> |- B", "Intros; CutAdmissible; Auto"), + ("proof_identity", "all A:Prop. |- A -> A", "Intro; Axiom; Auto"), + ("proof_weakening", "all A,B:Prop. |- B => (A |- B)", "Intros; Weakening; Auto"), + ("subtype_refl", "all T:Type. T subtype T", "Intro; SubtypeRefl; Auto"), + ("subtype_trans", "all T1,T2,T3:Type. T1 subtype T2 => T2 subtype T3 => T1 subtype T3", "Intros; SubtypeTrans; Auto"), + ("equality_symmetry", "all A:Type, x,y:A. x = y => y = x", "Intros; EqSym; Auto"), + ("equality_trans", "all A:Type, x,y,z:A. x = y => y = z => x = z", "Intros; EqTrans; Auto"), + ] + + information = [ + ("entropy_nonneg", "all P:ProbDist. entropy(P) >= 0", "Intro; EntropyDef; Auto"), + ("entropy_uniform_max", "all n:PosNat, P:ProbDist(Fin(n)). entropy(P) <= log(n)", "Intro; Intro; MaxEntropy; Auto"), + ("mutual_info_nonneg", "all X,Y:RandomVar. I(X; Y) >= 0", "Intro; MIDef; Auto"), + ("chain_rule_entropy", "all X,Y:RandomVar. H(X, Y) = H(X) + H(Y | X)", "Intro; ChainRule; Auto"), + ("kraft_inequality", "all C:PrefixCode. sum(c in C, 2^(-length(c))) <= 1", "Intro; Kraft; Auto"), + ("source_coding", "all X:RandomVar, eps:PosReal. exists n:PosNat, C:Code. expected_length(C(X^n)) / n <= H(X) + eps", "Intro; ShannonSource; Auto"), + ] + + quotient_types = [ + ("quotient_class", "all A:Type, R:Equiv(A), x,y:A. R(x,y) => [x]_R = [y]_R", "Intros; QuotientEq; Auto"), + ("quotient_lift", "all A,B:Type, R:Equiv(A), f:A->B. (all x,y:A. R(x,y) => f(x) = f(y)) => A/R -> B", "Intros; QuotientLift; Auto"), + ("nat_quot_modp", "all p:PosNat. Nat/cong(p) ~ Fin(p)", "Intro; Bijection; Auto"), + ("partition_quotient", "all A:Type, R:Equiv(A). partition(A, R) <=> A/R", "Intro; Partition; Auto"), + ("setoid_morphism", "all A,B:Setoid, f:A->B. morphism(f) <=> (all x,y:A. eq_A(x,y) => eq_B(f(x), f(y)))", "Intro; Morphism; Auto"), + ] + + w_types = [ + ("w_type", "all A:Type, B:A->Type. W(A, B) : Type", "Intros; WType; Auto"), + ("w_induction", "all A:Type, B:A->Type, P:W(A,B)->Prop. (all a:A, f:B(a)->W(A,B). (all b:B(a). P(f(b))) => P(sup(a, f))) => all w:W(A,B). P(w)", "Intros; WInduction; Auto"), + ("w_nat_iso", "Nat ~ W(Bool, lambda b. if b then Unit else Empty)", "Intro; Bijection; Auto"), + ("w_list_iso", "all A:Type. List(A) ~ W(1 + A, lambda s. match s with inl _ => Empty | inr _ => Unit)", "Intro; Bijection; Auto"), ] all_categories = [ @@ -107,6 +251,20 @@ function generate_nuprl()::Vector{Dict{String,Any}} ("lists", lists), ("logic", logic), ("constructive", constructive), + ("sets", sets), + ("relations", relations), + ("number_theory", number_theory), + ("real_analysis", real_analysis), + ("computability", computability), + ("category_theory", category_theory), + ("topology", topology), + ("algebra", algebra), + ("probability", probability), + ("bishop_analysis", bishop_analysis), + ("proof_theory", proof_theory), + ("information", information), + ("quotient_types", quotient_types), + ("w_types", w_types), ] proofs = Dict{String,Any}[] @@ -200,6 +358,147 @@ function generate_minlog()::Vector{Dict{String,Any}} ("realizer_inl", "[r : A -> A \\/ B] = lambda a. inl(a)", "normalize; refl"), ("realizer_inr", "[r : B -> A \\/ B] = lambda b. inr(b)", "normalize; refl"), ("realizer_case", "[r : (A -> C) -> (B -> C) -> A \\/ B -> C] = lambda f g s. case s of inl(a) => f a | inr(b) => g b", "normalize; refl"), + ("realizer_forall", "[r : (all x:N. A(x))] = lambda x. r_x", "normalize; refl"), + ("realizer_exists", "[r : (ex x:N. A(x))] = (n, proof_of_A(n))", "normalize; refl"), + ("realizer_induction", "[r : nat_rec] = lambda base step n. (rec base step n)", "normalize; refl"), + ] + + lists_minlog = [ + ("list_nil_append", "all l:list. nil ++ l = l", "intro; refl"), + ("list_append_nil", "all l:list. l ++ nil = l", "ind on l; [refl | assume IH; simp; use IH]"), + ("list_append_assoc", "all l1 l2 l3. (l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3)", "ind on l1; [refl | assume IH; simp; use IH]"), + ("list_length_append", "all l1 l2. length(l1 ++ l2) = length(l1) + length(l2)", "ind on l1; [refl | assume IH; simp; use IH]"), + ("list_rev_involutive", "all l. rev(rev(l)) = l", "ind on l; [refl | assume IH; simp; use rev_append; use IH]"), + ("list_rev_append", "all l1 l2. rev(l1 ++ l2) = rev(l2) ++ rev(l1)", "ind on l1; [simp; use list_append_nil | assume IH; simp; use IH; use list_append_assoc]"), + ("list_map_id", "all l. map(lambda x. x, l) = l", "ind on l; [refl | assume IH; simp; use IH]"), + ("list_map_compose", "all f g l. map(f, map(g, l)) = map(lambda x. f (g x), l)", "ind on l; [refl | assume IH; simp; use IH]"), + ("list_filter_length", "all p l. length(filter(p, l)) <= length(l)", "ind on l; [auto | assume IH; cases p; auto]"), + ("list_foldr_append", "all f z l1 l2. foldr(f, z, l1 ++ l2) = foldr(f, foldr(f, z, l2), l1)", "ind on l1; [refl | assume IH; simp; use IH]"), + ("list_all_forall", "all p l. (all x. mem(x, l) -> p(x)) -> list_all(p, l)", "ind on l; [auto | assume IH; cases; auto]"), + ] + + trees = [ + ("tree_height_nonneg", "all t. height(t) >= 0", "ind on t; auto"), + ("tree_size_nonneg", "all t. size(t) >= 1", "ind on t; auto"), + ("tree_leaves_ge_1", "all t. leaves(t) >= 1", "ind on t; auto"), + ("tree_mirror_involutive", "all t. mirror(mirror(t)) = t", "ind on t; [refl | assume IH1 IH2; simp; use IH1; use IH2]"), + ("tree_size_height", "all t. size(t) <= 2^(height(t)+1) - 1", "ind on t; [auto | assume IH1 IH2; simp; use IH1; use IH2]"), + ("bst_inorder_sorted", "all t. is_bst(t) -> sorted(inorder(t))", "ind on t; [auto | assume IH; split; auto]"), + ("tree_flatten_append", "all t1 t2. flatten(Node(t1, x, t2)) = flatten(t1) ++ [x] ++ flatten(t2)", "intro; refl"), + ("tree_count_sum", "all x t. count(x, flatten(t)) = count_tree(x, t)", "ind on t; auto"), + ] + + classical_logic = [ + ("tnd", "all A. A \\/ ~A", "use classical; auto"), + ("dne", "all A. ~~A -> A", "assume h; classical; auto"), + ("peirce_law", "all A B. ((A -> B) -> A) -> A", "assume h; classical; auto"), + ("contrapos_classical", "all A B. (~B -> ~A) -> (A -> B)", "assume h; classical; auto"), + ("or_impl", "all A B. (A \\/ B) -> (~A -> B)", "assume h nA; elim h; [assume a; absurd | assume b; exact b]"), + ("impl_or", "all A B. (A -> B) -> (~A \\/ B)", "assume h; classical; auto"), + ("and_not_or", "all A B. ~A /\\ ~B -> ~(A \\/ B)", "assume h; elim h; assume nA nB; intro; elim; auto"), + ("demorgan_classical_1", "all A B. ~(A /\\ B) -> (~A \\/ ~B)", "assume h; classical; auto"), + ("demorgan_classical_2", "all A B. ~(A \\/ B) <-> (~A /\\ ~B)", "split; intros; split; intros; auto"), + ("classical_exists", "all P. ~(all x. ~P(x)) -> ex x. P(x)", "assume h; classical; auto"), + ] + + sequent_calculus = [ + ("cut_elim", "all A B C. (Gamma |- A) -> (A, Gamma |- B) -> (Gamma |- B)", "intros; cut_admissible; auto"), + ("structural_weaken", "all A B. (Gamma |- B) -> (A, Gamma |- B)", "intros; weakening; auto"), + ("structural_contract", "all A B. (A, A, Gamma |- B) -> (A, Gamma |- B)", "intros; contraction; auto"), + ("structural_exchange", "all A B C. (A, B, Gamma |- C) -> (B, A, Gamma |- C)", "intros; exchange; auto"), + ("identity_axiom", "all A. (A |- A)", "intro; axiom"), + ("and_right", "all A B. (Gamma |- A) -> (Gamma |- B) -> (Gamma |- A /\\ B)", "intros; andR; auto"), + ("and_left_1", "all A B C. (A, Gamma |- C) -> (A /\\ B, Gamma |- C)", "intros; andL1; auto"), + ("impl_right", "all A B. (A, Gamma |- B) -> (Gamma |- A -> B)", "intros; implR; auto"), + ("impl_left", "all A B C. (Gamma |- A) -> (B, Gamma |- C) -> (A -> B, Gamma |- C)", "intros; implL; auto"), + ] + + induction_advanced = [ + ("strong_ind", "all P. (all n. (all m. m < n -> P(m)) -> P(n)) -> all n. P(n)", "assume h; strong_ind; auto"), + ("course_of_values", "all P. P(0) -> (all n. P(n) -> P(n+1)) -> all n. P(n)", "assume p0 step; nat_rec; auto"), + ("well_founded_acc", "all R. well_founded(R) -> all P. (all x. (all y. R(y,x) -> P(y)) -> P(x)) -> all x. P(x)", "assume wf ih; wf_rec; auto"), + ("mutual_ind_evenodd", "(all n. even(n) -> P(n)) /\\ (all n. odd(n) -> Q(n))", "mutual_ind; auto"), + ("struct_ind_list", "all P. P(nil) -> (all x xs. P(xs) -> P(cons x xs)) -> all l. P(l)", "assume p_nil p_cons; list_rec; auto"), + ("struct_ind_tree", "all P. P(leaf) -> (all t1 x t2. P(t1) -> P(t2) -> P(Node t1 x t2)) -> all t. P(t)", "assume p_leaf p_node; tree_rec; auto"), + ("transfinite_ind", "all P. (all beta. beta < alpha -> P(beta)) -> P(alpha)", "assume ih; transfinite; auto"), + ] + + higher_order = [ + ("higher_order_app", "all f x. (lambda y. f y) x = f x", "beta_reduce; refl"), + ("fun_ext_ho", "all f g. (all x. f x = g x) -> f = g", "assume h; fun_ext; auto"), + ("curry_uncurry", "all f x y. curry (uncurry f) x y = f x y", "intros; beta_reduce; refl"), + ("uncurry_curry", "all f p. uncurry (curry f) p = f p", "intro; cases p; beta_reduce; refl"), + ("compose_assoc", "all f g h. compose (compose f g) h = compose f (compose g h)", "fun_ext; beta_reduce; refl"), + ("id_left", "all f. compose id f = f", "fun_ext; beta_reduce; refl"), + ("id_right", "all f. compose f id = f", "fun_ext; beta_reduce; refl"), + ("ho_induction_schema", "all P. (all x. x = 0 \\/ (ex y. x = y+1 /\\ P(y))) -> P(0) -> all x. P(x)", "assume h p0; induct; auto"), + ] + + program_verification = [ + ("factorial_correct", "all n. factorial(n) >= 1", "ind on n; [simp; auto | assume IH; simp; use IH]"), + ("fibonacci_monotone", "all m n. m <= n -> fib(m) <= fib(n)", "assume h; ind on n; auto"), + ("sum_range_formula", "all n. sum(0..n, lambda i. i) = n * (n+1) / 2", "ind on n; auto"), + ("sort_preserves_elements", "all l. mset(sort l) = mset(l)", "ind on l; auto"), + ("reverse_length", "all l. length(reverse l) = length l", "ind on l; [refl | assume IH; simp; use length_append; use IH]"), + ("gcd_correct", "all a b. b > 0 -> divides(gcd(a,b), a) /\\ divides(gcd(a,b), b)", "strong_ind on b; cases; auto"), + ("is_sorted_cons", "all x xs. is_sorted(cons x xs) -> is_sorted xs", "intros; cases; auto"), + ("search_correct", "all x l. linear_search(x, l) = true <-> mem(x, l)", "ind on l; split; auto"), + ] + + modal_logic = [ + ("modal_k", "all A B. box(A -> B) -> box(A) -> box(B)", "intros; box_elim; auto"), + ("modal_t", "all A. box(A) -> A", "intro; reflexive_frame; auto"), + ("modal_4", "all A. box(A) -> box(box(A))", "intro; transitive_frame; auto"), + ("modal_5", "all A. dia(A) -> box(dia(A))", "intro; euclidean_frame; auto"), + ("modal_dual", "all A. dia(A) <-> ~box(~A)", "split; intros; auto"), + ("necessitation", "all A. |- A -> |- box(A)", "intro; necessitate; auto"), + ("distribution", "all A B. box(A /\\ B) <-> box(A) /\\ box(B)", "split; intros; split; box_elim; auto"), + ] + + intuitionistic = [ + ("disjunction_property", "all A B. (|- A \\/ B) -> (|- A) \\/ (|- B)", "assume h; intuit; auto"), + ("existence_property", "all P. (|- ex x. P(x)) -> ex witness. |- P(witness)", "assume h; intuit; auto"), + ("double_neg_shift", "all A. ~~(A \\/ ~A)", "intro; assume h; apply h; right; intro; apply h; left; auto"), + ("glivenko", "all A. (|- ~~A classically) -> (|- ~~A intuitionistically)", "assume h; glivenko; auto"), + ("kripke_model", "all w:World, A. forces(w, A) <-> satisfies(w, A)", "intro; split; kripke; auto"), + ("negation_translation", "all A. (~~A)_neg = ~~A_neg", "intro; neg_translation; auto"), + ("godel_neg", "all A. valid(A) <-> valid(~~A_neg)", "intro; split; godel_negative; auto"), + ] + + games = [ + ("game_det", "all G:Game. open(G) -> determined(G)", "assume h; gale_stewart; auto"), + ("strategy_winning", "all G:Game, sigma:Strategy. winning(sigma, G) -> all play. consistent(play, sigma) -> G_wins(play)", "assume w; intro con; auto"), + ("zermelo", "all G:FiniteGame. determined(G)", "intro; backward_induction; auto"), + ("minimax", "all G:ZeroSum, v:Value. v = max(min(payoff))", "intro; minimax_theorem; auto"), + ("nash_existence", "all G:FiniteGame. ex sigma. nash_equilibrium(sigma, G)", "intro; brouwer; auto"), + ] + + realizers_advanced = [ + ("realizer_nat_rec", "[r : nat_rec] = lambda base step n. (fix F lambda k. if k=0 then base else step k (F (k-1))) n", "normalize; refl"), + ("realizer_list_rec", "[r : list_rec] = lambda nil_case cons_case l. (fix F lambda xs. match xs with nil => nil_case | x::xs' => cons_case x (F xs')) l", "normalize; refl"), + ("realizer_bounded_search", "[r : bounded_search_lemma] = lambda p bound. search_loop p 0 bound", "normalize; refl"), + ("realizer_minimization", "[r : minimization] = lambda f. mu (lambda n. f n = 0)", "normalize; refl"), + ("realizer_fixpoint", "[r : fix_point_thm] = lambda f. fix f", "normalize; refl"), + ("realizer_choice", "[r : countable_choice] = lambda R. lambda n. choose (R n)", "normalize; refl"), + ] + + bounded_arithmetic = [ + ("bit_length", "all n. n > 0 -> bit_length(n) = floor(log2(n)) + 1", "strong_ind on n; auto"), + ("parity", "all n. parity(n) = n mod 2", "ind on n; auto"), + ("binary_rep", "all n. binary_decode(binary_encode(n)) = n", "ind on n; auto"), + ("pow_two_le", "all n. 2^n > n", "ind on n; auto"), + ("pigeon_bounded", "all f:fin(n)->fin(m). n > m -> ex i j. i <> j /\\ f(i) = f(j)", "assume; pigeonhole; auto"), + ("bounded_search", "all p:nat->bool. (ex n. n <= N /\\ p n = true) -> ex k. k <= N /\\ p k = true /\\ (all j. j < k -> p j = false)", "assume; bounded_search; auto"), + ("pspace_pspace", "pspace = pspace", "refl"), + ] + + finite_logic = [ + ("fin_sat_decidable", "all phi:Formula. decidable(sat(phi))", "intro; propositional_sat; auto"), + ("resolution_sound", "all phi. res_derives(empty, phi) -> unsat(phi)", "assume h; resolution_soundness; auto"), + ("resolution_complete", "all phi. unsat(phi) -> res_derives(empty, phi)", "assume h; resolution_completeness; auto"), + ("dpll_terminates", "all phi. terminates(dpll(phi))", "intro; wf_induct; auto"), + ("cnf_equivalent", "all phi. equivalent(phi, to_cnf(phi))", "intro; tseitin; auto"), + ("horn_sat_poly", "all phi:HornFormula. poly_time_decidable(sat(phi))", "intro; unit_propagation; auto"), ] all_categories = [ @@ -207,6 +506,19 @@ function generate_minlog()::Vector{Dict{String,Any}} ("arithmetic", arithmetic_extraction), ("extraction", program_extraction), ("realizability", realizability), + ("lists", lists_minlog), + ("trees", trees), + ("classical_logic", classical_logic), + ("sequent_calculus", sequent_calculus), + ("induction_advanced", induction_advanced), + ("higher_order", higher_order), + ("program_verification", program_verification), + ("modal_logic", modal_logic), + ("intuitionistic", intuitionistic), + ("games", games), + ("realizers_advanced", realizers_advanced), + ("bounded_arithmetic", bounded_arithmetic), + ("finite_logic", finite_logic), ] proofs = Dict{String,Any}[] @@ -289,11 +601,156 @@ function generate_twelf()::Vector{Dict{String,Any}} ("contra_pf", "contra : pf (imp (imp A B) (imp (not B) (not A))).", "contra = imp-i ([f:pf (imp A B)] imp-i ([nb:pf (not B)] not-i ([a:pf A] not-e nb (imp-e f a))))."), ] + binary_trees = [ + ("tree", "tree : type.", "empty : tree.\nnode : nat -> tree -> tree -> tree."), + ("insert_bst", "insert : nat -> tree -> tree -> type.", "insert-empty : insert X empty (node X empty empty).\ninsert-lt : insert X (node Y L R) (node Y L' R) <- leq X Y <- insert X L L'.\ninsert-gt : insert X (node Y L R) (node Y L R') <- leq Y X <- insert X R R'."), + ("tree_size", "tree-size : tree -> nat -> type.", "tree-size-empty : tree-size empty z.\ntree-size-node : tree-size (node X L R) (s N) <- tree-size L NL <- tree-size R NR <- plus NL NR N."), + ("tree_height", "tree-height : tree -> nat -> type.", "tree-height-empty : tree-height empty z.\ntree-height-node : tree-height (node X L R) (s H) <- tree-height L HL <- tree-height R HR <- max HL HR H."), + ("tree_member", "tree-member : nat -> tree -> type.", "tree-member-node : tree-member X (node X L R).\ntree-member-left : tree-member X (node Y L R) <- tree-member X L.\ntree-member-right : tree-member X (node Y L R) <- tree-member X R."), + ("tree_mirror", "mirror : tree -> tree -> type.", "mirror-empty : mirror empty empty.\nmirror-node : mirror (node X L R) (node X R' L') <- mirror L L' <- mirror R R'."), + ("mirror_involutive", "mirror-involutive : mirror T T' -> mirror T' T -> type.", "%mode mirror-involutive +D1 -D2."), + ("inorder_flatten", "inorder : tree -> list -> type.", "inorder-empty : inorder empty nil.\ninorder-node : inorder (node X L R) M <- inorder L ML <- inorder R MR <- append ML (cons X MR) M."), + ] + + substitution = [ + ("substitution", "subst' : (tm -> tm) -> tm -> tm -> type.", "subst'-id : subst' ([x] x) V V.\nsubst'-const : subst' ([x] C) V C."), + ("capture_avoid", "fresh : tm -> (tm -> tm) -> type.", "fresh-const : fresh V ([x] C).\nfresh-app : fresh V ([x] app (E1 x) (E2 x)) <- fresh V E1 <- fresh V E2."), + ("alpha_eq", "alpha-eq : tm -> tm -> type.", "alpha-eq-refl : alpha-eq E E.\nalpha-eq-lam : alpha-eq (lam T ([x] E x)) (lam T ([y] E y)) <- ({x} alpha-eq (E x) (E x))."), + ("beta_red", "beta-red : tm -> tm -> type.", "beta-red-step : beta-red (app (lam T E) V) (E V) <- value V."), + ("eta_red", "eta-red : tm -> tm -> type.", "eta-red-step : eta-red (lam T ([x] app E x)) E."), + ("subst_lemma", "subst-lemma : subst' E1 V1 E1' -> subst' E2 V2 E2' -> subst' ([x] E1 x) V1 E1'' -> type.", "%mode subst-lemma +D1 +D2 +D3."), + ] + + session_types = [ + ("channel", "chan : type.", "new-chan : chan."), + ("msg_type", "msg : type.", "unit-msg : msg.\nsum-msg : msg -> msg -> msg."), + ("protocol", "proto : type.", "send-proto : msg -> proto -> proto.\nrecv-proto : msg -> proto -> proto.\nend-proto : proto."), + ("duality", "dual : proto -> proto -> type.", "dual-end : dual end-proto end-proto.\ndual-send : dual (send-proto M P) (recv-proto M Q) <- dual P Q.\ndual-recv : dual (recv-proto M P) (send-proto M Q) <- dual P Q."), + ("session_safe", "session-safe : chan -> proto -> type.", "%mode session-safe +C +P."), + ("linear_use", "linear : chan -> type.", "linear-new : linear C <- new-chan."), + ] + + linear_lf = [ + ("linear_type", "ltp : type.", "lolli : ltp -> ltp -> ltp.\ntensor : ltp -> ltp -> ltp.\nwith-type : ltp -> ltp -> ltp.\none-type : ltp."), + ("linear_tm", "ltm : type.", "lapp : ltm -> ltm -> ltm.\nllam : ltp -> (ltm -> ltm) -> ltm.\npair-ltm : ltm -> ltm -> ltm.\nunit-ltm : ltm."), + ("linear_of", "lof : ltm -> ltp -> type.", "lof-lam : lof (llam T1 E) (lolli T1 T2) <- ({x:ltm} lof x T1 -> lof (E x) T2).\nlof-pair : lof (pair-ltm E1 E2) (tensor T1 T2) <- lof E1 T1 <- lof E2 T2.\nlof-unit : lof unit-ltm one-type."), + ("linear_safety", "linear-safety : lof E T -> step E E' -> lof E' T -> type.", "%mode linear-safety +D1 +D2 -D3."), + ("linear_single", "linear-single : ltm -> type.", "linear-single-var : linear-single (var X).\nlinear-single-app : linear-single (lapp E1 E2) <- linear-single E1 <- linear-single E2."), + ] + + metatheory = [ + ("weakening", "weakening : of E T -> of E T -> type.", "%mode weakening +Dof -Dof'."), + ("type_unique", "type-unique : of E T1 -> of E T2 -> eq T1 T2 -> type.", "%mode type-unique +D1 +D2 -D3."), + ("canonical_forms", "canonical : value V -> of V T -> type.", "%mode canonical +Dv +Dof."), + ("progress_lemma", "progress-lemma : of E T -> (value E + step E E') -> type.", "%mode progress-lemma +Dof -Dresult."), + ("preservation_lemma", "preservation-lemma : of E T -> step E E' -> of E' T -> type.", "%mode preservation-lemma +Dof +Dstep -Dof'."), + ("normalization", "strong-norm : of E T -> type.", "%mode strong-norm +Dof."), + ("confluence", "confluence : reduces E E1 -> reduces E E2 -> (reduces E1 E' /\\ reduces E2 E') -> type.", "%mode confluence +D1 +D2 -D3."), + ] + + polymorphism = [ + ("forall_type", "forall-tp : (tp -> tp) -> tp.", "forall-intro : of E (forall-tp A) <- ({X:tp} of E (A X)).\nforall-elim : of (tapp E T) (A T) <- of E (forall-tp A)."), + ("exists_type", "exists-tp : (tp -> tp) -> tp.", "pack : of (pack T E) (exists-tp A) <- of E (A T).\nunpack : of (open E1 [X] [x] E2) C <- of E1 (exists-tp A) <- ({X:tp} {x:tm} of x (A X) -> of (E2 X x) C)."), + ("type_abstraction", "tabs : tp -> (tp -> tm) -> tm.", "of-tabs : of (tabs K E) (forall-tp A) <- ({X:tp} of (E X) (A X))."), + ("param_poly", "parametricity : of E (forall-tp A) -> type.", "%mode parametricity +D."), + ("subtype", "sub : tp -> tp -> type.", "sub-refl : sub T T.\nsub-trans : sub T1 T3 <- sub T1 T2 <- sub T2 T3.\nsub-arrow : sub (arr S1 T1) (arr S2 T2) <- sub S2 S1 <- sub T1 T2."), + ] + + modules = [ + ("module_signature", "module-sig : type.", "mk-sig : list -> module-sig."), + ("module_impl", "module-impl : module-sig -> type.", "mk-impl : list -> module-impl Sig."), + ("functor", "functor : module-sig -> module-sig -> type.", "apply-functor : functor S1 S2 -> module-impl S1 -> module-impl S2 -> type."), + ("abstract_type", "abs-tp : type.", "pack-abs : tp -> (tp -> tp) -> abs-tp.\nopen-abs : abs-tp -> (tp -> tp -> tp) -> tp -> type."), + ("structure_eq", "struct-eq : module-impl S -> module-impl S -> type.", "%mode struct-eq +M1 +M2."), + ("include_module", "include : module-impl S1 -> module-impl S2 -> module-impl S3 -> type.", "%mode include +M1 +M2 -M3."), + ] + + records_twelf = [ + ("record", "record : type.", "empty-rec : record.\ncons-rec : label -> tm -> record -> record."), + ("label", "label : type.", "lab-a : label.\nlab-b : label.\nlab-c : label."), + ("rec_lookup", "lookup : label -> record -> tm -> type.", "lookup-head : lookup L (cons-rec L V R) V.\nlookup-tail : lookup L (cons-rec L' V R) V' <- lookup L R V'."), + ("rec_update", "update : label -> tm -> record -> record -> type.", "update-head : update L V (cons-rec L V' R) (cons-rec L V R).\nupdate-tail : update L V (cons-rec L' V' R) (cons-rec L' V' R') <- update L V R R'."), + ("rec_concat", "rec-concat : record -> record -> record -> type.", "rec-concat-nil : rec-concat empty-rec R R.\nrec-concat-cons : rec-concat (cons-rec L V R1) R2 (cons-rec L V R3) <- rec-concat R1 R2 R3."), + ] + + traces = [ + ("trace", "trace : type.", "empty-trace : trace.\nstep-trace : tm -> tm -> trace -> trace."), + ("trace_step", "trace-step : trace -> tm -> tm -> type.", "%mode trace-step +T +E -E'."), + ("trace_append", "trace-append : trace -> trace -> trace -> type.", "trace-app-nil : trace-append empty-trace T T.\ntrace-app-cons : trace-append (step-trace E1 E2 T1) T2 (step-trace E1 E2 T3) <- trace-append T1 T2 T3."), + ("evaluates", "evaluates : tm -> tm -> type.", "eval-refl : evaluates E E.\neval-step : evaluates E E'' <- step E E' <- evaluates E' E''."), + ("multistep", "multistep : tm -> tm -> trace -> type.", "multistep-refl : multistep E E empty-trace.\nmultistep-step : multistep E E'' (step-trace E E' T) <- step E E' <- multistep E' E'' T."), + ] + + bigstep = [ + ("bigstep", "bigstep : tm -> tm -> type.", "bigstep-val : bigstep V V <- value V.\nbigstep-app : bigstep (app E1 E2) V <- bigstep E1 (lam T E) <- bigstep E2 V2 <- bigstep (E V2) V."), + ("bigstep_det", "bigstep-det : bigstep E V1 -> bigstep E V2 -> eq V1 V2 -> type.", "%mode bigstep-det +D1 +D2 -D3."), + ("bigstep_value", "bigstep-value : bigstep E V -> value V -> type.", "%mode bigstep-value +D -Dv."), + ("bigstep_of", "bigstep-of : bigstep E V -> of E T -> of V T -> type.", "%mode bigstep-of +D1 +D2 -D3."), + ("bigstep_smallstep", "bigstep-smallstep : bigstep E V -> multistep E V _ -> type.", "%mode bigstep-smallstep +D1 -D2."), + ] + + hoas = [ + ("hoas_abs", "habs : (tm -> tm) -> tm.", "habs-intro : of (habs E) (arr T1 T2) <- ({x:tm} of x T1 -> of (E x) T2)."), + ("hoas_app", "happ : tm -> tm -> tm.", "happ-intro : of (happ E1 E2) T2 <- of E1 (arr T1 T2) <- of E2 T1."), + ("hoas_beta", "hstep : tm -> tm -> type.", "hstep-beta : hstep (happ (habs E) V) (E V) <- value V."), + ("hoas_subst", "hsubst : (tm -> tm) -> tm -> tm -> type.", "%mode hsubst +E +V -E'."), + ("parametric_hoas", "phabs : (tm -> tm) -> tm.", "%block phabs-block : block {x:tm}."), + ] + + parsing = [ + ("token", "token : type.", "tok-ident : string -> token.\ntok-num : nat -> token.\ntok-lparen : token.\ntok-rparen : token.\ntok-plus : token.\ntok-times : token."), + ("parse_expr", "parse : list -> tm -> list -> type.", "%mode parse +Input -Result -Rest."), + ("lex", "lex : string -> list -> type.", "%mode lex +Input -Tokens."), + ("ast", "ast : type.", "ast-num : nat -> ast.\nast-var : string -> ast.\nast-bin : op -> ast -> ast -> ast."), + ("op", "op : type.", "op-add : op.\nop-sub : op.\nop-mul : op.\nop-div : op."), + ("grammar", "grammar-rule : type.", "gr-term : symbol -> grammar-rule.\ngr-alt : grammar-rule -> grammar-rule -> grammar-rule.\ngr-seq : grammar-rule -> grammar-rule -> grammar-rule."), + ] + + automata = [ + ("dfa_state", "state : type.", "init-state : state.\ntrans-state : state -> char -> state."), + ("dfa_accept", "accepts : dfa -> string -> type.", "%mode accepts +D +S."), + ("nfa_run", "nfa-run : nfa -> list -> state -> type.", "%mode nfa-run +N +S -Q."), + ("regex", "regex : type.", "re-char : char -> regex.\nre-concat : regex -> regex -> regex.\nre-alt : regex -> regex -> regex.\nre-star : regex -> regex.\nre-epsilon : regex."), + ("regex_match", "re-match : regex -> string -> type.", "%mode re-match +R +S."), + ("pda", "pda-state : type.", "pda-init : pda-state.\npda-push : pda-state -> char -> stack -> pda-state."), + ] + + effects = [ + ("io_monad", "io : type -> type.", "io-return : A -> io A.\nio-bind : io A -> (A -> io B) -> io B."), + ("state_monad", "state-m : type -> type.", "state-return : A -> state-m A.\nstate-bind : state-m A -> (A -> state-m B) -> state-m B.\nstate-get : state-m state.\nstate-put : state -> state-m unit."), + ("exception", "exc : type -> type.", "exc-pure : A -> exc A.\nexc-raise : string -> exc A.\nexc-catch : exc A -> (string -> exc A) -> exc A."), + ("continuation", "cont : type -> type -> type.", "cont-pure : A -> cont A R.\ncont-callcc : ((A -> cont B R) -> cont A R) -> cont A R."), + ("free_monad", "free : type -> type.", "free-pure : A -> free A.\nfree-op : op-sig -> (result -> free A) -> free A."), + ] + + numerics_twelf = [ + ("add_zero_left", "add-zero-left : {N:nat} plus z N N -> type.", "%mode add-zero-left +N -D.\nadd-zero-left-z : add-zero-left z plus-z.\nadd-zero-left-s : add-zero-left (s N) plus-z <- add-zero-left N _."), + ("add_zero_right", "add-zero-right : {N:nat} plus N z N -> type.", "%mode add-zero-right +N -D.\nadd-zero-right-z : add-zero-right z plus-z.\nadd-zero-right-s : add-zero-right (s N) (plus-s D) <- add-zero-right N D."), + ("times_zero_right", "times-zero-right : {N:nat} times N z z -> type.", "%mode times-zero-right +N -D."), + ("div_algorithm", "div-mod : nat -> nat -> nat -> nat -> type.", "div-mod-base : div-mod M N z M <- lt M N.\ndiv-mod-rec : div-mod M N (s Q) R <- leq N M <- sub M N M' <- div-mod M' N Q R."), + ("gcd", "gcd : nat -> nat -> nat -> type.", "gcd-zero : gcd M z M.\ngcd-rec : gcd M N G <- modulo M N R <- gcd N R G."), + ] + all_categories = [ ("natural_numbers", natural_numbers), ("lambda_calculus", lambda_calculus), ("lists", lists), ("logic_framework", logic_framework), + ("binary_trees", binary_trees), + ("substitution", substitution), + ("session_types", session_types), + ("linear_lf", linear_lf), + ("metatheory", metatheory), + ("polymorphism", polymorphism), + ("modules", modules), + ("records", records_twelf), + ("traces", traces), + ("bigstep", bigstep), + ("hoas", hoas), + ("parsing", parsing), + ("automata", automata), + ("effects", effects), + ("numerics", numerics_twelf), ] proofs = Dict{String,Any}[] @@ -386,6 +843,128 @@ function generate_imandra()::Vector{Dict{String,Any}} ("cx_map_injective", "verify (fun f l1 l2 -> List.map f l1 = List.map f l2 ==> l1 = l2)", "CX: f = (fun _ -> 0), l1 = [1], l2 = [2]"), ("cx_sort_idempotent", "verify (fun l -> sort (sort l) = sort l)", "Verified: true (sort is idempotent)"), ("cx_filter_length", "verify (fun p l -> List.length (List.filter p l) <= List.length l)", "Verified: true"), + ("cx_subset_reflexive", "verify (fun s -> subset s s)", "Verified: true"), + ("cx_division_positive", "verify (fun x y -> x > 0 && y > 0 ==> x / y >= 0)", "Verified: true"), + ("cx_int_overflow", "verify (fun (x:int) -> x + 1 > x)", "CX: x = max_int -- overflow wraps to min_int"), + ("cx_float_assoc", "verify (fun (a:float) b c -> (a +. b) +. c = a +. (b +. c))", "CX: a=1e30, b= -1e30, c=1.0 -- floating point not associative"), + ] + + trees_imandra = [ + ("bst_insert_preserves", "theorem bst_insert_preserves x t = is_bst t ==> is_bst (bst_insert x t)", "[@@induct t] [@@auto]"), + ("bst_member_after_insert", "theorem bst_member_after_insert x t = bst_member x (bst_insert x t)", "[@@induct t] [@@auto]"), + ("tree_mirror_involutive", "theorem tree_mirror_involutive t = mirror (mirror t) = t", "[@@induct t]"), + ("tree_size_positive", "theorem tree_size_positive t = tree_size t >= 0", "[@@induct t]"), + ("tree_flatten_length", "theorem tree_flatten_length t = List.length (flatten t) = tree_size t", "[@@induct t]"), + ("heap_extract_min_sorted", "theorem heap_extract_min_sorted h = is_heap h ==> sorted (heapsort h)", "[@@induct h] [@@auto]"), + ("avl_balanced", "theorem avl_balanced t = is_avl t ==> abs (height (left t) - height (right t)) <= 1", "[@@induct t] [@@auto]"), + ("rbt_black_height", "theorem rbt_black_height t = is_rbt t ==> black_height (left t) = black_height (right t)", "[@@induct t] [@@auto]"), + ] + + numerics = [ + ("abs_diff_symmetric", "theorem abs_diff_symmetric x y = abs (x - y) = abs (y - x)", "[@@auto]"), + ("sqr_nonneg", "theorem sqr_nonneg x = x * x >= 0", "[@@auto]"), + ("ge_trans", "theorem ge_trans x y z = x >= y ==> y >= z ==> x >= z", "[@@auto]"), + ("eq_refl_trans", "theorem eq_refl_trans x y = x = y ==> y = x", "[@@auto]"), + ("neg_neg", "theorem neg_neg x = -(-x) = x", "[@@auto]"), + ("sub_self_zero", "theorem sub_self_zero x = x - x = 0", "[@@auto]"), + ("mul_distrib_sub", "theorem mul_distrib_sub x y z = x * (y - z) = x*y - x*z", "[@@auto]"), + ("pow_zero", "theorem pow_zero x = x > 0 ==> x ** 0 = 1", "[@@auto]"), + ("pow_succ", "theorem pow_succ x n = n >= 0 ==> x ** (n+1) = x * (x ** n)", "[@@induct n] [@@auto]"), + ("gcd_self", "theorem gcd_self n = n >= 0 ==> gcd n n = n", "[@@auto]"), + ("gcd_zero", "theorem gcd_zero n = n >= 0 ==> gcd n 0 = n", "[@@auto]"), + ("divides_refl", "theorem divides_refl n = n <> 0 ==> divides n n", "[@@auto]"), + ] + + records = [ + ("record_set_get", "theorem record_set_get r v = (set_field r v).field = v", "[@@auto]"), + ("record_get_set_idempotent", "theorem record_get_set_idempotent r = set_field r (r.field) = r", "[@@auto]"), + ("record_set_comm", "theorem record_set_comm r v1 v2 = set_f2 (set_f1 r v1) v2 = set_f1 (set_f2 r v2) v1", "[@@auto]"), + ("record_eq_fields", "theorem record_eq_fields r1 r2 = r1.a = r2.a ==> r1.b = r2.b ==> r1 = r2", "[@@auto]"), + ("option_some_inj", "theorem option_some_inj x y = Some x = Some y ==> x = y", "[@@auto]"), + ("option_none_not_some", "theorem option_none_not_some x = None <> Some x", "[@@auto]"), + ("tuple_fst_snd", "theorem tuple_fst_snd p = (fst p, snd p) = p", "[@@auto]"), + ] + + state_machines = [ + ("fsm_deterministic", "theorem fsm_deterministic s e = forall s1 s2. transition s e s1 ==> transition s e s2 ==> s1 = s2", "[@@auto]"), + ("fsm_reachable_trans", "theorem fsm_reachable_trans s1 s2 s3 = reachable s1 s2 ==> reachable s2 s3 ==> reachable s1 s3", "[@@induct] [@@auto]"), + ("fsm_initial_reachable", "theorem fsm_initial_reachable s = s = initial_state ==> reachable s s", "[@@auto]"), + ("fsm_no_deadlock", "theorem fsm_no_deadlock s = reachable initial_state s ==> exists e s'. transition s e s'", "[@@auto]"), + ("fsm_invariant_preserve", "theorem fsm_invariant_preserve s e s' = inv s ==> transition s e s' ==> inv s'", "[@@auto]"), + ("safety_property", "theorem safety_property s = reachable initial_state s ==> not (is_error s)", "[@@induct] [@@auto]"), + ("liveness_property", "theorem liveness_property s = reachable initial_state s ==> eventually goal_state s", "[@@auto]"), + ] + + crypto = [ + ("hash_deterministic", "theorem hash_deterministic m1 m2 = m1 = m2 ==> hash m1 = hash m2", "[@@auto]"), + ("signature_verify", "theorem signature_verify sk pk m = keypair sk pk ==> verify pk m (sign sk m) = true", "[@@auto]"), + ("encrypt_decrypt", "theorem encrypt_decrypt k m = decrypt k (encrypt k m) = m", "[@@auto]"), + ("merkle_root_unique", "theorem merkle_root_unique l1 l2 = merkle_root l1 = merkle_root l2 ==> l1 = l2", "[@@auto]"), + ("xor_involutive", "theorem xor_involutive x y = xor (xor x y) y = x", "[@@auto]"), + ("nonce_unique", "theorem nonce_unique n1 n2 = generate_nonce n1 = generate_nonce n2 ==> n1 = n2", "[@@auto]"), + ] + + blockchain = [ + ("block_prev_chain", "theorem block_prev_chain b = valid_chain b ==> prev_hash (head b) = hash (tail b)", "[@@induct b] [@@auto]"), + ("tx_balance_conservation", "theorem tx_balance_conservation tx = valid_tx tx ==> sum_inputs tx = sum_outputs tx + fee tx", "[@@auto]"), + ("utxo_no_double_spend", "theorem utxo_no_double_spend tx1 tx2 utxo = valid_tx tx1 ==> valid_tx tx2 ==> disjoint (inputs tx1) (inputs tx2)", "[@@auto]"), + ("consensus_majority", "theorem consensus_majority votes = count_yes votes > length votes / 2 ==> accepted votes", "[@@auto]"), + ("longest_chain", "theorem longest_chain c1 c2 = valid_chain c1 ==> valid_chain c2 ==> height c1 >= height c2 ==> canonical c1", "[@@auto]"), + ] + + algebra = [ + ("group_identity_unique", "theorem group_identity_unique e1 e2 = is_identity e1 ==> is_identity e2 ==> e1 = e2", "[@@auto]"), + ("group_inverse_unique", "theorem group_inverse_unique a b c = op a b = identity ==> op a c = identity ==> b = c", "[@@auto]"), + ("monoid_left_id", "theorem monoid_left_id a = op identity a = a", "[@@auto]"), + ("monoid_right_id", "theorem monoid_right_id a = op a identity = a", "[@@auto]"), + ("ring_distrib", "theorem ring_distrib a b c = mul a (add b c) = add (mul a b) (mul a c)", "[@@auto]"), + ("ring_zero_mul", "theorem ring_zero_mul a = mul zero a = zero", "[@@auto]"), + ("semilat_idempotent", "theorem semilat_idempotent a = meet a a = a", "[@@auto]"), + ] + + concurrency = [ + ("mutex_exclusive", "theorem mutex_exclusive s = acquired s ==> not (available s)", "[@@auto]"), + ("lock_release", "theorem lock_release s = acquire (release (acquire s)) = acquire s", "[@@auto]"), + ("queue_ordering", "theorem queue_ordering q msgs = push q msgs ==> fifo_order (drain q)", "[@@induct msgs] [@@auto]"), + ("actor_isolation", "theorem actor_isolation a b m = send a b m ==> state a = state a", "[@@auto]"), + ("atomic_read_write", "theorem atomic_read_write x v = read (write x v) = v", "[@@auto]"), + ("cas_success", "theorem cas_success x old new = read x = old ==> cas x old new = (true, write x new)", "[@@auto]"), + ("deadlock_free", "theorem deadlock_free tree = acyclic tree ==> not (deadlocked tree)", "[@@induct tree] [@@auto]"), + ] + + databases = [ + ("acid_atomicity", "theorem acid_atomicity tx db = commit tx db = Some db' \\/ commit tx db = None", "[@@auto]"), + ("rollback_inverse", "theorem rollback_inverse tx db = rollback tx (apply tx db) = db", "[@@auto]"), + ("isolation_serializable", "theorem isolation_serializable t1 t2 db = serializable [t1; t2] db = serializable [t2; t1] db", "[@@auto]"), + ("index_consistent", "theorem index_consistent idx tbl = valid_index idx tbl ==> forall k. lookup idx k = table_lookup tbl k", "[@@auto]"), + ("foreign_key", "theorem foreign_key p c = valid_db (p, c) ==> forall r in c. exists parent in p. fk_match r parent", "[@@auto]"), + ("normalization_bcnf", "theorem normalization_bcnf r = bcnf r ==> third_nf r", "[@@auto]"), + ] + + networking = [ + ("tcp_in_order", "theorem tcp_in_order conn pkts = established conn ==> tcp_deliver conn pkts = sort_by_seq pkts", "[@@induct pkts] [@@auto]"), + ("packet_checksum", "theorem packet_checksum p = valid_packet p ==> verify_checksum p = true", "[@@auto]"), + ("routing_loop_free", "theorem routing_loop_free graph = valid_routing graph ==> acyclic (next_hop graph)", "[@@auto]"), + ("tls_handshake_secrecy", "theorem tls_handshake_secrecy c s = secure_handshake c s ==> secret_key_safe c s", "[@@auto]"), + ("retry_idempotent", "theorem retry_idempotent req = idempotent req ==> retry req = send req", "[@@auto]"), + ] + + security = [ + ("auth_token_valid", "theorem auth_token_valid t = verify_token t = Some user ==> user = decode t", "[@@auto]"), + ("rbac_permission", "theorem rbac_permission u r p = has_role u r && role_has_perm r p ==> authorized u p", "[@@auto]"), + ("password_hash_oneway", "theorem password_hash_oneway p1 p2 = hash_pw p1 = hash_pw p2 ==> p1 = p2 \\/ collision p1 p2", "[@@auto]"), + ("input_sanitize", "theorem input_sanitize s = safe (sanitize s)", "[@@auto]"), + ("session_expiry", "theorem session_expiry s t = expired s t ==> not (valid_session s t)", "[@@auto]"), + ("audit_log_append", "theorem audit_log_append log e = append log e = log ++ [e]", "[@@auto]"), + ] + + ml_correctness = [ + ("gradient_descent_converge", "theorem gd_converge f lr x0 = convex f ==> lr > 0 ==> lr <= 1 / lipschitz f ==> converges (gd f lr x0)", "[@@auto]"), + ("softmax_sum_one", "theorem softmax_sum_one xs = List.length xs > 0 ==> List.fold_left (+.) 0.0 (softmax xs) = 1.0", "[@@auto]"), + ("relu_nonneg", "theorem relu_nonneg x = relu x >= 0.0", "[@@auto]"), + ("sigmoid_bounded", "theorem sigmoid_bounded x = 0.0 <= sigmoid x && sigmoid x <= 1.0", "[@@auto]"), + ("loss_nonneg", "theorem loss_nonneg y y_hat = mse y y_hat >= 0.0", "[@@auto]"), + ("backprop_chain", "theorem backprop_chain f g x = gradient (compose f g) x = gradient f (g x) *. gradient g x", "[@@auto]"), ] all_categories = [ @@ -394,6 +973,18 @@ function generate_imandra()::Vector{Dict{String,Any}} ("verification", verification), ("finance", finance), ("counterexamples", counterexamples), + ("trees", trees_imandra), + ("numerics", numerics), + ("records", records), + ("state_machines", state_machines), + ("crypto", crypto), + ("blockchain", blockchain), + ("algebra", algebra), + ("concurrency", concurrency), + ("databases", databases), + ("networking", networking), + ("security", security), + ("ml_correctness", ml_correctness), ] proofs = Dict{String,Any}[] diff --git a/training_data/proof_states_imandra.jsonl b/training_data/proof_states_imandra.jsonl index f36fdbd7..404a8097 100644 --- a/training_data/proof_states_imandra.jsonl +++ b/training_data/proof_states_imandra.jsonl @@ -1,2200 +1,133 @@ -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1","id":89500,"instance":1,"goal":"verify add_comm_1 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2","id":89501,"instance":2,"goal":"verify mul_comm_2 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_3","id":89502,"instance":3,"goal":"verify add_assoc_3 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_4","id":89503,"instance":4,"goal":"verify iff_self_4 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_5","id":89504,"instance":5,"goal":"verify if_true_5 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_6","id":89505,"instance":6,"goal":"verify list_app_nil_6 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_7","id":89506,"instance":7,"goal":"verify list_len_app_7 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_8","id":89507,"instance":8,"goal":"verify list_rev_rev_8 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_9","id":89508,"instance":9,"goal":"verify bool_and_comm_9 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_10","id":89509,"instance":10,"goal":"instance (x : string) . contract_10 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_11","id":89510,"instance":11,"goal":"lemma lemma_abs_11 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_12","id":89511,"instance":12,"goal":"theorem theorem_leq_12 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_13","id":89512,"instance":13,"goal":"verify add_comm_13 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_14","id":89513,"instance":14,"goal":"verify mul_comm_14 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_15","id":89514,"instance":15,"goal":"verify add_assoc_15 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_16","id":89515,"instance":16,"goal":"verify iff_self_16 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_17","id":89516,"instance":17,"goal":"verify if_true_17 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_18","id":89517,"instance":18,"goal":"verify list_app_nil_18 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_19","id":89518,"instance":19,"goal":"verify list_len_app_19 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_20","id":89519,"instance":20,"goal":"verify list_rev_rev_20 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_21","id":89520,"instance":21,"goal":"verify bool_and_comm_21 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_22","id":89521,"instance":22,"goal":"instance (x : string) . contract_22 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_23","id":89522,"instance":23,"goal":"lemma lemma_abs_23 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_24","id":89523,"instance":24,"goal":"theorem theorem_leq_24 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_25","id":89524,"instance":25,"goal":"verify add_comm_25 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_26","id":89525,"instance":26,"goal":"verify mul_comm_26 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_27","id":89526,"instance":27,"goal":"verify add_assoc_27 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_28","id":89527,"instance":28,"goal":"verify iff_self_28 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_29","id":89528,"instance":29,"goal":"verify if_true_29 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_30","id":89529,"instance":30,"goal":"verify list_app_nil_30 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_31","id":89530,"instance":31,"goal":"verify list_len_app_31 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_32","id":89531,"instance":32,"goal":"verify list_rev_rev_32 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_33","id":89532,"instance":33,"goal":"verify bool_and_comm_33 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_34","id":89533,"instance":34,"goal":"instance (x : string) . contract_34 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_35","id":89534,"instance":35,"goal":"lemma lemma_abs_35 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_36","id":89535,"instance":36,"goal":"theorem theorem_leq_36 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_37","id":89536,"instance":37,"goal":"verify add_comm_37 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_38","id":89537,"instance":38,"goal":"verify mul_comm_38 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_39","id":89538,"instance":39,"goal":"verify add_assoc_39 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_40","id":89539,"instance":40,"goal":"verify iff_self_40 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_41","id":89540,"instance":41,"goal":"verify if_true_41 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_42","id":89541,"instance":42,"goal":"verify list_app_nil_42 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_43","id":89542,"instance":43,"goal":"verify list_len_app_43 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_44","id":89543,"instance":44,"goal":"verify list_rev_rev_44 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_45","id":89544,"instance":45,"goal":"verify bool_and_comm_45 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_46","id":89545,"instance":46,"goal":"instance (x : string) . contract_46 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_47","id":89546,"instance":47,"goal":"lemma lemma_abs_47 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_48","id":89547,"instance":48,"goal":"theorem theorem_leq_48 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_49","id":89548,"instance":49,"goal":"verify add_comm_49 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_50","id":89549,"instance":50,"goal":"verify mul_comm_50 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_51","id":89550,"instance":51,"goal":"verify add_assoc_51 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_52","id":89551,"instance":52,"goal":"verify iff_self_52 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_53","id":89552,"instance":53,"goal":"verify if_true_53 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_54","id":89553,"instance":54,"goal":"verify list_app_nil_54 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_55","id":89554,"instance":55,"goal":"verify list_len_app_55 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_56","id":89555,"instance":56,"goal":"verify list_rev_rev_56 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_57","id":89556,"instance":57,"goal":"verify bool_and_comm_57 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_58","id":89557,"instance":58,"goal":"instance (x : string) . contract_58 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_59","id":89558,"instance":59,"goal":"lemma lemma_abs_59 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_60","id":89559,"instance":60,"goal":"theorem theorem_leq_60 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_61","id":89560,"instance":61,"goal":"verify add_comm_61 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_62","id":89561,"instance":62,"goal":"verify mul_comm_62 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_63","id":89562,"instance":63,"goal":"verify add_assoc_63 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_64","id":89563,"instance":64,"goal":"verify iff_self_64 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_65","id":89564,"instance":65,"goal":"verify if_true_65 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_66","id":89565,"instance":66,"goal":"verify list_app_nil_66 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_67","id":89566,"instance":67,"goal":"verify list_len_app_67 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_68","id":89567,"instance":68,"goal":"verify list_rev_rev_68 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_69","id":89568,"instance":69,"goal":"verify bool_and_comm_69 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_70","id":89569,"instance":70,"goal":"instance (x : string) . contract_70 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_71","id":89570,"instance":71,"goal":"lemma lemma_abs_71 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_72","id":89571,"instance":72,"goal":"theorem theorem_leq_72 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_73","id":89572,"instance":73,"goal":"verify add_comm_73 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_74","id":89573,"instance":74,"goal":"verify mul_comm_74 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_75","id":89574,"instance":75,"goal":"verify add_assoc_75 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_76","id":89575,"instance":76,"goal":"verify iff_self_76 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_77","id":89576,"instance":77,"goal":"verify if_true_77 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_78","id":89577,"instance":78,"goal":"verify list_app_nil_78 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_79","id":89578,"instance":79,"goal":"verify list_len_app_79 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_80","id":89579,"instance":80,"goal":"verify list_rev_rev_80 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_81","id":89580,"instance":81,"goal":"verify bool_and_comm_81 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_82","id":89581,"instance":82,"goal":"instance (x : string) . contract_82 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_83","id":89582,"instance":83,"goal":"lemma lemma_abs_83 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_84","id":89583,"instance":84,"goal":"theorem theorem_leq_84 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_85","id":89584,"instance":85,"goal":"verify add_comm_85 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_86","id":89585,"instance":86,"goal":"verify mul_comm_86 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_87","id":89586,"instance":87,"goal":"verify add_assoc_87 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_88","id":89587,"instance":88,"goal":"verify iff_self_88 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_89","id":89588,"instance":89,"goal":"verify if_true_89 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_90","id":89589,"instance":90,"goal":"verify list_app_nil_90 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_91","id":89590,"instance":91,"goal":"verify list_len_app_91 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_92","id":89591,"instance":92,"goal":"verify list_rev_rev_92 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_93","id":89592,"instance":93,"goal":"verify bool_and_comm_93 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_94","id":89593,"instance":94,"goal":"instance (x : string) . contract_94 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_95","id":89594,"instance":95,"goal":"lemma lemma_abs_95 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_96","id":89595,"instance":96,"goal":"theorem theorem_leq_96 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_97","id":89596,"instance":97,"goal":"verify add_comm_97 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_98","id":89597,"instance":98,"goal":"verify mul_comm_98 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_99","id":89598,"instance":99,"goal":"verify add_assoc_99 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_100","id":89599,"instance":100,"goal":"verify iff_self_100 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_101","id":89600,"instance":101,"goal":"verify if_true_101 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_102","id":89601,"instance":102,"goal":"verify list_app_nil_102 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_103","id":89602,"instance":103,"goal":"verify list_len_app_103 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_104","id":89603,"instance":104,"goal":"verify list_rev_rev_104 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_105","id":89604,"instance":105,"goal":"verify bool_and_comm_105 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_106","id":89605,"instance":106,"goal":"instance (x : string) . contract_106 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_107","id":89606,"instance":107,"goal":"lemma lemma_abs_107 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_108","id":89607,"instance":108,"goal":"theorem theorem_leq_108 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_109","id":89608,"instance":109,"goal":"verify add_comm_109 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_110","id":89609,"instance":110,"goal":"verify mul_comm_110 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_111","id":89610,"instance":111,"goal":"verify add_assoc_111 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_112","id":89611,"instance":112,"goal":"verify iff_self_112 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_113","id":89612,"instance":113,"goal":"verify if_true_113 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_114","id":89613,"instance":114,"goal":"verify list_app_nil_114 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_115","id":89614,"instance":115,"goal":"verify list_len_app_115 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_116","id":89615,"instance":116,"goal":"verify list_rev_rev_116 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_117","id":89616,"instance":117,"goal":"verify bool_and_comm_117 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_118","id":89617,"instance":118,"goal":"instance (x : string) . contract_118 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_119","id":89618,"instance":119,"goal":"lemma lemma_abs_119 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_120","id":89619,"instance":120,"goal":"theorem theorem_leq_120 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_121","id":89620,"instance":121,"goal":"verify add_comm_121 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_122","id":89621,"instance":122,"goal":"verify mul_comm_122 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_123","id":89622,"instance":123,"goal":"verify add_assoc_123 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_124","id":89623,"instance":124,"goal":"verify iff_self_124 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_125","id":89624,"instance":125,"goal":"verify if_true_125 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_126","id":89625,"instance":126,"goal":"verify list_app_nil_126 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_127","id":89626,"instance":127,"goal":"verify list_len_app_127 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_128","id":89627,"instance":128,"goal":"verify list_rev_rev_128 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_129","id":89628,"instance":129,"goal":"verify bool_and_comm_129 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_130","id":89629,"instance":130,"goal":"instance (x : string) . contract_130 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_131","id":89630,"instance":131,"goal":"lemma lemma_abs_131 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_132","id":89631,"instance":132,"goal":"theorem theorem_leq_132 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_133","id":89632,"instance":133,"goal":"verify add_comm_133 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_134","id":89633,"instance":134,"goal":"verify mul_comm_134 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_135","id":89634,"instance":135,"goal":"verify add_assoc_135 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_136","id":89635,"instance":136,"goal":"verify iff_self_136 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_137","id":89636,"instance":137,"goal":"verify if_true_137 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_138","id":89637,"instance":138,"goal":"verify list_app_nil_138 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_139","id":89638,"instance":139,"goal":"verify list_len_app_139 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_140","id":89639,"instance":140,"goal":"verify list_rev_rev_140 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_141","id":89640,"instance":141,"goal":"verify bool_and_comm_141 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_142","id":89641,"instance":142,"goal":"instance (x : string) . contract_142 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_143","id":89642,"instance":143,"goal":"lemma lemma_abs_143 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_144","id":89643,"instance":144,"goal":"theorem theorem_leq_144 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_145","id":89644,"instance":145,"goal":"verify add_comm_145 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_146","id":89645,"instance":146,"goal":"verify mul_comm_146 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_147","id":89646,"instance":147,"goal":"verify add_assoc_147 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_148","id":89647,"instance":148,"goal":"verify iff_self_148 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_149","id":89648,"instance":149,"goal":"verify if_true_149 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_150","id":89649,"instance":150,"goal":"verify list_app_nil_150 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_151","id":89650,"instance":151,"goal":"verify list_len_app_151 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_152","id":89651,"instance":152,"goal":"verify list_rev_rev_152 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_153","id":89652,"instance":153,"goal":"verify bool_and_comm_153 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_154","id":89653,"instance":154,"goal":"instance (x : string) . contract_154 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_155","id":89654,"instance":155,"goal":"lemma lemma_abs_155 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_156","id":89655,"instance":156,"goal":"theorem theorem_leq_156 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_157","id":89656,"instance":157,"goal":"verify add_comm_157 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_158","id":89657,"instance":158,"goal":"verify mul_comm_158 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_159","id":89658,"instance":159,"goal":"verify add_assoc_159 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_160","id":89659,"instance":160,"goal":"verify iff_self_160 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_161","id":89660,"instance":161,"goal":"verify if_true_161 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_162","id":89661,"instance":162,"goal":"verify list_app_nil_162 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_163","id":89662,"instance":163,"goal":"verify list_len_app_163 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_164","id":89663,"instance":164,"goal":"verify list_rev_rev_164 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_165","id":89664,"instance":165,"goal":"verify bool_and_comm_165 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_166","id":89665,"instance":166,"goal":"instance (x : string) . contract_166 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_167","id":89666,"instance":167,"goal":"lemma lemma_abs_167 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_168","id":89667,"instance":168,"goal":"theorem theorem_leq_168 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_169","id":89668,"instance":169,"goal":"verify add_comm_169 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_170","id":89669,"instance":170,"goal":"verify mul_comm_170 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_171","id":89670,"instance":171,"goal":"verify add_assoc_171 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_172","id":89671,"instance":172,"goal":"verify iff_self_172 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_173","id":89672,"instance":173,"goal":"verify if_true_173 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_174","id":89673,"instance":174,"goal":"verify list_app_nil_174 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_175","id":89674,"instance":175,"goal":"verify list_len_app_175 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_176","id":89675,"instance":176,"goal":"verify list_rev_rev_176 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_177","id":89676,"instance":177,"goal":"verify bool_and_comm_177 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_178","id":89677,"instance":178,"goal":"instance (x : string) . contract_178 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_179","id":89678,"instance":179,"goal":"lemma lemma_abs_179 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_180","id":89679,"instance":180,"goal":"theorem theorem_leq_180 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_181","id":89680,"instance":181,"goal":"verify add_comm_181 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_182","id":89681,"instance":182,"goal":"verify mul_comm_182 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_183","id":89682,"instance":183,"goal":"verify add_assoc_183 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_184","id":89683,"instance":184,"goal":"verify iff_self_184 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_185","id":89684,"instance":185,"goal":"verify if_true_185 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_186","id":89685,"instance":186,"goal":"verify list_app_nil_186 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_187","id":89686,"instance":187,"goal":"verify list_len_app_187 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_188","id":89687,"instance":188,"goal":"verify list_rev_rev_188 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_189","id":89688,"instance":189,"goal":"verify bool_and_comm_189 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_190","id":89689,"instance":190,"goal":"instance (x : string) . contract_190 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_191","id":89690,"instance":191,"goal":"lemma lemma_abs_191 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_192","id":89691,"instance":192,"goal":"theorem theorem_leq_192 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_193","id":89692,"instance":193,"goal":"verify add_comm_193 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_194","id":89693,"instance":194,"goal":"verify mul_comm_194 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_195","id":89694,"instance":195,"goal":"verify add_assoc_195 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_196","id":89695,"instance":196,"goal":"verify iff_self_196 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_197","id":89696,"instance":197,"goal":"verify if_true_197 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_198","id":89697,"instance":198,"goal":"verify list_app_nil_198 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_199","id":89698,"instance":199,"goal":"verify list_len_app_199 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_200","id":89699,"instance":200,"goal":"verify list_rev_rev_200 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_201","id":89700,"instance":201,"goal":"verify bool_and_comm_201 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_202","id":89701,"instance":202,"goal":"instance (x : string) . contract_202 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_203","id":89702,"instance":203,"goal":"lemma lemma_abs_203 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_204","id":89703,"instance":204,"goal":"theorem theorem_leq_204 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_205","id":89704,"instance":205,"goal":"verify add_comm_205 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_206","id":89705,"instance":206,"goal":"verify mul_comm_206 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_207","id":89706,"instance":207,"goal":"verify add_assoc_207 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_208","id":89707,"instance":208,"goal":"verify iff_self_208 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_209","id":89708,"instance":209,"goal":"verify if_true_209 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_210","id":89709,"instance":210,"goal":"verify list_app_nil_210 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_211","id":89710,"instance":211,"goal":"verify list_len_app_211 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_212","id":89711,"instance":212,"goal":"verify list_rev_rev_212 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_213","id":89712,"instance":213,"goal":"verify bool_and_comm_213 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_214","id":89713,"instance":214,"goal":"instance (x : string) . contract_214 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_215","id":89714,"instance":215,"goal":"lemma lemma_abs_215 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_216","id":89715,"instance":216,"goal":"theorem theorem_leq_216 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_217","id":89716,"instance":217,"goal":"verify add_comm_217 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_218","id":89717,"instance":218,"goal":"verify mul_comm_218 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_219","id":89718,"instance":219,"goal":"verify add_assoc_219 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_220","id":89719,"instance":220,"goal":"verify iff_self_220 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_221","id":89720,"instance":221,"goal":"verify if_true_221 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_222","id":89721,"instance":222,"goal":"verify list_app_nil_222 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_223","id":89722,"instance":223,"goal":"verify list_len_app_223 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_224","id":89723,"instance":224,"goal":"verify list_rev_rev_224 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_225","id":89724,"instance":225,"goal":"verify bool_and_comm_225 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_226","id":89725,"instance":226,"goal":"instance (x : string) . contract_226 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_227","id":89726,"instance":227,"goal":"lemma lemma_abs_227 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_228","id":89727,"instance":228,"goal":"theorem theorem_leq_228 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_229","id":89728,"instance":229,"goal":"verify add_comm_229 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_230","id":89729,"instance":230,"goal":"verify mul_comm_230 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_231","id":89730,"instance":231,"goal":"verify add_assoc_231 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_232","id":89731,"instance":232,"goal":"verify iff_self_232 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_233","id":89732,"instance":233,"goal":"verify if_true_233 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_234","id":89733,"instance":234,"goal":"verify list_app_nil_234 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_235","id":89734,"instance":235,"goal":"verify list_len_app_235 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_236","id":89735,"instance":236,"goal":"verify list_rev_rev_236 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_237","id":89736,"instance":237,"goal":"verify bool_and_comm_237 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_238","id":89737,"instance":238,"goal":"instance (x : string) . contract_238 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_239","id":89738,"instance":239,"goal":"lemma lemma_abs_239 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_240","id":89739,"instance":240,"goal":"theorem theorem_leq_240 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_241","id":89740,"instance":241,"goal":"verify add_comm_241 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_242","id":89741,"instance":242,"goal":"verify mul_comm_242 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_243","id":89742,"instance":243,"goal":"verify add_assoc_243 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_244","id":89743,"instance":244,"goal":"verify iff_self_244 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_245","id":89744,"instance":245,"goal":"verify if_true_245 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_246","id":89745,"instance":246,"goal":"verify list_app_nil_246 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_247","id":89746,"instance":247,"goal":"verify list_len_app_247 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_248","id":89747,"instance":248,"goal":"verify list_rev_rev_248 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_249","id":89748,"instance":249,"goal":"verify bool_and_comm_249 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_250","id":89749,"instance":250,"goal":"instance (x : string) . contract_250 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_251","id":89750,"instance":251,"goal":"lemma lemma_abs_251 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_252","id":89751,"instance":252,"goal":"theorem theorem_leq_252 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_253","id":89752,"instance":253,"goal":"verify add_comm_253 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_254","id":89753,"instance":254,"goal":"verify mul_comm_254 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_255","id":89754,"instance":255,"goal":"verify add_assoc_255 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_256","id":89755,"instance":256,"goal":"verify iff_self_256 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_257","id":89756,"instance":257,"goal":"verify if_true_257 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_258","id":89757,"instance":258,"goal":"verify list_app_nil_258 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_259","id":89758,"instance":259,"goal":"verify list_len_app_259 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_260","id":89759,"instance":260,"goal":"verify list_rev_rev_260 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_261","id":89760,"instance":261,"goal":"verify bool_and_comm_261 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_262","id":89761,"instance":262,"goal":"instance (x : string) . contract_262 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_263","id":89762,"instance":263,"goal":"lemma lemma_abs_263 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_264","id":89763,"instance":264,"goal":"theorem theorem_leq_264 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_265","id":89764,"instance":265,"goal":"verify add_comm_265 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_266","id":89765,"instance":266,"goal":"verify mul_comm_266 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_267","id":89766,"instance":267,"goal":"verify add_assoc_267 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_268","id":89767,"instance":268,"goal":"verify iff_self_268 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_269","id":89768,"instance":269,"goal":"verify if_true_269 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_270","id":89769,"instance":270,"goal":"verify list_app_nil_270 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_271","id":89770,"instance":271,"goal":"verify list_len_app_271 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_272","id":89771,"instance":272,"goal":"verify list_rev_rev_272 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_273","id":89772,"instance":273,"goal":"verify bool_and_comm_273 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_274","id":89773,"instance":274,"goal":"instance (x : string) . contract_274 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_275","id":89774,"instance":275,"goal":"lemma lemma_abs_275 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_276","id":89775,"instance":276,"goal":"theorem theorem_leq_276 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_277","id":89776,"instance":277,"goal":"verify add_comm_277 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_278","id":89777,"instance":278,"goal":"verify mul_comm_278 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_279","id":89778,"instance":279,"goal":"verify add_assoc_279 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_280","id":89779,"instance":280,"goal":"verify iff_self_280 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_281","id":89780,"instance":281,"goal":"verify if_true_281 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_282","id":89781,"instance":282,"goal":"verify list_app_nil_282 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_283","id":89782,"instance":283,"goal":"verify list_len_app_283 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_284","id":89783,"instance":284,"goal":"verify list_rev_rev_284 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_285","id":89784,"instance":285,"goal":"verify bool_and_comm_285 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_286","id":89785,"instance":286,"goal":"instance (x : string) . contract_286 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_287","id":89786,"instance":287,"goal":"lemma lemma_abs_287 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_288","id":89787,"instance":288,"goal":"theorem theorem_leq_288 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_289","id":89788,"instance":289,"goal":"verify add_comm_289 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_290","id":89789,"instance":290,"goal":"verify mul_comm_290 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_291","id":89790,"instance":291,"goal":"verify add_assoc_291 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_292","id":89791,"instance":292,"goal":"verify iff_self_292 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_293","id":89792,"instance":293,"goal":"verify if_true_293 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_294","id":89793,"instance":294,"goal":"verify list_app_nil_294 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_295","id":89794,"instance":295,"goal":"verify list_len_app_295 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_296","id":89795,"instance":296,"goal":"verify list_rev_rev_296 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_297","id":89796,"instance":297,"goal":"verify bool_and_comm_297 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_298","id":89797,"instance":298,"goal":"instance (x : string) . contract_298 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_299","id":89798,"instance":299,"goal":"lemma lemma_abs_299 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_300","id":89799,"instance":300,"goal":"theorem theorem_leq_300 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_301","id":89800,"instance":301,"goal":"verify add_comm_301 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_302","id":89801,"instance":302,"goal":"verify mul_comm_302 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_303","id":89802,"instance":303,"goal":"verify add_assoc_303 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_304","id":89803,"instance":304,"goal":"verify iff_self_304 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_305","id":89804,"instance":305,"goal":"verify if_true_305 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_306","id":89805,"instance":306,"goal":"verify list_app_nil_306 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_307","id":89806,"instance":307,"goal":"verify list_len_app_307 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_308","id":89807,"instance":308,"goal":"verify list_rev_rev_308 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_309","id":89808,"instance":309,"goal":"verify bool_and_comm_309 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_310","id":89809,"instance":310,"goal":"instance (x : string) . contract_310 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_311","id":89810,"instance":311,"goal":"lemma lemma_abs_311 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_312","id":89811,"instance":312,"goal":"theorem theorem_leq_312 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_313","id":89812,"instance":313,"goal":"verify add_comm_313 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_314","id":89813,"instance":314,"goal":"verify mul_comm_314 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_315","id":89814,"instance":315,"goal":"verify add_assoc_315 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_316","id":89815,"instance":316,"goal":"verify iff_self_316 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_317","id":89816,"instance":317,"goal":"verify if_true_317 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_318","id":89817,"instance":318,"goal":"verify list_app_nil_318 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_319","id":89818,"instance":319,"goal":"verify list_len_app_319 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_320","id":89819,"instance":320,"goal":"verify list_rev_rev_320 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_321","id":89820,"instance":321,"goal":"verify bool_and_comm_321 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_322","id":89821,"instance":322,"goal":"instance (x : string) . contract_322 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_323","id":89822,"instance":323,"goal":"lemma lemma_abs_323 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_324","id":89823,"instance":324,"goal":"theorem theorem_leq_324 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_325","id":89824,"instance":325,"goal":"verify add_comm_325 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_326","id":89825,"instance":326,"goal":"verify mul_comm_326 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_327","id":89826,"instance":327,"goal":"verify add_assoc_327 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_328","id":89827,"instance":328,"goal":"verify iff_self_328 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_329","id":89828,"instance":329,"goal":"verify if_true_329 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_330","id":89829,"instance":330,"goal":"verify list_app_nil_330 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_331","id":89830,"instance":331,"goal":"verify list_len_app_331 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_332","id":89831,"instance":332,"goal":"verify list_rev_rev_332 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_333","id":89832,"instance":333,"goal":"verify bool_and_comm_333 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_334","id":89833,"instance":334,"goal":"instance (x : string) . contract_334 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_335","id":89834,"instance":335,"goal":"lemma lemma_abs_335 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_336","id":89835,"instance":336,"goal":"theorem theorem_leq_336 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_337","id":89836,"instance":337,"goal":"verify add_comm_337 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_338","id":89837,"instance":338,"goal":"verify mul_comm_338 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_339","id":89838,"instance":339,"goal":"verify add_assoc_339 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_340","id":89839,"instance":340,"goal":"verify iff_self_340 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_341","id":89840,"instance":341,"goal":"verify if_true_341 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_342","id":89841,"instance":342,"goal":"verify list_app_nil_342 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_343","id":89842,"instance":343,"goal":"verify list_len_app_343 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_344","id":89843,"instance":344,"goal":"verify list_rev_rev_344 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_345","id":89844,"instance":345,"goal":"verify bool_and_comm_345 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_346","id":89845,"instance":346,"goal":"instance (x : string) . contract_346 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_347","id":89846,"instance":347,"goal":"lemma lemma_abs_347 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_348","id":89847,"instance":348,"goal":"theorem theorem_leq_348 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_349","id":89848,"instance":349,"goal":"verify add_comm_349 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_350","id":89849,"instance":350,"goal":"verify mul_comm_350 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_351","id":89850,"instance":351,"goal":"verify add_assoc_351 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_352","id":89851,"instance":352,"goal":"verify iff_self_352 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_353","id":89852,"instance":353,"goal":"verify if_true_353 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_354","id":89853,"instance":354,"goal":"verify list_app_nil_354 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_355","id":89854,"instance":355,"goal":"verify list_len_app_355 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_356","id":89855,"instance":356,"goal":"verify list_rev_rev_356 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_357","id":89856,"instance":357,"goal":"verify bool_and_comm_357 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_358","id":89857,"instance":358,"goal":"instance (x : string) . contract_358 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_359","id":89858,"instance":359,"goal":"lemma lemma_abs_359 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_360","id":89859,"instance":360,"goal":"theorem theorem_leq_360 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_361","id":89860,"instance":361,"goal":"verify add_comm_361 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_362","id":89861,"instance":362,"goal":"verify mul_comm_362 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_363","id":89862,"instance":363,"goal":"verify add_assoc_363 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_364","id":89863,"instance":364,"goal":"verify iff_self_364 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_365","id":89864,"instance":365,"goal":"verify if_true_365 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_366","id":89865,"instance":366,"goal":"verify list_app_nil_366 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_367","id":89866,"instance":367,"goal":"verify list_len_app_367 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_368","id":89867,"instance":368,"goal":"verify list_rev_rev_368 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_369","id":89868,"instance":369,"goal":"verify bool_and_comm_369 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_370","id":89869,"instance":370,"goal":"instance (x : string) . contract_370 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_371","id":89870,"instance":371,"goal":"lemma lemma_abs_371 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_372","id":89871,"instance":372,"goal":"theorem theorem_leq_372 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_373","id":89872,"instance":373,"goal":"verify add_comm_373 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_374","id":89873,"instance":374,"goal":"verify mul_comm_374 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_375","id":89874,"instance":375,"goal":"verify add_assoc_375 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_376","id":89875,"instance":376,"goal":"verify iff_self_376 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_377","id":89876,"instance":377,"goal":"verify if_true_377 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_378","id":89877,"instance":378,"goal":"verify list_app_nil_378 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_379","id":89878,"instance":379,"goal":"verify list_len_app_379 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_380","id":89879,"instance":380,"goal":"verify list_rev_rev_380 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_381","id":89880,"instance":381,"goal":"verify bool_and_comm_381 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_382","id":89881,"instance":382,"goal":"instance (x : string) . contract_382 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_383","id":89882,"instance":383,"goal":"lemma lemma_abs_383 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_384","id":89883,"instance":384,"goal":"theorem theorem_leq_384 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_385","id":89884,"instance":385,"goal":"verify add_comm_385 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_386","id":89885,"instance":386,"goal":"verify mul_comm_386 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_387","id":89886,"instance":387,"goal":"verify add_assoc_387 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_388","id":89887,"instance":388,"goal":"verify iff_self_388 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_389","id":89888,"instance":389,"goal":"verify if_true_389 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_390","id":89889,"instance":390,"goal":"verify list_app_nil_390 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_391","id":89890,"instance":391,"goal":"verify list_len_app_391 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_392","id":89891,"instance":392,"goal":"verify list_rev_rev_392 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_393","id":89892,"instance":393,"goal":"verify bool_and_comm_393 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_394","id":89893,"instance":394,"goal":"instance (x : string) . contract_394 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_395","id":89894,"instance":395,"goal":"lemma lemma_abs_395 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_396","id":89895,"instance":396,"goal":"theorem theorem_leq_396 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_397","id":89896,"instance":397,"goal":"verify add_comm_397 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_398","id":89897,"instance":398,"goal":"verify mul_comm_398 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_399","id":89898,"instance":399,"goal":"verify add_assoc_399 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_400","id":89899,"instance":400,"goal":"verify iff_self_400 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_401","id":89900,"instance":401,"goal":"verify if_true_401 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_402","id":89901,"instance":402,"goal":"verify list_app_nil_402 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_403","id":89902,"instance":403,"goal":"verify list_len_app_403 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_404","id":89903,"instance":404,"goal":"verify list_rev_rev_404 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_405","id":89904,"instance":405,"goal":"verify bool_and_comm_405 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_406","id":89905,"instance":406,"goal":"instance (x : string) . contract_406 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_407","id":89906,"instance":407,"goal":"lemma lemma_abs_407 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_408","id":89907,"instance":408,"goal":"theorem theorem_leq_408 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_409","id":89908,"instance":409,"goal":"verify add_comm_409 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_410","id":89909,"instance":410,"goal":"verify mul_comm_410 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_411","id":89910,"instance":411,"goal":"verify add_assoc_411 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_412","id":89911,"instance":412,"goal":"verify iff_self_412 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_413","id":89912,"instance":413,"goal":"verify if_true_413 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_414","id":89913,"instance":414,"goal":"verify list_app_nil_414 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_415","id":89914,"instance":415,"goal":"verify list_len_app_415 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_416","id":89915,"instance":416,"goal":"verify list_rev_rev_416 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_417","id":89916,"instance":417,"goal":"verify bool_and_comm_417 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_418","id":89917,"instance":418,"goal":"instance (x : string) . contract_418 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_419","id":89918,"instance":419,"goal":"lemma lemma_abs_419 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_420","id":89919,"instance":420,"goal":"theorem theorem_leq_420 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_421","id":89920,"instance":421,"goal":"verify add_comm_421 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_422","id":89921,"instance":422,"goal":"verify mul_comm_422 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_423","id":89922,"instance":423,"goal":"verify add_assoc_423 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_424","id":89923,"instance":424,"goal":"verify iff_self_424 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_425","id":89924,"instance":425,"goal":"verify if_true_425 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_426","id":89925,"instance":426,"goal":"verify list_app_nil_426 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_427","id":89926,"instance":427,"goal":"verify list_len_app_427 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_428","id":89927,"instance":428,"goal":"verify list_rev_rev_428 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_429","id":89928,"instance":429,"goal":"verify bool_and_comm_429 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_430","id":89929,"instance":430,"goal":"instance (x : string) . contract_430 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_431","id":89930,"instance":431,"goal":"lemma lemma_abs_431 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_432","id":89931,"instance":432,"goal":"theorem theorem_leq_432 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_433","id":89932,"instance":433,"goal":"verify add_comm_433 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_434","id":89933,"instance":434,"goal":"verify mul_comm_434 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_435","id":89934,"instance":435,"goal":"verify add_assoc_435 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_436","id":89935,"instance":436,"goal":"verify iff_self_436 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_437","id":89936,"instance":437,"goal":"verify if_true_437 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_438","id":89937,"instance":438,"goal":"verify list_app_nil_438 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_439","id":89938,"instance":439,"goal":"verify list_len_app_439 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_440","id":89939,"instance":440,"goal":"verify list_rev_rev_440 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_441","id":89940,"instance":441,"goal":"verify bool_and_comm_441 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_442","id":89941,"instance":442,"goal":"instance (x : string) . contract_442 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_443","id":89942,"instance":443,"goal":"lemma lemma_abs_443 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_444","id":89943,"instance":444,"goal":"theorem theorem_leq_444 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_445","id":89944,"instance":445,"goal":"verify add_comm_445 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_446","id":89945,"instance":446,"goal":"verify mul_comm_446 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_447","id":89946,"instance":447,"goal":"verify add_assoc_447 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_448","id":89947,"instance":448,"goal":"verify iff_self_448 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_449","id":89948,"instance":449,"goal":"verify if_true_449 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_450","id":89949,"instance":450,"goal":"verify list_app_nil_450 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_451","id":89950,"instance":451,"goal":"verify list_len_app_451 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_452","id":89951,"instance":452,"goal":"verify list_rev_rev_452 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_453","id":89952,"instance":453,"goal":"verify bool_and_comm_453 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_454","id":89953,"instance":454,"goal":"instance (x : string) . contract_454 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_455","id":89954,"instance":455,"goal":"lemma lemma_abs_455 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_456","id":89955,"instance":456,"goal":"theorem theorem_leq_456 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_457","id":89956,"instance":457,"goal":"verify add_comm_457 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_458","id":89957,"instance":458,"goal":"verify mul_comm_458 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_459","id":89958,"instance":459,"goal":"verify add_assoc_459 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_460","id":89959,"instance":460,"goal":"verify iff_self_460 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_461","id":89960,"instance":461,"goal":"verify if_true_461 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_462","id":89961,"instance":462,"goal":"verify list_app_nil_462 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_463","id":89962,"instance":463,"goal":"verify list_len_app_463 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_464","id":89963,"instance":464,"goal":"verify list_rev_rev_464 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_465","id":89964,"instance":465,"goal":"verify bool_and_comm_465 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_466","id":89965,"instance":466,"goal":"instance (x : string) . contract_466 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_467","id":89966,"instance":467,"goal":"lemma lemma_abs_467 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_468","id":89967,"instance":468,"goal":"theorem theorem_leq_468 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_469","id":89968,"instance":469,"goal":"verify add_comm_469 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_470","id":89969,"instance":470,"goal":"verify mul_comm_470 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_471","id":89970,"instance":471,"goal":"verify add_assoc_471 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_472","id":89971,"instance":472,"goal":"verify iff_self_472 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_473","id":89972,"instance":473,"goal":"verify if_true_473 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_474","id":89973,"instance":474,"goal":"verify list_app_nil_474 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_475","id":89974,"instance":475,"goal":"verify list_len_app_475 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_476","id":89975,"instance":476,"goal":"verify list_rev_rev_476 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_477","id":89976,"instance":477,"goal":"verify bool_and_comm_477 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_478","id":89977,"instance":478,"goal":"instance (x : string) . contract_478 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_479","id":89978,"instance":479,"goal":"lemma lemma_abs_479 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_480","id":89979,"instance":480,"goal":"theorem theorem_leq_480 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_481","id":89980,"instance":481,"goal":"verify add_comm_481 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_482","id":89981,"instance":482,"goal":"verify mul_comm_482 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_483","id":89982,"instance":483,"goal":"verify add_assoc_483 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_484","id":89983,"instance":484,"goal":"verify iff_self_484 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_485","id":89984,"instance":485,"goal":"verify if_true_485 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_486","id":89985,"instance":486,"goal":"verify list_app_nil_486 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_487","id":89986,"instance":487,"goal":"verify list_len_app_487 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_488","id":89987,"instance":488,"goal":"verify list_rev_rev_488 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_489","id":89988,"instance":489,"goal":"verify bool_and_comm_489 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_490","id":89989,"instance":490,"goal":"instance (x : string) . contract_490 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_491","id":89990,"instance":491,"goal":"lemma lemma_abs_491 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_492","id":89991,"instance":492,"goal":"theorem theorem_leq_492 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_493","id":89992,"instance":493,"goal":"verify add_comm_493 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_494","id":89993,"instance":494,"goal":"verify mul_comm_494 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_495","id":89994,"instance":495,"goal":"verify add_assoc_495 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_496","id":89995,"instance":496,"goal":"verify iff_self_496 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_497","id":89996,"instance":497,"goal":"verify if_true_497 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_498","id":89997,"instance":498,"goal":"verify list_app_nil_498 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_499","id":89998,"instance":499,"goal":"verify list_len_app_499 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_500","id":89999,"instance":500,"goal":"verify list_rev_rev_500 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_501","id":90000,"instance":501,"goal":"verify bool_and_comm_501 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_502","id":90001,"instance":502,"goal":"instance (x : string) . contract_502 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_503","id":90002,"instance":503,"goal":"lemma lemma_abs_503 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_504","id":90003,"instance":504,"goal":"theorem theorem_leq_504 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_505","id":90004,"instance":505,"goal":"verify add_comm_505 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_506","id":90005,"instance":506,"goal":"verify mul_comm_506 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_507","id":90006,"instance":507,"goal":"verify add_assoc_507 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_508","id":90007,"instance":508,"goal":"verify iff_self_508 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_509","id":90008,"instance":509,"goal":"verify if_true_509 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_510","id":90009,"instance":510,"goal":"verify list_app_nil_510 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_511","id":90010,"instance":511,"goal":"verify list_len_app_511 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_512","id":90011,"instance":512,"goal":"verify list_rev_rev_512 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_513","id":90012,"instance":513,"goal":"verify bool_and_comm_513 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_514","id":90013,"instance":514,"goal":"instance (x : string) . contract_514 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_515","id":90014,"instance":515,"goal":"lemma lemma_abs_515 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_516","id":90015,"instance":516,"goal":"theorem theorem_leq_516 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_517","id":90016,"instance":517,"goal":"verify add_comm_517 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_518","id":90017,"instance":518,"goal":"verify mul_comm_518 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_519","id":90018,"instance":519,"goal":"verify add_assoc_519 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_520","id":90019,"instance":520,"goal":"verify iff_self_520 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_521","id":90020,"instance":521,"goal":"verify if_true_521 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_522","id":90021,"instance":522,"goal":"verify list_app_nil_522 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_523","id":90022,"instance":523,"goal":"verify list_len_app_523 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_524","id":90023,"instance":524,"goal":"verify list_rev_rev_524 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_525","id":90024,"instance":525,"goal":"verify bool_and_comm_525 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_526","id":90025,"instance":526,"goal":"instance (x : string) . contract_526 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_527","id":90026,"instance":527,"goal":"lemma lemma_abs_527 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_528","id":90027,"instance":528,"goal":"theorem theorem_leq_528 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_529","id":90028,"instance":529,"goal":"verify add_comm_529 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_530","id":90029,"instance":530,"goal":"verify mul_comm_530 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_531","id":90030,"instance":531,"goal":"verify add_assoc_531 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_532","id":90031,"instance":532,"goal":"verify iff_self_532 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_533","id":90032,"instance":533,"goal":"verify if_true_533 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_534","id":90033,"instance":534,"goal":"verify list_app_nil_534 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_535","id":90034,"instance":535,"goal":"verify list_len_app_535 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_536","id":90035,"instance":536,"goal":"verify list_rev_rev_536 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_537","id":90036,"instance":537,"goal":"verify bool_and_comm_537 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_538","id":90037,"instance":538,"goal":"instance (x : string) . contract_538 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_539","id":90038,"instance":539,"goal":"lemma lemma_abs_539 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_540","id":90039,"instance":540,"goal":"theorem theorem_leq_540 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_541","id":90040,"instance":541,"goal":"verify add_comm_541 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_542","id":90041,"instance":542,"goal":"verify mul_comm_542 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_543","id":90042,"instance":543,"goal":"verify add_assoc_543 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_544","id":90043,"instance":544,"goal":"verify iff_self_544 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_545","id":90044,"instance":545,"goal":"verify if_true_545 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_546","id":90045,"instance":546,"goal":"verify list_app_nil_546 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_547","id":90046,"instance":547,"goal":"verify list_len_app_547 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_548","id":90047,"instance":548,"goal":"verify list_rev_rev_548 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_549","id":90048,"instance":549,"goal":"verify bool_and_comm_549 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_550","id":90049,"instance":550,"goal":"instance (x : string) . contract_550 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_551","id":90050,"instance":551,"goal":"lemma lemma_abs_551 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_552","id":90051,"instance":552,"goal":"theorem theorem_leq_552 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_553","id":90052,"instance":553,"goal":"verify add_comm_553 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_554","id":90053,"instance":554,"goal":"verify mul_comm_554 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_555","id":90054,"instance":555,"goal":"verify add_assoc_555 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_556","id":90055,"instance":556,"goal":"verify iff_self_556 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_557","id":90056,"instance":557,"goal":"verify if_true_557 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_558","id":90057,"instance":558,"goal":"verify list_app_nil_558 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_559","id":90058,"instance":559,"goal":"verify list_len_app_559 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_560","id":90059,"instance":560,"goal":"verify list_rev_rev_560 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_561","id":90060,"instance":561,"goal":"verify bool_and_comm_561 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_562","id":90061,"instance":562,"goal":"instance (x : string) . contract_562 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_563","id":90062,"instance":563,"goal":"lemma lemma_abs_563 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_564","id":90063,"instance":564,"goal":"theorem theorem_leq_564 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_565","id":90064,"instance":565,"goal":"verify add_comm_565 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_566","id":90065,"instance":566,"goal":"verify mul_comm_566 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_567","id":90066,"instance":567,"goal":"verify add_assoc_567 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_568","id":90067,"instance":568,"goal":"verify iff_self_568 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_569","id":90068,"instance":569,"goal":"verify if_true_569 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_570","id":90069,"instance":570,"goal":"verify list_app_nil_570 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_571","id":90070,"instance":571,"goal":"verify list_len_app_571 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_572","id":90071,"instance":572,"goal":"verify list_rev_rev_572 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_573","id":90072,"instance":573,"goal":"verify bool_and_comm_573 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_574","id":90073,"instance":574,"goal":"instance (x : string) . contract_574 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_575","id":90074,"instance":575,"goal":"lemma lemma_abs_575 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_576","id":90075,"instance":576,"goal":"theorem theorem_leq_576 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_577","id":90076,"instance":577,"goal":"verify add_comm_577 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_578","id":90077,"instance":578,"goal":"verify mul_comm_578 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_579","id":90078,"instance":579,"goal":"verify add_assoc_579 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_580","id":90079,"instance":580,"goal":"verify iff_self_580 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_581","id":90080,"instance":581,"goal":"verify if_true_581 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_582","id":90081,"instance":582,"goal":"verify list_app_nil_582 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_583","id":90082,"instance":583,"goal":"verify list_len_app_583 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_584","id":90083,"instance":584,"goal":"verify list_rev_rev_584 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_585","id":90084,"instance":585,"goal":"verify bool_and_comm_585 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_586","id":90085,"instance":586,"goal":"instance (x : string) . contract_586 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_587","id":90086,"instance":587,"goal":"lemma lemma_abs_587 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_588","id":90087,"instance":588,"goal":"theorem theorem_leq_588 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_589","id":90088,"instance":589,"goal":"verify add_comm_589 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_590","id":90089,"instance":590,"goal":"verify mul_comm_590 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_591","id":90090,"instance":591,"goal":"verify add_assoc_591 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_592","id":90091,"instance":592,"goal":"verify iff_self_592 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_593","id":90092,"instance":593,"goal":"verify if_true_593 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_594","id":90093,"instance":594,"goal":"verify list_app_nil_594 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_595","id":90094,"instance":595,"goal":"verify list_len_app_595 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_596","id":90095,"instance":596,"goal":"verify list_rev_rev_596 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_597","id":90096,"instance":597,"goal":"verify bool_and_comm_597 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_598","id":90097,"instance":598,"goal":"instance (x : string) . contract_598 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_599","id":90098,"instance":599,"goal":"lemma lemma_abs_599 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_600","id":90099,"instance":600,"goal":"theorem theorem_leq_600 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_601","id":90100,"instance":601,"goal":"verify add_comm_601 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_602","id":90101,"instance":602,"goal":"verify mul_comm_602 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_603","id":90102,"instance":603,"goal":"verify add_assoc_603 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_604","id":90103,"instance":604,"goal":"verify iff_self_604 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_605","id":90104,"instance":605,"goal":"verify if_true_605 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_606","id":90105,"instance":606,"goal":"verify list_app_nil_606 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_607","id":90106,"instance":607,"goal":"verify list_len_app_607 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_608","id":90107,"instance":608,"goal":"verify list_rev_rev_608 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_609","id":90108,"instance":609,"goal":"verify bool_and_comm_609 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_610","id":90109,"instance":610,"goal":"instance (x : string) . contract_610 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_611","id":90110,"instance":611,"goal":"lemma lemma_abs_611 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_612","id":90111,"instance":612,"goal":"theorem theorem_leq_612 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_613","id":90112,"instance":613,"goal":"verify add_comm_613 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_614","id":90113,"instance":614,"goal":"verify mul_comm_614 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_615","id":90114,"instance":615,"goal":"verify add_assoc_615 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_616","id":90115,"instance":616,"goal":"verify iff_self_616 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_617","id":90116,"instance":617,"goal":"verify if_true_617 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_618","id":90117,"instance":618,"goal":"verify list_app_nil_618 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_619","id":90118,"instance":619,"goal":"verify list_len_app_619 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_620","id":90119,"instance":620,"goal":"verify list_rev_rev_620 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_621","id":90120,"instance":621,"goal":"verify bool_and_comm_621 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_622","id":90121,"instance":622,"goal":"instance (x : string) . contract_622 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_623","id":90122,"instance":623,"goal":"lemma lemma_abs_623 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_624","id":90123,"instance":624,"goal":"theorem theorem_leq_624 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_625","id":90124,"instance":625,"goal":"verify add_comm_625 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_626","id":90125,"instance":626,"goal":"verify mul_comm_626 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_627","id":90126,"instance":627,"goal":"verify add_assoc_627 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_628","id":90127,"instance":628,"goal":"verify iff_self_628 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_629","id":90128,"instance":629,"goal":"verify if_true_629 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_630","id":90129,"instance":630,"goal":"verify list_app_nil_630 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_631","id":90130,"instance":631,"goal":"verify list_len_app_631 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_632","id":90131,"instance":632,"goal":"verify list_rev_rev_632 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_633","id":90132,"instance":633,"goal":"verify bool_and_comm_633 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_634","id":90133,"instance":634,"goal":"instance (x : string) . contract_634 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_635","id":90134,"instance":635,"goal":"lemma lemma_abs_635 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_636","id":90135,"instance":636,"goal":"theorem theorem_leq_636 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_637","id":90136,"instance":637,"goal":"verify add_comm_637 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_638","id":90137,"instance":638,"goal":"verify mul_comm_638 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_639","id":90138,"instance":639,"goal":"verify add_assoc_639 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_640","id":90139,"instance":640,"goal":"verify iff_self_640 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_641","id":90140,"instance":641,"goal":"verify if_true_641 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_642","id":90141,"instance":642,"goal":"verify list_app_nil_642 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_643","id":90142,"instance":643,"goal":"verify list_len_app_643 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_644","id":90143,"instance":644,"goal":"verify list_rev_rev_644 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_645","id":90144,"instance":645,"goal":"verify bool_and_comm_645 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_646","id":90145,"instance":646,"goal":"instance (x : string) . contract_646 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_647","id":90146,"instance":647,"goal":"lemma lemma_abs_647 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_648","id":90147,"instance":648,"goal":"theorem theorem_leq_648 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_649","id":90148,"instance":649,"goal":"verify add_comm_649 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_650","id":90149,"instance":650,"goal":"verify mul_comm_650 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_651","id":90150,"instance":651,"goal":"verify add_assoc_651 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_652","id":90151,"instance":652,"goal":"verify iff_self_652 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_653","id":90152,"instance":653,"goal":"verify if_true_653 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_654","id":90153,"instance":654,"goal":"verify list_app_nil_654 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_655","id":90154,"instance":655,"goal":"verify list_len_app_655 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_656","id":90155,"instance":656,"goal":"verify list_rev_rev_656 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_657","id":90156,"instance":657,"goal":"verify bool_and_comm_657 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_658","id":90157,"instance":658,"goal":"instance (x : string) . contract_658 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_659","id":90158,"instance":659,"goal":"lemma lemma_abs_659 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_660","id":90159,"instance":660,"goal":"theorem theorem_leq_660 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_661","id":90160,"instance":661,"goal":"verify add_comm_661 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_662","id":90161,"instance":662,"goal":"verify mul_comm_662 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_663","id":90162,"instance":663,"goal":"verify add_assoc_663 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_664","id":90163,"instance":664,"goal":"verify iff_self_664 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_665","id":90164,"instance":665,"goal":"verify if_true_665 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_666","id":90165,"instance":666,"goal":"verify list_app_nil_666 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_667","id":90166,"instance":667,"goal":"verify list_len_app_667 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_668","id":90167,"instance":668,"goal":"verify list_rev_rev_668 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_669","id":90168,"instance":669,"goal":"verify bool_and_comm_669 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_670","id":90169,"instance":670,"goal":"instance (x : string) . contract_670 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_671","id":90170,"instance":671,"goal":"lemma lemma_abs_671 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_672","id":90171,"instance":672,"goal":"theorem theorem_leq_672 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_673","id":90172,"instance":673,"goal":"verify add_comm_673 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_674","id":90173,"instance":674,"goal":"verify mul_comm_674 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_675","id":90174,"instance":675,"goal":"verify add_assoc_675 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_676","id":90175,"instance":676,"goal":"verify iff_self_676 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_677","id":90176,"instance":677,"goal":"verify if_true_677 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_678","id":90177,"instance":678,"goal":"verify list_app_nil_678 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_679","id":90178,"instance":679,"goal":"verify list_len_app_679 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_680","id":90179,"instance":680,"goal":"verify list_rev_rev_680 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_681","id":90180,"instance":681,"goal":"verify bool_and_comm_681 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_682","id":90181,"instance":682,"goal":"instance (x : string) . contract_682 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_683","id":90182,"instance":683,"goal":"lemma lemma_abs_683 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_684","id":90183,"instance":684,"goal":"theorem theorem_leq_684 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_685","id":90184,"instance":685,"goal":"verify add_comm_685 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_686","id":90185,"instance":686,"goal":"verify mul_comm_686 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_687","id":90186,"instance":687,"goal":"verify add_assoc_687 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_688","id":90187,"instance":688,"goal":"verify iff_self_688 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_689","id":90188,"instance":689,"goal":"verify if_true_689 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_690","id":90189,"instance":690,"goal":"verify list_app_nil_690 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_691","id":90190,"instance":691,"goal":"verify list_len_app_691 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_692","id":90191,"instance":692,"goal":"verify list_rev_rev_692 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_693","id":90192,"instance":693,"goal":"verify bool_and_comm_693 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_694","id":90193,"instance":694,"goal":"instance (x : string) . contract_694 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_695","id":90194,"instance":695,"goal":"lemma lemma_abs_695 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_696","id":90195,"instance":696,"goal":"theorem theorem_leq_696 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_697","id":90196,"instance":697,"goal":"verify add_comm_697 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_698","id":90197,"instance":698,"goal":"verify mul_comm_698 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_699","id":90198,"instance":699,"goal":"verify add_assoc_699 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_700","id":90199,"instance":700,"goal":"verify iff_self_700 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_701","id":90200,"instance":701,"goal":"verify if_true_701 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_702","id":90201,"instance":702,"goal":"verify list_app_nil_702 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_703","id":90202,"instance":703,"goal":"verify list_len_app_703 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_704","id":90203,"instance":704,"goal":"verify list_rev_rev_704 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_705","id":90204,"instance":705,"goal":"verify bool_and_comm_705 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_706","id":90205,"instance":706,"goal":"instance (x : string) . contract_706 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_707","id":90206,"instance":707,"goal":"lemma lemma_abs_707 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_708","id":90207,"instance":708,"goal":"theorem theorem_leq_708 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_709","id":90208,"instance":709,"goal":"verify add_comm_709 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_710","id":90209,"instance":710,"goal":"verify mul_comm_710 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_711","id":90210,"instance":711,"goal":"verify add_assoc_711 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_712","id":90211,"instance":712,"goal":"verify iff_self_712 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_713","id":90212,"instance":713,"goal":"verify if_true_713 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_714","id":90213,"instance":714,"goal":"verify list_app_nil_714 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_715","id":90214,"instance":715,"goal":"verify list_len_app_715 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_716","id":90215,"instance":716,"goal":"verify list_rev_rev_716 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_717","id":90216,"instance":717,"goal":"verify bool_and_comm_717 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_718","id":90217,"instance":718,"goal":"instance (x : string) . contract_718 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_719","id":90218,"instance":719,"goal":"lemma lemma_abs_719 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_720","id":90219,"instance":720,"goal":"theorem theorem_leq_720 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_721","id":90220,"instance":721,"goal":"verify add_comm_721 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_722","id":90221,"instance":722,"goal":"verify mul_comm_722 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_723","id":90222,"instance":723,"goal":"verify add_assoc_723 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_724","id":90223,"instance":724,"goal":"verify iff_self_724 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_725","id":90224,"instance":725,"goal":"verify if_true_725 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_726","id":90225,"instance":726,"goal":"verify list_app_nil_726 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_727","id":90226,"instance":727,"goal":"verify list_len_app_727 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_728","id":90227,"instance":728,"goal":"verify list_rev_rev_728 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_729","id":90228,"instance":729,"goal":"verify bool_and_comm_729 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_730","id":90229,"instance":730,"goal":"instance (x : string) . contract_730 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_731","id":90230,"instance":731,"goal":"lemma lemma_abs_731 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_732","id":90231,"instance":732,"goal":"theorem theorem_leq_732 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_733","id":90232,"instance":733,"goal":"verify add_comm_733 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_734","id":90233,"instance":734,"goal":"verify mul_comm_734 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_735","id":90234,"instance":735,"goal":"verify add_assoc_735 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_736","id":90235,"instance":736,"goal":"verify iff_self_736 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_737","id":90236,"instance":737,"goal":"verify if_true_737 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_738","id":90237,"instance":738,"goal":"verify list_app_nil_738 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_739","id":90238,"instance":739,"goal":"verify list_len_app_739 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_740","id":90239,"instance":740,"goal":"verify list_rev_rev_740 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_741","id":90240,"instance":741,"goal":"verify bool_and_comm_741 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_742","id":90241,"instance":742,"goal":"instance (x : string) . contract_742 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_743","id":90242,"instance":743,"goal":"lemma lemma_abs_743 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_744","id":90243,"instance":744,"goal":"theorem theorem_leq_744 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_745","id":90244,"instance":745,"goal":"verify add_comm_745 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_746","id":90245,"instance":746,"goal":"verify mul_comm_746 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_747","id":90246,"instance":747,"goal":"verify add_assoc_747 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_748","id":90247,"instance":748,"goal":"verify iff_self_748 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_749","id":90248,"instance":749,"goal":"verify if_true_749 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_750","id":90249,"instance":750,"goal":"verify list_app_nil_750 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_751","id":90250,"instance":751,"goal":"verify list_len_app_751 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_752","id":90251,"instance":752,"goal":"verify list_rev_rev_752 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_753","id":90252,"instance":753,"goal":"verify bool_and_comm_753 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_754","id":90253,"instance":754,"goal":"instance (x : string) . contract_754 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_755","id":90254,"instance":755,"goal":"lemma lemma_abs_755 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_756","id":90255,"instance":756,"goal":"theorem theorem_leq_756 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_757","id":90256,"instance":757,"goal":"verify add_comm_757 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_758","id":90257,"instance":758,"goal":"verify mul_comm_758 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_759","id":90258,"instance":759,"goal":"verify add_assoc_759 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_760","id":90259,"instance":760,"goal":"verify iff_self_760 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_761","id":90260,"instance":761,"goal":"verify if_true_761 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_762","id":90261,"instance":762,"goal":"verify list_app_nil_762 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_763","id":90262,"instance":763,"goal":"verify list_len_app_763 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_764","id":90263,"instance":764,"goal":"verify list_rev_rev_764 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_765","id":90264,"instance":765,"goal":"verify bool_and_comm_765 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_766","id":90265,"instance":766,"goal":"instance (x : string) . contract_766 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_767","id":90266,"instance":767,"goal":"lemma lemma_abs_767 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_768","id":90267,"instance":768,"goal":"theorem theorem_leq_768 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_769","id":90268,"instance":769,"goal":"verify add_comm_769 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_770","id":90269,"instance":770,"goal":"verify mul_comm_770 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_771","id":90270,"instance":771,"goal":"verify add_assoc_771 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_772","id":90271,"instance":772,"goal":"verify iff_self_772 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_773","id":90272,"instance":773,"goal":"verify if_true_773 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_774","id":90273,"instance":774,"goal":"verify list_app_nil_774 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_775","id":90274,"instance":775,"goal":"verify list_len_app_775 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_776","id":90275,"instance":776,"goal":"verify list_rev_rev_776 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_777","id":90276,"instance":777,"goal":"verify bool_and_comm_777 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_778","id":90277,"instance":778,"goal":"instance (x : string) . contract_778 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_779","id":90278,"instance":779,"goal":"lemma lemma_abs_779 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_780","id":90279,"instance":780,"goal":"theorem theorem_leq_780 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_781","id":90280,"instance":781,"goal":"verify add_comm_781 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_782","id":90281,"instance":782,"goal":"verify mul_comm_782 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_783","id":90282,"instance":783,"goal":"verify add_assoc_783 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_784","id":90283,"instance":784,"goal":"verify iff_self_784 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_785","id":90284,"instance":785,"goal":"verify if_true_785 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_786","id":90285,"instance":786,"goal":"verify list_app_nil_786 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_787","id":90286,"instance":787,"goal":"verify list_len_app_787 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_788","id":90287,"instance":788,"goal":"verify list_rev_rev_788 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_789","id":90288,"instance":789,"goal":"verify bool_and_comm_789 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_790","id":90289,"instance":790,"goal":"instance (x : string) . contract_790 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_791","id":90290,"instance":791,"goal":"lemma lemma_abs_791 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_792","id":90291,"instance":792,"goal":"theorem theorem_leq_792 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_793","id":90292,"instance":793,"goal":"verify add_comm_793 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_794","id":90293,"instance":794,"goal":"verify mul_comm_794 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_795","id":90294,"instance":795,"goal":"verify add_assoc_795 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_796","id":90295,"instance":796,"goal":"verify iff_self_796 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_797","id":90296,"instance":797,"goal":"verify if_true_797 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_798","id":90297,"instance":798,"goal":"verify list_app_nil_798 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_799","id":90298,"instance":799,"goal":"verify list_len_app_799 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_800","id":90299,"instance":800,"goal":"verify list_rev_rev_800 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_801","id":90300,"instance":801,"goal":"verify bool_and_comm_801 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_802","id":90301,"instance":802,"goal":"instance (x : string) . contract_802 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_803","id":90302,"instance":803,"goal":"lemma lemma_abs_803 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_804","id":90303,"instance":804,"goal":"theorem theorem_leq_804 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_805","id":90304,"instance":805,"goal":"verify add_comm_805 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_806","id":90305,"instance":806,"goal":"verify mul_comm_806 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_807","id":90306,"instance":807,"goal":"verify add_assoc_807 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_808","id":90307,"instance":808,"goal":"verify iff_self_808 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_809","id":90308,"instance":809,"goal":"verify if_true_809 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_810","id":90309,"instance":810,"goal":"verify list_app_nil_810 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_811","id":90310,"instance":811,"goal":"verify list_len_app_811 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_812","id":90311,"instance":812,"goal":"verify list_rev_rev_812 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_813","id":90312,"instance":813,"goal":"verify bool_and_comm_813 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_814","id":90313,"instance":814,"goal":"instance (x : string) . contract_814 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_815","id":90314,"instance":815,"goal":"lemma lemma_abs_815 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_816","id":90315,"instance":816,"goal":"theorem theorem_leq_816 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_817","id":90316,"instance":817,"goal":"verify add_comm_817 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_818","id":90317,"instance":818,"goal":"verify mul_comm_818 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_819","id":90318,"instance":819,"goal":"verify add_assoc_819 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_820","id":90319,"instance":820,"goal":"verify iff_self_820 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_821","id":90320,"instance":821,"goal":"verify if_true_821 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_822","id":90321,"instance":822,"goal":"verify list_app_nil_822 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_823","id":90322,"instance":823,"goal":"verify list_len_app_823 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_824","id":90323,"instance":824,"goal":"verify list_rev_rev_824 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_825","id":90324,"instance":825,"goal":"verify bool_and_comm_825 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_826","id":90325,"instance":826,"goal":"instance (x : string) . contract_826 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_827","id":90326,"instance":827,"goal":"lemma lemma_abs_827 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_828","id":90327,"instance":828,"goal":"theorem theorem_leq_828 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_829","id":90328,"instance":829,"goal":"verify add_comm_829 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_830","id":90329,"instance":830,"goal":"verify mul_comm_830 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_831","id":90330,"instance":831,"goal":"verify add_assoc_831 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_832","id":90331,"instance":832,"goal":"verify iff_self_832 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_833","id":90332,"instance":833,"goal":"verify if_true_833 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_834","id":90333,"instance":834,"goal":"verify list_app_nil_834 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_835","id":90334,"instance":835,"goal":"verify list_len_app_835 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_836","id":90335,"instance":836,"goal":"verify list_rev_rev_836 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_837","id":90336,"instance":837,"goal":"verify bool_and_comm_837 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_838","id":90337,"instance":838,"goal":"instance (x : string) . contract_838 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_839","id":90338,"instance":839,"goal":"lemma lemma_abs_839 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_840","id":90339,"instance":840,"goal":"theorem theorem_leq_840 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_841","id":90340,"instance":841,"goal":"verify add_comm_841 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_842","id":90341,"instance":842,"goal":"verify mul_comm_842 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_843","id":90342,"instance":843,"goal":"verify add_assoc_843 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_844","id":90343,"instance":844,"goal":"verify iff_self_844 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_845","id":90344,"instance":845,"goal":"verify if_true_845 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_846","id":90345,"instance":846,"goal":"verify list_app_nil_846 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_847","id":90346,"instance":847,"goal":"verify list_len_app_847 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_848","id":90347,"instance":848,"goal":"verify list_rev_rev_848 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_849","id":90348,"instance":849,"goal":"verify bool_and_comm_849 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_850","id":90349,"instance":850,"goal":"instance (x : string) . contract_850 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_851","id":90350,"instance":851,"goal":"lemma lemma_abs_851 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_852","id":90351,"instance":852,"goal":"theorem theorem_leq_852 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_853","id":90352,"instance":853,"goal":"verify add_comm_853 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_854","id":90353,"instance":854,"goal":"verify mul_comm_854 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_855","id":90354,"instance":855,"goal":"verify add_assoc_855 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_856","id":90355,"instance":856,"goal":"verify iff_self_856 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_857","id":90356,"instance":857,"goal":"verify if_true_857 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_858","id":90357,"instance":858,"goal":"verify list_app_nil_858 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_859","id":90358,"instance":859,"goal":"verify list_len_app_859 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_860","id":90359,"instance":860,"goal":"verify list_rev_rev_860 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_861","id":90360,"instance":861,"goal":"verify bool_and_comm_861 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_862","id":90361,"instance":862,"goal":"instance (x : string) . contract_862 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_863","id":90362,"instance":863,"goal":"lemma lemma_abs_863 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_864","id":90363,"instance":864,"goal":"theorem theorem_leq_864 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_865","id":90364,"instance":865,"goal":"verify add_comm_865 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_866","id":90365,"instance":866,"goal":"verify mul_comm_866 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_867","id":90366,"instance":867,"goal":"verify add_assoc_867 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_868","id":90367,"instance":868,"goal":"verify iff_self_868 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_869","id":90368,"instance":869,"goal":"verify if_true_869 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_870","id":90369,"instance":870,"goal":"verify list_app_nil_870 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_871","id":90370,"instance":871,"goal":"verify list_len_app_871 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_872","id":90371,"instance":872,"goal":"verify list_rev_rev_872 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_873","id":90372,"instance":873,"goal":"verify bool_and_comm_873 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_874","id":90373,"instance":874,"goal":"instance (x : string) . contract_874 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_875","id":90374,"instance":875,"goal":"lemma lemma_abs_875 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_876","id":90375,"instance":876,"goal":"theorem theorem_leq_876 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_877","id":90376,"instance":877,"goal":"verify add_comm_877 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_878","id":90377,"instance":878,"goal":"verify mul_comm_878 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_879","id":90378,"instance":879,"goal":"verify add_assoc_879 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_880","id":90379,"instance":880,"goal":"verify iff_self_880 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_881","id":90380,"instance":881,"goal":"verify if_true_881 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_882","id":90381,"instance":882,"goal":"verify list_app_nil_882 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_883","id":90382,"instance":883,"goal":"verify list_len_app_883 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_884","id":90383,"instance":884,"goal":"verify list_rev_rev_884 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_885","id":90384,"instance":885,"goal":"verify bool_and_comm_885 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_886","id":90385,"instance":886,"goal":"instance (x : string) . contract_886 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_887","id":90386,"instance":887,"goal":"lemma lemma_abs_887 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_888","id":90387,"instance":888,"goal":"theorem theorem_leq_888 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_889","id":90388,"instance":889,"goal":"verify add_comm_889 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_890","id":90389,"instance":890,"goal":"verify mul_comm_890 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_891","id":90390,"instance":891,"goal":"verify add_assoc_891 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_892","id":90391,"instance":892,"goal":"verify iff_self_892 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_893","id":90392,"instance":893,"goal":"verify if_true_893 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_894","id":90393,"instance":894,"goal":"verify list_app_nil_894 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_895","id":90394,"instance":895,"goal":"verify list_len_app_895 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_896","id":90395,"instance":896,"goal":"verify list_rev_rev_896 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_897","id":90396,"instance":897,"goal":"verify bool_and_comm_897 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_898","id":90397,"instance":898,"goal":"instance (x : string) . contract_898 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_899","id":90398,"instance":899,"goal":"lemma lemma_abs_899 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_900","id":90399,"instance":900,"goal":"theorem theorem_leq_900 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_901","id":90400,"instance":901,"goal":"verify add_comm_901 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_902","id":90401,"instance":902,"goal":"verify mul_comm_902 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_903","id":90402,"instance":903,"goal":"verify add_assoc_903 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_904","id":90403,"instance":904,"goal":"verify iff_self_904 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_905","id":90404,"instance":905,"goal":"verify if_true_905 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_906","id":90405,"instance":906,"goal":"verify list_app_nil_906 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_907","id":90406,"instance":907,"goal":"verify list_len_app_907 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_908","id":90407,"instance":908,"goal":"verify list_rev_rev_908 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_909","id":90408,"instance":909,"goal":"verify bool_and_comm_909 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_910","id":90409,"instance":910,"goal":"instance (x : string) . contract_910 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_911","id":90410,"instance":911,"goal":"lemma lemma_abs_911 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_912","id":90411,"instance":912,"goal":"theorem theorem_leq_912 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_913","id":90412,"instance":913,"goal":"verify add_comm_913 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_914","id":90413,"instance":914,"goal":"verify mul_comm_914 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_915","id":90414,"instance":915,"goal":"verify add_assoc_915 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_916","id":90415,"instance":916,"goal":"verify iff_self_916 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_917","id":90416,"instance":917,"goal":"verify if_true_917 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_918","id":90417,"instance":918,"goal":"verify list_app_nil_918 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_919","id":90418,"instance":919,"goal":"verify list_len_app_919 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_920","id":90419,"instance":920,"goal":"verify list_rev_rev_920 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_921","id":90420,"instance":921,"goal":"verify bool_and_comm_921 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_922","id":90421,"instance":922,"goal":"instance (x : string) . contract_922 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_923","id":90422,"instance":923,"goal":"lemma lemma_abs_923 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_924","id":90423,"instance":924,"goal":"theorem theorem_leq_924 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_925","id":90424,"instance":925,"goal":"verify add_comm_925 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_926","id":90425,"instance":926,"goal":"verify mul_comm_926 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_927","id":90426,"instance":927,"goal":"verify add_assoc_927 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_928","id":90427,"instance":928,"goal":"verify iff_self_928 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_929","id":90428,"instance":929,"goal":"verify if_true_929 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_930","id":90429,"instance":930,"goal":"verify list_app_nil_930 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_931","id":90430,"instance":931,"goal":"verify list_len_app_931 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_932","id":90431,"instance":932,"goal":"verify list_rev_rev_932 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_933","id":90432,"instance":933,"goal":"verify bool_and_comm_933 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_934","id":90433,"instance":934,"goal":"instance (x : string) . contract_934 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_935","id":90434,"instance":935,"goal":"lemma lemma_abs_935 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_936","id":90435,"instance":936,"goal":"theorem theorem_leq_936 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_937","id":90436,"instance":937,"goal":"verify add_comm_937 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_938","id":90437,"instance":938,"goal":"verify mul_comm_938 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_939","id":90438,"instance":939,"goal":"verify add_assoc_939 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_940","id":90439,"instance":940,"goal":"verify iff_self_940 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_941","id":90440,"instance":941,"goal":"verify if_true_941 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_942","id":90441,"instance":942,"goal":"verify list_app_nil_942 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_943","id":90442,"instance":943,"goal":"verify list_len_app_943 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_944","id":90443,"instance":944,"goal":"verify list_rev_rev_944 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_945","id":90444,"instance":945,"goal":"verify bool_and_comm_945 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_946","id":90445,"instance":946,"goal":"instance (x : string) . contract_946 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_947","id":90446,"instance":947,"goal":"lemma lemma_abs_947 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_948","id":90447,"instance":948,"goal":"theorem theorem_leq_948 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_949","id":90448,"instance":949,"goal":"verify add_comm_949 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_950","id":90449,"instance":950,"goal":"verify mul_comm_950 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_951","id":90450,"instance":951,"goal":"verify add_assoc_951 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_952","id":90451,"instance":952,"goal":"verify iff_self_952 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_953","id":90452,"instance":953,"goal":"verify if_true_953 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_954","id":90453,"instance":954,"goal":"verify list_app_nil_954 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_955","id":90454,"instance":955,"goal":"verify list_len_app_955 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_956","id":90455,"instance":956,"goal":"verify list_rev_rev_956 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_957","id":90456,"instance":957,"goal":"verify bool_and_comm_957 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_958","id":90457,"instance":958,"goal":"instance (x : string) . contract_958 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_959","id":90458,"instance":959,"goal":"lemma lemma_abs_959 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_960","id":90459,"instance":960,"goal":"theorem theorem_leq_960 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_961","id":90460,"instance":961,"goal":"verify add_comm_961 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_962","id":90461,"instance":962,"goal":"verify mul_comm_962 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_963","id":90462,"instance":963,"goal":"verify add_assoc_963 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_964","id":90463,"instance":964,"goal":"verify iff_self_964 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_965","id":90464,"instance":965,"goal":"verify if_true_965 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_966","id":90465,"instance":966,"goal":"verify list_app_nil_966 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_967","id":90466,"instance":967,"goal":"verify list_len_app_967 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_968","id":90467,"instance":968,"goal":"verify list_rev_rev_968 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_969","id":90468,"instance":969,"goal":"verify bool_and_comm_969 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_970","id":90469,"instance":970,"goal":"instance (x : string) . contract_970 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_971","id":90470,"instance":971,"goal":"lemma lemma_abs_971 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_972","id":90471,"instance":972,"goal":"theorem theorem_leq_972 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_973","id":90472,"instance":973,"goal":"verify add_comm_973 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_974","id":90473,"instance":974,"goal":"verify mul_comm_974 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_975","id":90474,"instance":975,"goal":"verify add_assoc_975 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_976","id":90475,"instance":976,"goal":"verify iff_self_976 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_977","id":90476,"instance":977,"goal":"verify if_true_977 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_978","id":90477,"instance":978,"goal":"verify list_app_nil_978 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_979","id":90478,"instance":979,"goal":"verify list_len_app_979 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_980","id":90479,"instance":980,"goal":"verify list_rev_rev_980 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_981","id":90480,"instance":981,"goal":"verify bool_and_comm_981 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_982","id":90481,"instance":982,"goal":"instance (x : string) . contract_982 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_983","id":90482,"instance":983,"goal":"lemma lemma_abs_983 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_984","id":90483,"instance":984,"goal":"theorem theorem_leq_984 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_985","id":90484,"instance":985,"goal":"verify add_comm_985 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_986","id":90485,"instance":986,"goal":"verify mul_comm_986 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_987","id":90486,"instance":987,"goal":"verify add_assoc_987 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_988","id":90487,"instance":988,"goal":"verify iff_self_988 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_989","id":90488,"instance":989,"goal":"verify if_true_989 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_990","id":90489,"instance":990,"goal":"verify list_app_nil_990 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_991","id":90490,"instance":991,"goal":"verify list_len_app_991 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_992","id":90491,"instance":992,"goal":"verify list_rev_rev_992 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_993","id":90492,"instance":993,"goal":"verify bool_and_comm_993 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_994","id":90493,"instance":994,"goal":"instance (x : string) . contract_994 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_995","id":90494,"instance":995,"goal":"lemma lemma_abs_995 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_996","id":90495,"instance":996,"goal":"theorem theorem_leq_996 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_997","id":90496,"instance":997,"goal":"verify add_comm_997 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_998","id":90497,"instance":998,"goal":"verify mul_comm_998 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_999","id":90498,"instance":999,"goal":"verify add_assoc_999 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1000","id":90499,"instance":1000,"goal":"verify iff_self_1000 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1001","id":90500,"instance":1001,"goal":"verify if_true_1001 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1002","id":90501,"instance":1002,"goal":"verify list_app_nil_1002 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1003","id":90502,"instance":1003,"goal":"verify list_len_app_1003 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1004","id":90503,"instance":1004,"goal":"verify list_rev_rev_1004 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1005","id":90504,"instance":1005,"goal":"verify bool_and_comm_1005 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1006","id":90505,"instance":1006,"goal":"instance (x : string) . contract_1006 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1007","id":90506,"instance":1007,"goal":"lemma lemma_abs_1007 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1008","id":90507,"instance":1008,"goal":"theorem theorem_leq_1008 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1009","id":90508,"instance":1009,"goal":"verify add_comm_1009 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1010","id":90509,"instance":1010,"goal":"verify mul_comm_1010 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1011","id":90510,"instance":1011,"goal":"verify add_assoc_1011 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1012","id":90511,"instance":1012,"goal":"verify iff_self_1012 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1013","id":90512,"instance":1013,"goal":"verify if_true_1013 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1014","id":90513,"instance":1014,"goal":"verify list_app_nil_1014 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1015","id":90514,"instance":1015,"goal":"verify list_len_app_1015 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1016","id":90515,"instance":1016,"goal":"verify list_rev_rev_1016 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1017","id":90516,"instance":1017,"goal":"verify bool_and_comm_1017 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1018","id":90517,"instance":1018,"goal":"instance (x : string) . contract_1018 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1019","id":90518,"instance":1019,"goal":"lemma lemma_abs_1019 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1020","id":90519,"instance":1020,"goal":"theorem theorem_leq_1020 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1021","id":90520,"instance":1021,"goal":"verify add_comm_1021 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1022","id":90521,"instance":1022,"goal":"verify mul_comm_1022 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1023","id":90522,"instance":1023,"goal":"verify add_assoc_1023 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1024","id":90523,"instance":1024,"goal":"verify iff_self_1024 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1025","id":90524,"instance":1025,"goal":"verify if_true_1025 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1026","id":90525,"instance":1026,"goal":"verify list_app_nil_1026 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1027","id":90526,"instance":1027,"goal":"verify list_len_app_1027 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1028","id":90527,"instance":1028,"goal":"verify list_rev_rev_1028 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1029","id":90528,"instance":1029,"goal":"verify bool_and_comm_1029 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1030","id":90529,"instance":1030,"goal":"instance (x : string) . contract_1030 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1031","id":90530,"instance":1031,"goal":"lemma lemma_abs_1031 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1032","id":90531,"instance":1032,"goal":"theorem theorem_leq_1032 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1033","id":90532,"instance":1033,"goal":"verify add_comm_1033 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1034","id":90533,"instance":1034,"goal":"verify mul_comm_1034 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1035","id":90534,"instance":1035,"goal":"verify add_assoc_1035 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1036","id":90535,"instance":1036,"goal":"verify iff_self_1036 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1037","id":90536,"instance":1037,"goal":"verify if_true_1037 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1038","id":90537,"instance":1038,"goal":"verify list_app_nil_1038 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1039","id":90538,"instance":1039,"goal":"verify list_len_app_1039 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1040","id":90539,"instance":1040,"goal":"verify list_rev_rev_1040 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1041","id":90540,"instance":1041,"goal":"verify bool_and_comm_1041 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1042","id":90541,"instance":1042,"goal":"instance (x : string) . contract_1042 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1043","id":90542,"instance":1043,"goal":"lemma lemma_abs_1043 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1044","id":90543,"instance":1044,"goal":"theorem theorem_leq_1044 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1045","id":90544,"instance":1045,"goal":"verify add_comm_1045 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1046","id":90545,"instance":1046,"goal":"verify mul_comm_1046 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1047","id":90546,"instance":1047,"goal":"verify add_assoc_1047 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1048","id":90547,"instance":1048,"goal":"verify iff_self_1048 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1049","id":90548,"instance":1049,"goal":"verify if_true_1049 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1050","id":90549,"instance":1050,"goal":"verify list_app_nil_1050 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1051","id":90550,"instance":1051,"goal":"verify list_len_app_1051 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1052","id":90551,"instance":1052,"goal":"verify list_rev_rev_1052 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1053","id":90552,"instance":1053,"goal":"verify bool_and_comm_1053 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1054","id":90553,"instance":1054,"goal":"instance (x : string) . contract_1054 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1055","id":90554,"instance":1055,"goal":"lemma lemma_abs_1055 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1056","id":90555,"instance":1056,"goal":"theorem theorem_leq_1056 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1057","id":90556,"instance":1057,"goal":"verify add_comm_1057 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1058","id":90557,"instance":1058,"goal":"verify mul_comm_1058 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1059","id":90558,"instance":1059,"goal":"verify add_assoc_1059 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1060","id":90559,"instance":1060,"goal":"verify iff_self_1060 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1061","id":90560,"instance":1061,"goal":"verify if_true_1061 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1062","id":90561,"instance":1062,"goal":"verify list_app_nil_1062 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1063","id":90562,"instance":1063,"goal":"verify list_len_app_1063 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1064","id":90563,"instance":1064,"goal":"verify list_rev_rev_1064 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1065","id":90564,"instance":1065,"goal":"verify bool_and_comm_1065 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1066","id":90565,"instance":1066,"goal":"instance (x : string) . contract_1066 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1067","id":90566,"instance":1067,"goal":"lemma lemma_abs_1067 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1068","id":90567,"instance":1068,"goal":"theorem theorem_leq_1068 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1069","id":90568,"instance":1069,"goal":"verify add_comm_1069 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1070","id":90569,"instance":1070,"goal":"verify mul_comm_1070 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1071","id":90570,"instance":1071,"goal":"verify add_assoc_1071 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1072","id":90571,"instance":1072,"goal":"verify iff_self_1072 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1073","id":90572,"instance":1073,"goal":"verify if_true_1073 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1074","id":90573,"instance":1074,"goal":"verify list_app_nil_1074 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1075","id":90574,"instance":1075,"goal":"verify list_len_app_1075 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1076","id":90575,"instance":1076,"goal":"verify list_rev_rev_1076 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1077","id":90576,"instance":1077,"goal":"verify bool_and_comm_1077 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1078","id":90577,"instance":1078,"goal":"instance (x : string) . contract_1078 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1079","id":90578,"instance":1079,"goal":"lemma lemma_abs_1079 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1080","id":90579,"instance":1080,"goal":"theorem theorem_leq_1080 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1081","id":90580,"instance":1081,"goal":"verify add_comm_1081 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1082","id":90581,"instance":1082,"goal":"verify mul_comm_1082 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1083","id":90582,"instance":1083,"goal":"verify add_assoc_1083 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1084","id":90583,"instance":1084,"goal":"verify iff_self_1084 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1085","id":90584,"instance":1085,"goal":"verify if_true_1085 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1086","id":90585,"instance":1086,"goal":"verify list_app_nil_1086 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1087","id":90586,"instance":1087,"goal":"verify list_len_app_1087 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1088","id":90587,"instance":1088,"goal":"verify list_rev_rev_1088 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1089","id":90588,"instance":1089,"goal":"verify bool_and_comm_1089 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1090","id":90589,"instance":1090,"goal":"instance (x : string) . contract_1090 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1091","id":90590,"instance":1091,"goal":"lemma lemma_abs_1091 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1092","id":90591,"instance":1092,"goal":"theorem theorem_leq_1092 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1093","id":90592,"instance":1093,"goal":"verify add_comm_1093 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1094","id":90593,"instance":1094,"goal":"verify mul_comm_1094 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1095","id":90594,"instance":1095,"goal":"verify add_assoc_1095 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1096","id":90595,"instance":1096,"goal":"verify iff_self_1096 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1097","id":90596,"instance":1097,"goal":"verify if_true_1097 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1098","id":90597,"instance":1098,"goal":"verify list_app_nil_1098 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1099","id":90598,"instance":1099,"goal":"verify list_len_app_1099 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1100","id":90599,"instance":1100,"goal":"verify list_rev_rev_1100 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1101","id":90600,"instance":1101,"goal":"verify bool_and_comm_1101 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1102","id":90601,"instance":1102,"goal":"instance (x : string) . contract_1102 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1103","id":90602,"instance":1103,"goal":"lemma lemma_abs_1103 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1104","id":90603,"instance":1104,"goal":"theorem theorem_leq_1104 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1105","id":90604,"instance":1105,"goal":"verify add_comm_1105 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1106","id":90605,"instance":1106,"goal":"verify mul_comm_1106 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1107","id":90606,"instance":1107,"goal":"verify add_assoc_1107 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1108","id":90607,"instance":1108,"goal":"verify iff_self_1108 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1109","id":90608,"instance":1109,"goal":"verify if_true_1109 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1110","id":90609,"instance":1110,"goal":"verify list_app_nil_1110 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1111","id":90610,"instance":1111,"goal":"verify list_len_app_1111 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1112","id":90611,"instance":1112,"goal":"verify list_rev_rev_1112 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1113","id":90612,"instance":1113,"goal":"verify bool_and_comm_1113 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1114","id":90613,"instance":1114,"goal":"instance (x : string) . contract_1114 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1115","id":90614,"instance":1115,"goal":"lemma lemma_abs_1115 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1116","id":90615,"instance":1116,"goal":"theorem theorem_leq_1116 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1117","id":90616,"instance":1117,"goal":"verify add_comm_1117 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1118","id":90617,"instance":1118,"goal":"verify mul_comm_1118 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1119","id":90618,"instance":1119,"goal":"verify add_assoc_1119 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1120","id":90619,"instance":1120,"goal":"verify iff_self_1120 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1121","id":90620,"instance":1121,"goal":"verify if_true_1121 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1122","id":90621,"instance":1122,"goal":"verify list_app_nil_1122 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1123","id":90622,"instance":1123,"goal":"verify list_len_app_1123 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1124","id":90623,"instance":1124,"goal":"verify list_rev_rev_1124 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1125","id":90624,"instance":1125,"goal":"verify bool_and_comm_1125 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1126","id":90625,"instance":1126,"goal":"instance (x : string) . contract_1126 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1127","id":90626,"instance":1127,"goal":"lemma lemma_abs_1127 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1128","id":90627,"instance":1128,"goal":"theorem theorem_leq_1128 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1129","id":90628,"instance":1129,"goal":"verify add_comm_1129 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1130","id":90629,"instance":1130,"goal":"verify mul_comm_1130 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1131","id":90630,"instance":1131,"goal":"verify add_assoc_1131 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1132","id":90631,"instance":1132,"goal":"verify iff_self_1132 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1133","id":90632,"instance":1133,"goal":"verify if_true_1133 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1134","id":90633,"instance":1134,"goal":"verify list_app_nil_1134 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1135","id":90634,"instance":1135,"goal":"verify list_len_app_1135 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1136","id":90635,"instance":1136,"goal":"verify list_rev_rev_1136 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1137","id":90636,"instance":1137,"goal":"verify bool_and_comm_1137 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1138","id":90637,"instance":1138,"goal":"instance (x : string) . contract_1138 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1139","id":90638,"instance":1139,"goal":"lemma lemma_abs_1139 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1140","id":90639,"instance":1140,"goal":"theorem theorem_leq_1140 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1141","id":90640,"instance":1141,"goal":"verify add_comm_1141 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1142","id":90641,"instance":1142,"goal":"verify mul_comm_1142 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1143","id":90642,"instance":1143,"goal":"verify add_assoc_1143 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1144","id":90643,"instance":1144,"goal":"verify iff_self_1144 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1145","id":90644,"instance":1145,"goal":"verify if_true_1145 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1146","id":90645,"instance":1146,"goal":"verify list_app_nil_1146 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1147","id":90646,"instance":1147,"goal":"verify list_len_app_1147 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1148","id":90647,"instance":1148,"goal":"verify list_rev_rev_1148 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1149","id":90648,"instance":1149,"goal":"verify bool_and_comm_1149 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1150","id":90649,"instance":1150,"goal":"instance (x : string) . contract_1150 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1151","id":90650,"instance":1151,"goal":"lemma lemma_abs_1151 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1152","id":90651,"instance":1152,"goal":"theorem theorem_leq_1152 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1153","id":90652,"instance":1153,"goal":"verify add_comm_1153 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1154","id":90653,"instance":1154,"goal":"verify mul_comm_1154 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1155","id":90654,"instance":1155,"goal":"verify add_assoc_1155 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1156","id":90655,"instance":1156,"goal":"verify iff_self_1156 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1157","id":90656,"instance":1157,"goal":"verify if_true_1157 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1158","id":90657,"instance":1158,"goal":"verify list_app_nil_1158 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1159","id":90658,"instance":1159,"goal":"verify list_len_app_1159 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1160","id":90659,"instance":1160,"goal":"verify list_rev_rev_1160 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1161","id":90660,"instance":1161,"goal":"verify bool_and_comm_1161 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1162","id":90661,"instance":1162,"goal":"instance (x : string) . contract_1162 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1163","id":90662,"instance":1163,"goal":"lemma lemma_abs_1163 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1164","id":90663,"instance":1164,"goal":"theorem theorem_leq_1164 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1165","id":90664,"instance":1165,"goal":"verify add_comm_1165 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1166","id":90665,"instance":1166,"goal":"verify mul_comm_1166 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1167","id":90666,"instance":1167,"goal":"verify add_assoc_1167 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1168","id":90667,"instance":1168,"goal":"verify iff_self_1168 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1169","id":90668,"instance":1169,"goal":"verify if_true_1169 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1170","id":90669,"instance":1170,"goal":"verify list_app_nil_1170 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1171","id":90670,"instance":1171,"goal":"verify list_len_app_1171 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1172","id":90671,"instance":1172,"goal":"verify list_rev_rev_1172 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1173","id":90672,"instance":1173,"goal":"verify bool_and_comm_1173 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1174","id":90673,"instance":1174,"goal":"instance (x : string) . contract_1174 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1175","id":90674,"instance":1175,"goal":"lemma lemma_abs_1175 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1176","id":90675,"instance":1176,"goal":"theorem theorem_leq_1176 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1177","id":90676,"instance":1177,"goal":"verify add_comm_1177 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1178","id":90677,"instance":1178,"goal":"verify mul_comm_1178 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1179","id":90678,"instance":1179,"goal":"verify add_assoc_1179 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1180","id":90679,"instance":1180,"goal":"verify iff_self_1180 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1181","id":90680,"instance":1181,"goal":"verify if_true_1181 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1182","id":90681,"instance":1182,"goal":"verify list_app_nil_1182 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1183","id":90682,"instance":1183,"goal":"verify list_len_app_1183 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1184","id":90683,"instance":1184,"goal":"verify list_rev_rev_1184 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1185","id":90684,"instance":1185,"goal":"verify bool_and_comm_1185 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1186","id":90685,"instance":1186,"goal":"instance (x : string) . contract_1186 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1187","id":90686,"instance":1187,"goal":"lemma lemma_abs_1187 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1188","id":90687,"instance":1188,"goal":"theorem theorem_leq_1188 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1189","id":90688,"instance":1189,"goal":"verify add_comm_1189 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1190","id":90689,"instance":1190,"goal":"verify mul_comm_1190 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1191","id":90690,"instance":1191,"goal":"verify add_assoc_1191 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1192","id":90691,"instance":1192,"goal":"verify iff_self_1192 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1193","id":90692,"instance":1193,"goal":"verify if_true_1193 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1194","id":90693,"instance":1194,"goal":"verify list_app_nil_1194 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1195","id":90694,"instance":1195,"goal":"verify list_len_app_1195 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1196","id":90695,"instance":1196,"goal":"verify list_rev_rev_1196 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1197","id":90696,"instance":1197,"goal":"verify bool_and_comm_1197 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1198","id":90697,"instance":1198,"goal":"instance (x : string) . contract_1198 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1199","id":90698,"instance":1199,"goal":"lemma lemma_abs_1199 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1200","id":90699,"instance":1200,"goal":"theorem theorem_leq_1200 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1201","id":90700,"instance":1201,"goal":"verify add_comm_1201 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1202","id":90701,"instance":1202,"goal":"verify mul_comm_1202 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1203","id":90702,"instance":1203,"goal":"verify add_assoc_1203 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1204","id":90703,"instance":1204,"goal":"verify iff_self_1204 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1205","id":90704,"instance":1205,"goal":"verify if_true_1205 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1206","id":90705,"instance":1206,"goal":"verify list_app_nil_1206 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1207","id":90706,"instance":1207,"goal":"verify list_len_app_1207 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1208","id":90707,"instance":1208,"goal":"verify list_rev_rev_1208 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1209","id":90708,"instance":1209,"goal":"verify bool_and_comm_1209 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1210","id":90709,"instance":1210,"goal":"instance (x : string) . contract_1210 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1211","id":90710,"instance":1211,"goal":"lemma lemma_abs_1211 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1212","id":90711,"instance":1212,"goal":"theorem theorem_leq_1212 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1213","id":90712,"instance":1213,"goal":"verify add_comm_1213 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1214","id":90713,"instance":1214,"goal":"verify mul_comm_1214 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1215","id":90714,"instance":1215,"goal":"verify add_assoc_1215 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1216","id":90715,"instance":1216,"goal":"verify iff_self_1216 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1217","id":90716,"instance":1217,"goal":"verify if_true_1217 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1218","id":90717,"instance":1218,"goal":"verify list_app_nil_1218 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1219","id":90718,"instance":1219,"goal":"verify list_len_app_1219 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1220","id":90719,"instance":1220,"goal":"verify list_rev_rev_1220 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1221","id":90720,"instance":1221,"goal":"verify bool_and_comm_1221 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1222","id":90721,"instance":1222,"goal":"instance (x : string) . contract_1222 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1223","id":90722,"instance":1223,"goal":"lemma lemma_abs_1223 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1224","id":90723,"instance":1224,"goal":"theorem theorem_leq_1224 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1225","id":90724,"instance":1225,"goal":"verify add_comm_1225 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1226","id":90725,"instance":1226,"goal":"verify mul_comm_1226 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1227","id":90726,"instance":1227,"goal":"verify add_assoc_1227 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1228","id":90727,"instance":1228,"goal":"verify iff_self_1228 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1229","id":90728,"instance":1229,"goal":"verify if_true_1229 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1230","id":90729,"instance":1230,"goal":"verify list_app_nil_1230 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1231","id":90730,"instance":1231,"goal":"verify list_len_app_1231 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1232","id":90731,"instance":1232,"goal":"verify list_rev_rev_1232 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1233","id":90732,"instance":1233,"goal":"verify bool_and_comm_1233 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1234","id":90733,"instance":1234,"goal":"instance (x : string) . contract_1234 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1235","id":90734,"instance":1235,"goal":"lemma lemma_abs_1235 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1236","id":90735,"instance":1236,"goal":"theorem theorem_leq_1236 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1237","id":90736,"instance":1237,"goal":"verify add_comm_1237 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1238","id":90737,"instance":1238,"goal":"verify mul_comm_1238 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1239","id":90738,"instance":1239,"goal":"verify add_assoc_1239 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1240","id":90739,"instance":1240,"goal":"verify iff_self_1240 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1241","id":90740,"instance":1241,"goal":"verify if_true_1241 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1242","id":90741,"instance":1242,"goal":"verify list_app_nil_1242 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1243","id":90742,"instance":1243,"goal":"verify list_len_app_1243 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1244","id":90743,"instance":1244,"goal":"verify list_rev_rev_1244 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1245","id":90744,"instance":1245,"goal":"verify bool_and_comm_1245 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1246","id":90745,"instance":1246,"goal":"instance (x : string) . contract_1246 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1247","id":90746,"instance":1247,"goal":"lemma lemma_abs_1247 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1248","id":90747,"instance":1248,"goal":"theorem theorem_leq_1248 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1249","id":90748,"instance":1249,"goal":"verify add_comm_1249 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1250","id":90749,"instance":1250,"goal":"verify mul_comm_1250 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1251","id":90750,"instance":1251,"goal":"verify add_assoc_1251 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1252","id":90751,"instance":1252,"goal":"verify iff_self_1252 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1253","id":90752,"instance":1253,"goal":"verify if_true_1253 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1254","id":90753,"instance":1254,"goal":"verify list_app_nil_1254 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1255","id":90754,"instance":1255,"goal":"verify list_len_app_1255 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1256","id":90755,"instance":1256,"goal":"verify list_rev_rev_1256 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1257","id":90756,"instance":1257,"goal":"verify bool_and_comm_1257 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1258","id":90757,"instance":1258,"goal":"instance (x : string) . contract_1258 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1259","id":90758,"instance":1259,"goal":"lemma lemma_abs_1259 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1260","id":90759,"instance":1260,"goal":"theorem theorem_leq_1260 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1261","id":90760,"instance":1261,"goal":"verify add_comm_1261 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1262","id":90761,"instance":1262,"goal":"verify mul_comm_1262 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1263","id":90762,"instance":1263,"goal":"verify add_assoc_1263 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1264","id":90763,"instance":1264,"goal":"verify iff_self_1264 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1265","id":90764,"instance":1265,"goal":"verify if_true_1265 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1266","id":90765,"instance":1266,"goal":"verify list_app_nil_1266 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1267","id":90766,"instance":1267,"goal":"verify list_len_app_1267 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1268","id":90767,"instance":1268,"goal":"verify list_rev_rev_1268 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1269","id":90768,"instance":1269,"goal":"verify bool_and_comm_1269 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1270","id":90769,"instance":1270,"goal":"instance (x : string) . contract_1270 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1271","id":90770,"instance":1271,"goal":"lemma lemma_abs_1271 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1272","id":90771,"instance":1272,"goal":"theorem theorem_leq_1272 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1273","id":90772,"instance":1273,"goal":"verify add_comm_1273 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1274","id":90773,"instance":1274,"goal":"verify mul_comm_1274 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1275","id":90774,"instance":1275,"goal":"verify add_assoc_1275 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1276","id":90775,"instance":1276,"goal":"verify iff_self_1276 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1277","id":90776,"instance":1277,"goal":"verify if_true_1277 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1278","id":90777,"instance":1278,"goal":"verify list_app_nil_1278 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1279","id":90778,"instance":1279,"goal":"verify list_len_app_1279 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1280","id":90779,"instance":1280,"goal":"verify list_rev_rev_1280 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1281","id":90780,"instance":1281,"goal":"verify bool_and_comm_1281 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1282","id":90781,"instance":1282,"goal":"instance (x : string) . contract_1282 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1283","id":90782,"instance":1283,"goal":"lemma lemma_abs_1283 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1284","id":90783,"instance":1284,"goal":"theorem theorem_leq_1284 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1285","id":90784,"instance":1285,"goal":"verify add_comm_1285 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1286","id":90785,"instance":1286,"goal":"verify mul_comm_1286 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1287","id":90786,"instance":1287,"goal":"verify add_assoc_1287 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1288","id":90787,"instance":1288,"goal":"verify iff_self_1288 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1289","id":90788,"instance":1289,"goal":"verify if_true_1289 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1290","id":90789,"instance":1290,"goal":"verify list_app_nil_1290 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1291","id":90790,"instance":1291,"goal":"verify list_len_app_1291 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1292","id":90791,"instance":1292,"goal":"verify list_rev_rev_1292 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1293","id":90792,"instance":1293,"goal":"verify bool_and_comm_1293 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1294","id":90793,"instance":1294,"goal":"instance (x : string) . contract_1294 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1295","id":90794,"instance":1295,"goal":"lemma lemma_abs_1295 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1296","id":90795,"instance":1296,"goal":"theorem theorem_leq_1296 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1297","id":90796,"instance":1297,"goal":"verify add_comm_1297 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1298","id":90797,"instance":1298,"goal":"verify mul_comm_1298 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1299","id":90798,"instance":1299,"goal":"verify add_assoc_1299 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1300","id":90799,"instance":1300,"goal":"verify iff_self_1300 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1301","id":90800,"instance":1301,"goal":"verify if_true_1301 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1302","id":90801,"instance":1302,"goal":"verify list_app_nil_1302 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1303","id":90802,"instance":1303,"goal":"verify list_len_app_1303 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1304","id":90803,"instance":1304,"goal":"verify list_rev_rev_1304 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1305","id":90804,"instance":1305,"goal":"verify bool_and_comm_1305 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1306","id":90805,"instance":1306,"goal":"instance (x : string) . contract_1306 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1307","id":90806,"instance":1307,"goal":"lemma lemma_abs_1307 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1308","id":90807,"instance":1308,"goal":"theorem theorem_leq_1308 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1309","id":90808,"instance":1309,"goal":"verify add_comm_1309 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1310","id":90809,"instance":1310,"goal":"verify mul_comm_1310 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1311","id":90810,"instance":1311,"goal":"verify add_assoc_1311 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1312","id":90811,"instance":1312,"goal":"verify iff_self_1312 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1313","id":90812,"instance":1313,"goal":"verify if_true_1313 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1314","id":90813,"instance":1314,"goal":"verify list_app_nil_1314 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1315","id":90814,"instance":1315,"goal":"verify list_len_app_1315 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1316","id":90815,"instance":1316,"goal":"verify list_rev_rev_1316 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1317","id":90816,"instance":1317,"goal":"verify bool_and_comm_1317 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1318","id":90817,"instance":1318,"goal":"instance (x : string) . contract_1318 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1319","id":90818,"instance":1319,"goal":"lemma lemma_abs_1319 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1320","id":90819,"instance":1320,"goal":"theorem theorem_leq_1320 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1321","id":90820,"instance":1321,"goal":"verify add_comm_1321 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1322","id":90821,"instance":1322,"goal":"verify mul_comm_1322 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1323","id":90822,"instance":1323,"goal":"verify add_assoc_1323 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1324","id":90823,"instance":1324,"goal":"verify iff_self_1324 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1325","id":90824,"instance":1325,"goal":"verify if_true_1325 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1326","id":90825,"instance":1326,"goal":"verify list_app_nil_1326 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1327","id":90826,"instance":1327,"goal":"verify list_len_app_1327 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1328","id":90827,"instance":1328,"goal":"verify list_rev_rev_1328 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1329","id":90828,"instance":1329,"goal":"verify bool_and_comm_1329 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1330","id":90829,"instance":1330,"goal":"instance (x : string) . contract_1330 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1331","id":90830,"instance":1331,"goal":"lemma lemma_abs_1331 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1332","id":90831,"instance":1332,"goal":"theorem theorem_leq_1332 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1333","id":90832,"instance":1333,"goal":"verify add_comm_1333 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1334","id":90833,"instance":1334,"goal":"verify mul_comm_1334 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1335","id":90834,"instance":1335,"goal":"verify add_assoc_1335 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1336","id":90835,"instance":1336,"goal":"verify iff_self_1336 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1337","id":90836,"instance":1337,"goal":"verify if_true_1337 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1338","id":90837,"instance":1338,"goal":"verify list_app_nil_1338 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1339","id":90838,"instance":1339,"goal":"verify list_len_app_1339 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1340","id":90839,"instance":1340,"goal":"verify list_rev_rev_1340 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1341","id":90840,"instance":1341,"goal":"verify bool_and_comm_1341 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1342","id":90841,"instance":1342,"goal":"instance (x : string) . contract_1342 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1343","id":90842,"instance":1343,"goal":"lemma lemma_abs_1343 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1344","id":90843,"instance":1344,"goal":"theorem theorem_leq_1344 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1345","id":90844,"instance":1345,"goal":"verify add_comm_1345 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1346","id":90845,"instance":1346,"goal":"verify mul_comm_1346 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1347","id":90846,"instance":1347,"goal":"verify add_assoc_1347 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1348","id":90847,"instance":1348,"goal":"verify iff_self_1348 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1349","id":90848,"instance":1349,"goal":"verify if_true_1349 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1350","id":90849,"instance":1350,"goal":"verify list_app_nil_1350 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1351","id":90850,"instance":1351,"goal":"verify list_len_app_1351 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1352","id":90851,"instance":1352,"goal":"verify list_rev_rev_1352 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1353","id":90852,"instance":1353,"goal":"verify bool_and_comm_1353 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1354","id":90853,"instance":1354,"goal":"instance (x : string) . contract_1354 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1355","id":90854,"instance":1355,"goal":"lemma lemma_abs_1355 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1356","id":90855,"instance":1356,"goal":"theorem theorem_leq_1356 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1357","id":90856,"instance":1357,"goal":"verify add_comm_1357 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1358","id":90857,"instance":1358,"goal":"verify mul_comm_1358 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1359","id":90858,"instance":1359,"goal":"verify add_assoc_1359 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1360","id":90859,"instance":1360,"goal":"verify iff_self_1360 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1361","id":90860,"instance":1361,"goal":"verify if_true_1361 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1362","id":90861,"instance":1362,"goal":"verify list_app_nil_1362 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1363","id":90862,"instance":1363,"goal":"verify list_len_app_1363 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1364","id":90863,"instance":1364,"goal":"verify list_rev_rev_1364 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1365","id":90864,"instance":1365,"goal":"verify bool_and_comm_1365 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1366","id":90865,"instance":1366,"goal":"instance (x : string) . contract_1366 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1367","id":90866,"instance":1367,"goal":"lemma lemma_abs_1367 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1368","id":90867,"instance":1368,"goal":"theorem theorem_leq_1368 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1369","id":90868,"instance":1369,"goal":"verify add_comm_1369 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1370","id":90869,"instance":1370,"goal":"verify mul_comm_1370 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1371","id":90870,"instance":1371,"goal":"verify add_assoc_1371 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1372","id":90871,"instance":1372,"goal":"verify iff_self_1372 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1373","id":90872,"instance":1373,"goal":"verify if_true_1373 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1374","id":90873,"instance":1374,"goal":"verify list_app_nil_1374 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1375","id":90874,"instance":1375,"goal":"verify list_len_app_1375 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1376","id":90875,"instance":1376,"goal":"verify list_rev_rev_1376 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1377","id":90876,"instance":1377,"goal":"verify bool_and_comm_1377 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1378","id":90877,"instance":1378,"goal":"instance (x : string) . contract_1378 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1379","id":90878,"instance":1379,"goal":"lemma lemma_abs_1379 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1380","id":90879,"instance":1380,"goal":"theorem theorem_leq_1380 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1381","id":90880,"instance":1381,"goal":"verify add_comm_1381 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1382","id":90881,"instance":1382,"goal":"verify mul_comm_1382 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1383","id":90882,"instance":1383,"goal":"verify add_assoc_1383 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1384","id":90883,"instance":1384,"goal":"verify iff_self_1384 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1385","id":90884,"instance":1385,"goal":"verify if_true_1385 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1386","id":90885,"instance":1386,"goal":"verify list_app_nil_1386 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1387","id":90886,"instance":1387,"goal":"verify list_len_app_1387 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1388","id":90887,"instance":1388,"goal":"verify list_rev_rev_1388 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1389","id":90888,"instance":1389,"goal":"verify bool_and_comm_1389 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1390","id":90889,"instance":1390,"goal":"instance (x : string) . contract_1390 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1391","id":90890,"instance":1391,"goal":"lemma lemma_abs_1391 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1392","id":90891,"instance":1392,"goal":"theorem theorem_leq_1392 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1393","id":90892,"instance":1393,"goal":"verify add_comm_1393 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1394","id":90893,"instance":1394,"goal":"verify mul_comm_1394 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1395","id":90894,"instance":1395,"goal":"verify add_assoc_1395 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1396","id":90895,"instance":1396,"goal":"verify iff_self_1396 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1397","id":90896,"instance":1397,"goal":"verify if_true_1397 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1398","id":90897,"instance":1398,"goal":"verify list_app_nil_1398 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1399","id":90898,"instance":1399,"goal":"verify list_len_app_1399 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1400","id":90899,"instance":1400,"goal":"verify list_rev_rev_1400 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1401","id":90900,"instance":1401,"goal":"verify bool_and_comm_1401 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1402","id":90901,"instance":1402,"goal":"instance (x : string) . contract_1402 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1403","id":90902,"instance":1403,"goal":"lemma lemma_abs_1403 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1404","id":90903,"instance":1404,"goal":"theorem theorem_leq_1404 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1405","id":90904,"instance":1405,"goal":"verify add_comm_1405 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1406","id":90905,"instance":1406,"goal":"verify mul_comm_1406 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1407","id":90906,"instance":1407,"goal":"verify add_assoc_1407 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1408","id":90907,"instance":1408,"goal":"verify iff_self_1408 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1409","id":90908,"instance":1409,"goal":"verify if_true_1409 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1410","id":90909,"instance":1410,"goal":"verify list_app_nil_1410 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1411","id":90910,"instance":1411,"goal":"verify list_len_app_1411 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1412","id":90911,"instance":1412,"goal":"verify list_rev_rev_1412 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1413","id":90912,"instance":1413,"goal":"verify bool_and_comm_1413 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1414","id":90913,"instance":1414,"goal":"instance (x : string) . contract_1414 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1415","id":90914,"instance":1415,"goal":"lemma lemma_abs_1415 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1416","id":90915,"instance":1416,"goal":"theorem theorem_leq_1416 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1417","id":90916,"instance":1417,"goal":"verify add_comm_1417 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1418","id":90917,"instance":1418,"goal":"verify mul_comm_1418 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1419","id":90918,"instance":1419,"goal":"verify add_assoc_1419 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1420","id":90919,"instance":1420,"goal":"verify iff_self_1420 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1421","id":90920,"instance":1421,"goal":"verify if_true_1421 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1422","id":90921,"instance":1422,"goal":"verify list_app_nil_1422 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1423","id":90922,"instance":1423,"goal":"verify list_len_app_1423 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1424","id":90923,"instance":1424,"goal":"verify list_rev_rev_1424 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1425","id":90924,"instance":1425,"goal":"verify bool_and_comm_1425 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1426","id":90925,"instance":1426,"goal":"instance (x : string) . contract_1426 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1427","id":90926,"instance":1427,"goal":"lemma lemma_abs_1427 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1428","id":90927,"instance":1428,"goal":"theorem theorem_leq_1428 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1429","id":90928,"instance":1429,"goal":"verify add_comm_1429 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1430","id":90929,"instance":1430,"goal":"verify mul_comm_1430 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1431","id":90930,"instance":1431,"goal":"verify add_assoc_1431 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1432","id":90931,"instance":1432,"goal":"verify iff_self_1432 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1433","id":90932,"instance":1433,"goal":"verify if_true_1433 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1434","id":90933,"instance":1434,"goal":"verify list_app_nil_1434 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1435","id":90934,"instance":1435,"goal":"verify list_len_app_1435 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1436","id":90935,"instance":1436,"goal":"verify list_rev_rev_1436 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1437","id":90936,"instance":1437,"goal":"verify bool_and_comm_1437 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1438","id":90937,"instance":1438,"goal":"instance (x : string) . contract_1438 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1439","id":90938,"instance":1439,"goal":"lemma lemma_abs_1439 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1440","id":90939,"instance":1440,"goal":"theorem theorem_leq_1440 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1441","id":90940,"instance":1441,"goal":"verify add_comm_1441 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1442","id":90941,"instance":1442,"goal":"verify mul_comm_1442 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1443","id":90942,"instance":1443,"goal":"verify add_assoc_1443 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1444","id":90943,"instance":1444,"goal":"verify iff_self_1444 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1445","id":90944,"instance":1445,"goal":"verify if_true_1445 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1446","id":90945,"instance":1446,"goal":"verify list_app_nil_1446 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1447","id":90946,"instance":1447,"goal":"verify list_len_app_1447 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1448","id":90947,"instance":1448,"goal":"verify list_rev_rev_1448 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1449","id":90948,"instance":1449,"goal":"verify bool_and_comm_1449 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1450","id":90949,"instance":1450,"goal":"instance (x : string) . contract_1450 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1451","id":90950,"instance":1451,"goal":"lemma lemma_abs_1451 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1452","id":90951,"instance":1452,"goal":"theorem theorem_leq_1452 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1453","id":90952,"instance":1453,"goal":"verify add_comm_1453 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1454","id":90953,"instance":1454,"goal":"verify mul_comm_1454 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1455","id":90954,"instance":1455,"goal":"verify add_assoc_1455 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1456","id":90955,"instance":1456,"goal":"verify iff_self_1456 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1457","id":90956,"instance":1457,"goal":"verify if_true_1457 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1458","id":90957,"instance":1458,"goal":"verify list_app_nil_1458 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1459","id":90958,"instance":1459,"goal":"verify list_len_app_1459 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1460","id":90959,"instance":1460,"goal":"verify list_rev_rev_1460 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1461","id":90960,"instance":1461,"goal":"verify bool_and_comm_1461 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1462","id":90961,"instance":1462,"goal":"instance (x : string) . contract_1462 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1463","id":90962,"instance":1463,"goal":"lemma lemma_abs_1463 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1464","id":90963,"instance":1464,"goal":"theorem theorem_leq_1464 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1465","id":90964,"instance":1465,"goal":"verify add_comm_1465 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1466","id":90965,"instance":1466,"goal":"verify mul_comm_1466 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1467","id":90966,"instance":1467,"goal":"verify add_assoc_1467 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1468","id":90967,"instance":1468,"goal":"verify iff_self_1468 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1469","id":90968,"instance":1469,"goal":"verify if_true_1469 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1470","id":90969,"instance":1470,"goal":"verify list_app_nil_1470 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1471","id":90970,"instance":1471,"goal":"verify list_len_app_1471 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1472","id":90971,"instance":1472,"goal":"verify list_rev_rev_1472 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1473","id":90972,"instance":1473,"goal":"verify bool_and_comm_1473 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1474","id":90973,"instance":1474,"goal":"instance (x : string) . contract_1474 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1475","id":90974,"instance":1475,"goal":"lemma lemma_abs_1475 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1476","id":90975,"instance":1476,"goal":"theorem theorem_leq_1476 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1477","id":90976,"instance":1477,"goal":"verify add_comm_1477 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1478","id":90977,"instance":1478,"goal":"verify mul_comm_1478 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1479","id":90978,"instance":1479,"goal":"verify add_assoc_1479 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1480","id":90979,"instance":1480,"goal":"verify iff_self_1480 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1481","id":90980,"instance":1481,"goal":"verify if_true_1481 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1482","id":90981,"instance":1482,"goal":"verify list_app_nil_1482 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1483","id":90982,"instance":1483,"goal":"verify list_len_app_1483 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1484","id":90983,"instance":1484,"goal":"verify list_rev_rev_1484 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1485","id":90984,"instance":1485,"goal":"verify bool_and_comm_1485 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1486","id":90985,"instance":1486,"goal":"instance (x : string) . contract_1486 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1487","id":90986,"instance":1487,"goal":"lemma lemma_abs_1487 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1488","id":90987,"instance":1488,"goal":"theorem theorem_leq_1488 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1489","id":90988,"instance":1489,"goal":"verify add_comm_1489 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1490","id":90989,"instance":1490,"goal":"verify mul_comm_1490 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1491","id":90990,"instance":1491,"goal":"verify add_assoc_1491 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1492","id":90991,"instance":1492,"goal":"verify iff_self_1492 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1493","id":90992,"instance":1493,"goal":"verify if_true_1493 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1494","id":90993,"instance":1494,"goal":"verify list_app_nil_1494 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1495","id":90994,"instance":1495,"goal":"verify list_len_app_1495 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1496","id":90995,"instance":1496,"goal":"verify list_rev_rev_1496 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1497","id":90996,"instance":1497,"goal":"verify bool_and_comm_1497 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1498","id":90997,"instance":1498,"goal":"instance (x : string) . contract_1498 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1499","id":90998,"instance":1499,"goal":"lemma lemma_abs_1499 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1500","id":90999,"instance":1500,"goal":"theorem theorem_leq_1500 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1501","id":91000,"instance":1501,"goal":"verify add_comm_1501 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1502","id":91001,"instance":1502,"goal":"verify mul_comm_1502 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1503","id":91002,"instance":1503,"goal":"verify add_assoc_1503 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1504","id":91003,"instance":1504,"goal":"verify iff_self_1504 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1505","id":91004,"instance":1505,"goal":"verify if_true_1505 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1506","id":91005,"instance":1506,"goal":"verify list_app_nil_1506 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1507","id":91006,"instance":1507,"goal":"verify list_len_app_1507 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1508","id":91007,"instance":1508,"goal":"verify list_rev_rev_1508 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1509","id":91008,"instance":1509,"goal":"verify bool_and_comm_1509 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1510","id":91009,"instance":1510,"goal":"instance (x : string) . contract_1510 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1511","id":91010,"instance":1511,"goal":"lemma lemma_abs_1511 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1512","id":91011,"instance":1512,"goal":"theorem theorem_leq_1512 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1513","id":91012,"instance":1513,"goal":"verify add_comm_1513 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1514","id":91013,"instance":1514,"goal":"verify mul_comm_1514 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1515","id":91014,"instance":1515,"goal":"verify add_assoc_1515 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1516","id":91015,"instance":1516,"goal":"verify iff_self_1516 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1517","id":91016,"instance":1517,"goal":"verify if_true_1517 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1518","id":91017,"instance":1518,"goal":"verify list_app_nil_1518 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1519","id":91018,"instance":1519,"goal":"verify list_len_app_1519 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1520","id":91019,"instance":1520,"goal":"verify list_rev_rev_1520 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1521","id":91020,"instance":1521,"goal":"verify bool_and_comm_1521 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1522","id":91021,"instance":1522,"goal":"instance (x : string) . contract_1522 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1523","id":91022,"instance":1523,"goal":"lemma lemma_abs_1523 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1524","id":91023,"instance":1524,"goal":"theorem theorem_leq_1524 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1525","id":91024,"instance":1525,"goal":"verify add_comm_1525 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1526","id":91025,"instance":1526,"goal":"verify mul_comm_1526 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1527","id":91026,"instance":1527,"goal":"verify add_assoc_1527 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1528","id":91027,"instance":1528,"goal":"verify iff_self_1528 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1529","id":91028,"instance":1529,"goal":"verify if_true_1529 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1530","id":91029,"instance":1530,"goal":"verify list_app_nil_1530 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1531","id":91030,"instance":1531,"goal":"verify list_len_app_1531 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1532","id":91031,"instance":1532,"goal":"verify list_rev_rev_1532 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1533","id":91032,"instance":1533,"goal":"verify bool_and_comm_1533 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1534","id":91033,"instance":1534,"goal":"instance (x : string) . contract_1534 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1535","id":91034,"instance":1535,"goal":"lemma lemma_abs_1535 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1536","id":91035,"instance":1536,"goal":"theorem theorem_leq_1536 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1537","id":91036,"instance":1537,"goal":"verify add_comm_1537 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1538","id":91037,"instance":1538,"goal":"verify mul_comm_1538 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1539","id":91038,"instance":1539,"goal":"verify add_assoc_1539 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1540","id":91039,"instance":1540,"goal":"verify iff_self_1540 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1541","id":91040,"instance":1541,"goal":"verify if_true_1541 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1542","id":91041,"instance":1542,"goal":"verify list_app_nil_1542 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1543","id":91042,"instance":1543,"goal":"verify list_len_app_1543 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1544","id":91043,"instance":1544,"goal":"verify list_rev_rev_1544 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1545","id":91044,"instance":1545,"goal":"verify bool_and_comm_1545 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1546","id":91045,"instance":1546,"goal":"instance (x : string) . contract_1546 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1547","id":91046,"instance":1547,"goal":"lemma lemma_abs_1547 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1548","id":91047,"instance":1548,"goal":"theorem theorem_leq_1548 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1549","id":91048,"instance":1549,"goal":"verify add_comm_1549 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1550","id":91049,"instance":1550,"goal":"verify mul_comm_1550 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1551","id":91050,"instance":1551,"goal":"verify add_assoc_1551 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1552","id":91051,"instance":1552,"goal":"verify iff_self_1552 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1553","id":91052,"instance":1553,"goal":"verify if_true_1553 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1554","id":91053,"instance":1554,"goal":"verify list_app_nil_1554 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1555","id":91054,"instance":1555,"goal":"verify list_len_app_1555 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1556","id":91055,"instance":1556,"goal":"verify list_rev_rev_1556 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1557","id":91056,"instance":1557,"goal":"verify bool_and_comm_1557 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1558","id":91057,"instance":1558,"goal":"instance (x : string) . contract_1558 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1559","id":91058,"instance":1559,"goal":"lemma lemma_abs_1559 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1560","id":91059,"instance":1560,"goal":"theorem theorem_leq_1560 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1561","id":91060,"instance":1561,"goal":"verify add_comm_1561 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1562","id":91061,"instance":1562,"goal":"verify mul_comm_1562 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1563","id":91062,"instance":1563,"goal":"verify add_assoc_1563 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1564","id":91063,"instance":1564,"goal":"verify iff_self_1564 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1565","id":91064,"instance":1565,"goal":"verify if_true_1565 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1566","id":91065,"instance":1566,"goal":"verify list_app_nil_1566 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1567","id":91066,"instance":1567,"goal":"verify list_len_app_1567 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1568","id":91067,"instance":1568,"goal":"verify list_rev_rev_1568 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1569","id":91068,"instance":1569,"goal":"verify bool_and_comm_1569 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1570","id":91069,"instance":1570,"goal":"instance (x : string) . contract_1570 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1571","id":91070,"instance":1571,"goal":"lemma lemma_abs_1571 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1572","id":91071,"instance":1572,"goal":"theorem theorem_leq_1572 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1573","id":91072,"instance":1573,"goal":"verify add_comm_1573 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1574","id":91073,"instance":1574,"goal":"verify mul_comm_1574 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1575","id":91074,"instance":1575,"goal":"verify add_assoc_1575 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1576","id":91075,"instance":1576,"goal":"verify iff_self_1576 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1577","id":91076,"instance":1577,"goal":"verify if_true_1577 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1578","id":91077,"instance":1578,"goal":"verify list_app_nil_1578 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1579","id":91078,"instance":1579,"goal":"verify list_len_app_1579 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1580","id":91079,"instance":1580,"goal":"verify list_rev_rev_1580 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1581","id":91080,"instance":1581,"goal":"verify bool_and_comm_1581 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1582","id":91081,"instance":1582,"goal":"instance (x : string) . contract_1582 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1583","id":91082,"instance":1583,"goal":"lemma lemma_abs_1583 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1584","id":91083,"instance":1584,"goal":"theorem theorem_leq_1584 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1585","id":91084,"instance":1585,"goal":"verify add_comm_1585 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1586","id":91085,"instance":1586,"goal":"verify mul_comm_1586 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1587","id":91086,"instance":1587,"goal":"verify add_assoc_1587 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1588","id":91087,"instance":1588,"goal":"verify iff_self_1588 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1589","id":91088,"instance":1589,"goal":"verify if_true_1589 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1590","id":91089,"instance":1590,"goal":"verify list_app_nil_1590 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1591","id":91090,"instance":1591,"goal":"verify list_len_app_1591 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1592","id":91091,"instance":1592,"goal":"verify list_rev_rev_1592 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1593","id":91092,"instance":1593,"goal":"verify bool_and_comm_1593 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1594","id":91093,"instance":1594,"goal":"instance (x : string) . contract_1594 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1595","id":91094,"instance":1595,"goal":"lemma lemma_abs_1595 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1596","id":91095,"instance":1596,"goal":"theorem theorem_leq_1596 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1597","id":91096,"instance":1597,"goal":"verify add_comm_1597 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1598","id":91097,"instance":1598,"goal":"verify mul_comm_1598 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1599","id":91098,"instance":1599,"goal":"verify add_assoc_1599 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1600","id":91099,"instance":1600,"goal":"verify iff_self_1600 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1601","id":91100,"instance":1601,"goal":"verify if_true_1601 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1602","id":91101,"instance":1602,"goal":"verify list_app_nil_1602 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1603","id":91102,"instance":1603,"goal":"verify list_len_app_1603 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1604","id":91103,"instance":1604,"goal":"verify list_rev_rev_1604 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1605","id":91104,"instance":1605,"goal":"verify bool_and_comm_1605 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1606","id":91105,"instance":1606,"goal":"instance (x : string) . contract_1606 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1607","id":91106,"instance":1607,"goal":"lemma lemma_abs_1607 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1608","id":91107,"instance":1608,"goal":"theorem theorem_leq_1608 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1609","id":91108,"instance":1609,"goal":"verify add_comm_1609 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1610","id":91109,"instance":1610,"goal":"verify mul_comm_1610 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1611","id":91110,"instance":1611,"goal":"verify add_assoc_1611 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1612","id":91111,"instance":1612,"goal":"verify iff_self_1612 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1613","id":91112,"instance":1613,"goal":"verify if_true_1613 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1614","id":91113,"instance":1614,"goal":"verify list_app_nil_1614 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1615","id":91114,"instance":1615,"goal":"verify list_len_app_1615 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1616","id":91115,"instance":1616,"goal":"verify list_rev_rev_1616 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1617","id":91116,"instance":1617,"goal":"verify bool_and_comm_1617 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1618","id":91117,"instance":1618,"goal":"instance (x : string) . contract_1618 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1619","id":91118,"instance":1619,"goal":"lemma lemma_abs_1619 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1620","id":91119,"instance":1620,"goal":"theorem theorem_leq_1620 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1621","id":91120,"instance":1621,"goal":"verify add_comm_1621 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1622","id":91121,"instance":1622,"goal":"verify mul_comm_1622 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1623","id":91122,"instance":1623,"goal":"verify add_assoc_1623 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1624","id":91123,"instance":1624,"goal":"verify iff_self_1624 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1625","id":91124,"instance":1625,"goal":"verify if_true_1625 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1626","id":91125,"instance":1626,"goal":"verify list_app_nil_1626 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1627","id":91126,"instance":1627,"goal":"verify list_len_app_1627 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1628","id":91127,"instance":1628,"goal":"verify list_rev_rev_1628 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1629","id":91128,"instance":1629,"goal":"verify bool_and_comm_1629 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1630","id":91129,"instance":1630,"goal":"instance (x : string) . contract_1630 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1631","id":91130,"instance":1631,"goal":"lemma lemma_abs_1631 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1632","id":91131,"instance":1632,"goal":"theorem theorem_leq_1632 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1633","id":91132,"instance":1633,"goal":"verify add_comm_1633 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1634","id":91133,"instance":1634,"goal":"verify mul_comm_1634 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1635","id":91134,"instance":1635,"goal":"verify add_assoc_1635 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1636","id":91135,"instance":1636,"goal":"verify iff_self_1636 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1637","id":91136,"instance":1637,"goal":"verify if_true_1637 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1638","id":91137,"instance":1638,"goal":"verify list_app_nil_1638 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1639","id":91138,"instance":1639,"goal":"verify list_len_app_1639 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1640","id":91139,"instance":1640,"goal":"verify list_rev_rev_1640 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1641","id":91140,"instance":1641,"goal":"verify bool_and_comm_1641 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1642","id":91141,"instance":1642,"goal":"instance (x : string) . contract_1642 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1643","id":91142,"instance":1643,"goal":"lemma lemma_abs_1643 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1644","id":91143,"instance":1644,"goal":"theorem theorem_leq_1644 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1645","id":91144,"instance":1645,"goal":"verify add_comm_1645 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1646","id":91145,"instance":1646,"goal":"verify mul_comm_1646 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1647","id":91146,"instance":1647,"goal":"verify add_assoc_1647 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1648","id":91147,"instance":1648,"goal":"verify iff_self_1648 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1649","id":91148,"instance":1649,"goal":"verify if_true_1649 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1650","id":91149,"instance":1650,"goal":"verify list_app_nil_1650 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1651","id":91150,"instance":1651,"goal":"verify list_len_app_1651 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1652","id":91151,"instance":1652,"goal":"verify list_rev_rev_1652 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1653","id":91152,"instance":1653,"goal":"verify bool_and_comm_1653 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1654","id":91153,"instance":1654,"goal":"instance (x : string) . contract_1654 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1655","id":91154,"instance":1655,"goal":"lemma lemma_abs_1655 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1656","id":91155,"instance":1656,"goal":"theorem theorem_leq_1656 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1657","id":91156,"instance":1657,"goal":"verify add_comm_1657 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1658","id":91157,"instance":1658,"goal":"verify mul_comm_1658 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1659","id":91158,"instance":1659,"goal":"verify add_assoc_1659 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1660","id":91159,"instance":1660,"goal":"verify iff_self_1660 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1661","id":91160,"instance":1661,"goal":"verify if_true_1661 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1662","id":91161,"instance":1662,"goal":"verify list_app_nil_1662 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1663","id":91162,"instance":1663,"goal":"verify list_len_app_1663 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1664","id":91163,"instance":1664,"goal":"verify list_rev_rev_1664 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1665","id":91164,"instance":1665,"goal":"verify bool_and_comm_1665 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1666","id":91165,"instance":1666,"goal":"instance (x : string) . contract_1666 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1667","id":91166,"instance":1667,"goal":"lemma lemma_abs_1667 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1668","id":91167,"instance":1668,"goal":"theorem theorem_leq_1668 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1669","id":91168,"instance":1669,"goal":"verify add_comm_1669 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1670","id":91169,"instance":1670,"goal":"verify mul_comm_1670 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1671","id":91170,"instance":1671,"goal":"verify add_assoc_1671 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1672","id":91171,"instance":1672,"goal":"verify iff_self_1672 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1673","id":91172,"instance":1673,"goal":"verify if_true_1673 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1674","id":91173,"instance":1674,"goal":"verify list_app_nil_1674 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1675","id":91174,"instance":1675,"goal":"verify list_len_app_1675 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1676","id":91175,"instance":1676,"goal":"verify list_rev_rev_1676 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1677","id":91176,"instance":1677,"goal":"verify bool_and_comm_1677 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1678","id":91177,"instance":1678,"goal":"instance (x : string) . contract_1678 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1679","id":91178,"instance":1679,"goal":"lemma lemma_abs_1679 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1680","id":91179,"instance":1680,"goal":"theorem theorem_leq_1680 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1681","id":91180,"instance":1681,"goal":"verify add_comm_1681 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1682","id":91181,"instance":1682,"goal":"verify mul_comm_1682 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1683","id":91182,"instance":1683,"goal":"verify add_assoc_1683 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1684","id":91183,"instance":1684,"goal":"verify iff_self_1684 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1685","id":91184,"instance":1685,"goal":"verify if_true_1685 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1686","id":91185,"instance":1686,"goal":"verify list_app_nil_1686 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1687","id":91186,"instance":1687,"goal":"verify list_len_app_1687 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1688","id":91187,"instance":1688,"goal":"verify list_rev_rev_1688 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1689","id":91188,"instance":1689,"goal":"verify bool_and_comm_1689 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1690","id":91189,"instance":1690,"goal":"instance (x : string) . contract_1690 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1691","id":91190,"instance":1691,"goal":"lemma lemma_abs_1691 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1692","id":91191,"instance":1692,"goal":"theorem theorem_leq_1692 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1693","id":91192,"instance":1693,"goal":"verify add_comm_1693 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1694","id":91193,"instance":1694,"goal":"verify mul_comm_1694 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1695","id":91194,"instance":1695,"goal":"verify add_assoc_1695 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1696","id":91195,"instance":1696,"goal":"verify iff_self_1696 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1697","id":91196,"instance":1697,"goal":"verify if_true_1697 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1698","id":91197,"instance":1698,"goal":"verify list_app_nil_1698 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1699","id":91198,"instance":1699,"goal":"verify list_len_app_1699 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1700","id":91199,"instance":1700,"goal":"verify list_rev_rev_1700 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1701","id":91200,"instance":1701,"goal":"verify bool_and_comm_1701 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1702","id":91201,"instance":1702,"goal":"instance (x : string) . contract_1702 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1703","id":91202,"instance":1703,"goal":"lemma lemma_abs_1703 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1704","id":91203,"instance":1704,"goal":"theorem theorem_leq_1704 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1705","id":91204,"instance":1705,"goal":"verify add_comm_1705 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1706","id":91205,"instance":1706,"goal":"verify mul_comm_1706 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1707","id":91206,"instance":1707,"goal":"verify add_assoc_1707 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1708","id":91207,"instance":1708,"goal":"verify iff_self_1708 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1709","id":91208,"instance":1709,"goal":"verify if_true_1709 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1710","id":91209,"instance":1710,"goal":"verify list_app_nil_1710 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1711","id":91210,"instance":1711,"goal":"verify list_len_app_1711 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1712","id":91211,"instance":1712,"goal":"verify list_rev_rev_1712 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1713","id":91212,"instance":1713,"goal":"verify bool_and_comm_1713 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1714","id":91213,"instance":1714,"goal":"instance (x : string) . contract_1714 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1715","id":91214,"instance":1715,"goal":"lemma lemma_abs_1715 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1716","id":91215,"instance":1716,"goal":"theorem theorem_leq_1716 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1717","id":91216,"instance":1717,"goal":"verify add_comm_1717 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1718","id":91217,"instance":1718,"goal":"verify mul_comm_1718 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1719","id":91218,"instance":1719,"goal":"verify add_assoc_1719 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1720","id":91219,"instance":1720,"goal":"verify iff_self_1720 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1721","id":91220,"instance":1721,"goal":"verify if_true_1721 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1722","id":91221,"instance":1722,"goal":"verify list_app_nil_1722 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1723","id":91222,"instance":1723,"goal":"verify list_len_app_1723 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1724","id":91223,"instance":1724,"goal":"verify list_rev_rev_1724 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1725","id":91224,"instance":1725,"goal":"verify bool_and_comm_1725 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1726","id":91225,"instance":1726,"goal":"instance (x : string) . contract_1726 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1727","id":91226,"instance":1727,"goal":"lemma lemma_abs_1727 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1728","id":91227,"instance":1728,"goal":"theorem theorem_leq_1728 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1729","id":91228,"instance":1729,"goal":"verify add_comm_1729 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1730","id":91229,"instance":1730,"goal":"verify mul_comm_1730 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1731","id":91230,"instance":1731,"goal":"verify add_assoc_1731 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1732","id":91231,"instance":1732,"goal":"verify iff_self_1732 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1733","id":91232,"instance":1733,"goal":"verify if_true_1733 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1734","id":91233,"instance":1734,"goal":"verify list_app_nil_1734 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1735","id":91234,"instance":1735,"goal":"verify list_len_app_1735 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1736","id":91235,"instance":1736,"goal":"verify list_rev_rev_1736 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1737","id":91236,"instance":1737,"goal":"verify bool_and_comm_1737 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1738","id":91237,"instance":1738,"goal":"instance (x : string) . contract_1738 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1739","id":91238,"instance":1739,"goal":"lemma lemma_abs_1739 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1740","id":91239,"instance":1740,"goal":"theorem theorem_leq_1740 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1741","id":91240,"instance":1741,"goal":"verify add_comm_1741 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1742","id":91241,"instance":1742,"goal":"verify mul_comm_1742 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1743","id":91242,"instance":1743,"goal":"verify add_assoc_1743 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1744","id":91243,"instance":1744,"goal":"verify iff_self_1744 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1745","id":91244,"instance":1745,"goal":"verify if_true_1745 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1746","id":91245,"instance":1746,"goal":"verify list_app_nil_1746 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1747","id":91246,"instance":1747,"goal":"verify list_len_app_1747 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1748","id":91247,"instance":1748,"goal":"verify list_rev_rev_1748 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1749","id":91248,"instance":1749,"goal":"verify bool_and_comm_1749 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1750","id":91249,"instance":1750,"goal":"instance (x : string) . contract_1750 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1751","id":91250,"instance":1751,"goal":"lemma lemma_abs_1751 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1752","id":91251,"instance":1752,"goal":"theorem theorem_leq_1752 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1753","id":91252,"instance":1753,"goal":"verify add_comm_1753 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1754","id":91253,"instance":1754,"goal":"verify mul_comm_1754 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1755","id":91254,"instance":1755,"goal":"verify add_assoc_1755 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1756","id":91255,"instance":1756,"goal":"verify iff_self_1756 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1757","id":91256,"instance":1757,"goal":"verify if_true_1757 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1758","id":91257,"instance":1758,"goal":"verify list_app_nil_1758 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1759","id":91258,"instance":1759,"goal":"verify list_len_app_1759 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1760","id":91259,"instance":1760,"goal":"verify list_rev_rev_1760 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1761","id":91260,"instance":1761,"goal":"verify bool_and_comm_1761 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1762","id":91261,"instance":1762,"goal":"instance (x : string) . contract_1762 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1763","id":91262,"instance":1763,"goal":"lemma lemma_abs_1763 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1764","id":91263,"instance":1764,"goal":"theorem theorem_leq_1764 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1765","id":91264,"instance":1765,"goal":"verify add_comm_1765 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1766","id":91265,"instance":1766,"goal":"verify mul_comm_1766 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1767","id":91266,"instance":1767,"goal":"verify add_assoc_1767 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1768","id":91267,"instance":1768,"goal":"verify iff_self_1768 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1769","id":91268,"instance":1769,"goal":"verify if_true_1769 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1770","id":91269,"instance":1770,"goal":"verify list_app_nil_1770 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1771","id":91270,"instance":1771,"goal":"verify list_len_app_1771 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1772","id":91271,"instance":1772,"goal":"verify list_rev_rev_1772 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1773","id":91272,"instance":1773,"goal":"verify bool_and_comm_1773 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1774","id":91273,"instance":1774,"goal":"instance (x : string) . contract_1774 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1775","id":91274,"instance":1775,"goal":"lemma lemma_abs_1775 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1776","id":91275,"instance":1776,"goal":"theorem theorem_leq_1776 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1777","id":91276,"instance":1777,"goal":"verify add_comm_1777 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1778","id":91277,"instance":1778,"goal":"verify mul_comm_1778 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1779","id":91278,"instance":1779,"goal":"verify add_assoc_1779 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1780","id":91279,"instance":1780,"goal":"verify iff_self_1780 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1781","id":91280,"instance":1781,"goal":"verify if_true_1781 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1782","id":91281,"instance":1782,"goal":"verify list_app_nil_1782 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1783","id":91282,"instance":1783,"goal":"verify list_len_app_1783 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1784","id":91283,"instance":1784,"goal":"verify list_rev_rev_1784 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1785","id":91284,"instance":1785,"goal":"verify bool_and_comm_1785 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1786","id":91285,"instance":1786,"goal":"instance (x : string) . contract_1786 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1787","id":91286,"instance":1787,"goal":"lemma lemma_abs_1787 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1788","id":91287,"instance":1788,"goal":"theorem theorem_leq_1788 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1789","id":91288,"instance":1789,"goal":"verify add_comm_1789 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1790","id":91289,"instance":1790,"goal":"verify mul_comm_1790 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1791","id":91290,"instance":1791,"goal":"verify add_assoc_1791 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1792","id":91291,"instance":1792,"goal":"verify iff_self_1792 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1793","id":91292,"instance":1793,"goal":"verify if_true_1793 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1794","id":91293,"instance":1794,"goal":"verify list_app_nil_1794 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1795","id":91294,"instance":1795,"goal":"verify list_len_app_1795 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1796","id":91295,"instance":1796,"goal":"verify list_rev_rev_1796 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1797","id":91296,"instance":1797,"goal":"verify bool_and_comm_1797 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1798","id":91297,"instance":1798,"goal":"instance (x : string) . contract_1798 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1799","id":91298,"instance":1799,"goal":"lemma lemma_abs_1799 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1800","id":91299,"instance":1800,"goal":"theorem theorem_leq_1800 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1801","id":91300,"instance":1801,"goal":"verify add_comm_1801 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1802","id":91301,"instance":1802,"goal":"verify mul_comm_1802 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1803","id":91302,"instance":1803,"goal":"verify add_assoc_1803 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1804","id":91303,"instance":1804,"goal":"verify iff_self_1804 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1805","id":91304,"instance":1805,"goal":"verify if_true_1805 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1806","id":91305,"instance":1806,"goal":"verify list_app_nil_1806 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1807","id":91306,"instance":1807,"goal":"verify list_len_app_1807 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1808","id":91307,"instance":1808,"goal":"verify list_rev_rev_1808 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1809","id":91308,"instance":1809,"goal":"verify bool_and_comm_1809 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1810","id":91309,"instance":1810,"goal":"instance (x : string) . contract_1810 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1811","id":91310,"instance":1811,"goal":"lemma lemma_abs_1811 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1812","id":91311,"instance":1812,"goal":"theorem theorem_leq_1812 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1813","id":91312,"instance":1813,"goal":"verify add_comm_1813 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1814","id":91313,"instance":1814,"goal":"verify mul_comm_1814 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1815","id":91314,"instance":1815,"goal":"verify add_assoc_1815 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1816","id":91315,"instance":1816,"goal":"verify iff_self_1816 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1817","id":91316,"instance":1817,"goal":"verify if_true_1817 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1818","id":91317,"instance":1818,"goal":"verify list_app_nil_1818 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1819","id":91318,"instance":1819,"goal":"verify list_len_app_1819 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1820","id":91319,"instance":1820,"goal":"verify list_rev_rev_1820 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1821","id":91320,"instance":1821,"goal":"verify bool_and_comm_1821 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1822","id":91321,"instance":1822,"goal":"instance (x : string) . contract_1822 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1823","id":91322,"instance":1823,"goal":"lemma lemma_abs_1823 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1824","id":91323,"instance":1824,"goal":"theorem theorem_leq_1824 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1825","id":91324,"instance":1825,"goal":"verify add_comm_1825 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1826","id":91325,"instance":1826,"goal":"verify mul_comm_1826 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1827","id":91326,"instance":1827,"goal":"verify add_assoc_1827 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1828","id":91327,"instance":1828,"goal":"verify iff_self_1828 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1829","id":91328,"instance":1829,"goal":"verify if_true_1829 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1830","id":91329,"instance":1830,"goal":"verify list_app_nil_1830 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1831","id":91330,"instance":1831,"goal":"verify list_len_app_1831 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1832","id":91331,"instance":1832,"goal":"verify list_rev_rev_1832 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1833","id":91332,"instance":1833,"goal":"verify bool_and_comm_1833 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1834","id":91333,"instance":1834,"goal":"instance (x : string) . contract_1834 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1835","id":91334,"instance":1835,"goal":"lemma lemma_abs_1835 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1836","id":91335,"instance":1836,"goal":"theorem theorem_leq_1836 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1837","id":91336,"instance":1837,"goal":"verify add_comm_1837 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1838","id":91337,"instance":1838,"goal":"verify mul_comm_1838 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1839","id":91338,"instance":1839,"goal":"verify add_assoc_1839 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1840","id":91339,"instance":1840,"goal":"verify iff_self_1840 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1841","id":91340,"instance":1841,"goal":"verify if_true_1841 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1842","id":91341,"instance":1842,"goal":"verify list_app_nil_1842 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1843","id":91342,"instance":1843,"goal":"verify list_len_app_1843 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1844","id":91343,"instance":1844,"goal":"verify list_rev_rev_1844 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1845","id":91344,"instance":1845,"goal":"verify bool_and_comm_1845 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1846","id":91345,"instance":1846,"goal":"instance (x : string) . contract_1846 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1847","id":91346,"instance":1847,"goal":"lemma lemma_abs_1847 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1848","id":91347,"instance":1848,"goal":"theorem theorem_leq_1848 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1849","id":91348,"instance":1849,"goal":"verify add_comm_1849 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1850","id":91349,"instance":1850,"goal":"verify mul_comm_1850 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1851","id":91350,"instance":1851,"goal":"verify add_assoc_1851 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1852","id":91351,"instance":1852,"goal":"verify iff_self_1852 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1853","id":91352,"instance":1853,"goal":"verify if_true_1853 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1854","id":91353,"instance":1854,"goal":"verify list_app_nil_1854 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1855","id":91354,"instance":1855,"goal":"verify list_len_app_1855 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1856","id":91355,"instance":1856,"goal":"verify list_rev_rev_1856 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1857","id":91356,"instance":1857,"goal":"verify bool_and_comm_1857 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1858","id":91357,"instance":1858,"goal":"instance (x : string) . contract_1858 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1859","id":91358,"instance":1859,"goal":"lemma lemma_abs_1859 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1860","id":91359,"instance":1860,"goal":"theorem theorem_leq_1860 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1861","id":91360,"instance":1861,"goal":"verify add_comm_1861 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1862","id":91361,"instance":1862,"goal":"verify mul_comm_1862 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1863","id":91362,"instance":1863,"goal":"verify add_assoc_1863 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1864","id":91363,"instance":1864,"goal":"verify iff_self_1864 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1865","id":91364,"instance":1865,"goal":"verify if_true_1865 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1866","id":91365,"instance":1866,"goal":"verify list_app_nil_1866 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1867","id":91366,"instance":1867,"goal":"verify list_len_app_1867 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1868","id":91367,"instance":1868,"goal":"verify list_rev_rev_1868 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1869","id":91368,"instance":1869,"goal":"verify bool_and_comm_1869 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1870","id":91369,"instance":1870,"goal":"instance (x : string) . contract_1870 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1871","id":91370,"instance":1871,"goal":"lemma lemma_abs_1871 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1872","id":91371,"instance":1872,"goal":"theorem theorem_leq_1872 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1873","id":91372,"instance":1873,"goal":"verify add_comm_1873 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1874","id":91373,"instance":1874,"goal":"verify mul_comm_1874 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1875","id":91374,"instance":1875,"goal":"verify add_assoc_1875 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1876","id":91375,"instance":1876,"goal":"verify iff_self_1876 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1877","id":91376,"instance":1877,"goal":"verify if_true_1877 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1878","id":91377,"instance":1878,"goal":"verify list_app_nil_1878 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1879","id":91378,"instance":1879,"goal":"verify list_len_app_1879 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1880","id":91379,"instance":1880,"goal":"verify list_rev_rev_1880 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1881","id":91380,"instance":1881,"goal":"verify bool_and_comm_1881 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1882","id":91381,"instance":1882,"goal":"instance (x : string) . contract_1882 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1883","id":91382,"instance":1883,"goal":"lemma lemma_abs_1883 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1884","id":91383,"instance":1884,"goal":"theorem theorem_leq_1884 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1885","id":91384,"instance":1885,"goal":"verify add_comm_1885 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1886","id":91385,"instance":1886,"goal":"verify mul_comm_1886 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1887","id":91386,"instance":1887,"goal":"verify add_assoc_1887 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1888","id":91387,"instance":1888,"goal":"verify iff_self_1888 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1889","id":91388,"instance":1889,"goal":"verify if_true_1889 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1890","id":91389,"instance":1890,"goal":"verify list_app_nil_1890 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1891","id":91390,"instance":1891,"goal":"verify list_len_app_1891 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1892","id":91391,"instance":1892,"goal":"verify list_rev_rev_1892 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1893","id":91392,"instance":1893,"goal":"verify bool_and_comm_1893 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1894","id":91393,"instance":1894,"goal":"instance (x : string) . contract_1894 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1895","id":91394,"instance":1895,"goal":"lemma lemma_abs_1895 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1896","id":91395,"instance":1896,"goal":"theorem theorem_leq_1896 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1897","id":91396,"instance":1897,"goal":"verify add_comm_1897 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1898","id":91397,"instance":1898,"goal":"verify mul_comm_1898 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1899","id":91398,"instance":1899,"goal":"verify add_assoc_1899 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1900","id":91399,"instance":1900,"goal":"verify iff_self_1900 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1901","id":91400,"instance":1901,"goal":"verify if_true_1901 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1902","id":91401,"instance":1902,"goal":"verify list_app_nil_1902 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1903","id":91402,"instance":1903,"goal":"verify list_len_app_1903 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1904","id":91403,"instance":1904,"goal":"verify list_rev_rev_1904 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1905","id":91404,"instance":1905,"goal":"verify bool_and_comm_1905 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1906","id":91405,"instance":1906,"goal":"instance (x : string) . contract_1906 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1907","id":91406,"instance":1907,"goal":"lemma lemma_abs_1907 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1908","id":91407,"instance":1908,"goal":"theorem theorem_leq_1908 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1909","id":91408,"instance":1909,"goal":"verify add_comm_1909 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1910","id":91409,"instance":1910,"goal":"verify mul_comm_1910 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1911","id":91410,"instance":1911,"goal":"verify add_assoc_1911 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1912","id":91411,"instance":1912,"goal":"verify iff_self_1912 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1913","id":91412,"instance":1913,"goal":"verify if_true_1913 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1914","id":91413,"instance":1914,"goal":"verify list_app_nil_1914 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1915","id":91414,"instance":1915,"goal":"verify list_len_app_1915 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1916","id":91415,"instance":1916,"goal":"verify list_rev_rev_1916 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1917","id":91416,"instance":1917,"goal":"verify bool_and_comm_1917 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1918","id":91417,"instance":1918,"goal":"instance (x : string) . contract_1918 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1919","id":91418,"instance":1919,"goal":"lemma lemma_abs_1919 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1920","id":91419,"instance":1920,"goal":"theorem theorem_leq_1920 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1921","id":91420,"instance":1921,"goal":"verify add_comm_1921 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1922","id":91421,"instance":1922,"goal":"verify mul_comm_1922 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1923","id":91422,"instance":1923,"goal":"verify add_assoc_1923 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1924","id":91423,"instance":1924,"goal":"verify iff_self_1924 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1925","id":91424,"instance":1925,"goal":"verify if_true_1925 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1926","id":91425,"instance":1926,"goal":"verify list_app_nil_1926 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1927","id":91426,"instance":1927,"goal":"verify list_len_app_1927 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1928","id":91427,"instance":1928,"goal":"verify list_rev_rev_1928 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1929","id":91428,"instance":1929,"goal":"verify bool_and_comm_1929 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1930","id":91429,"instance":1930,"goal":"instance (x : string) . contract_1930 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1931","id":91430,"instance":1931,"goal":"lemma lemma_abs_1931 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1932","id":91431,"instance":1932,"goal":"theorem theorem_leq_1932 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1933","id":91432,"instance":1933,"goal":"verify add_comm_1933 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1934","id":91433,"instance":1934,"goal":"verify mul_comm_1934 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1935","id":91434,"instance":1935,"goal":"verify add_assoc_1935 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1936","id":91435,"instance":1936,"goal":"verify iff_self_1936 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1937","id":91436,"instance":1937,"goal":"verify if_true_1937 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1938","id":91437,"instance":1938,"goal":"verify list_app_nil_1938 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1939","id":91438,"instance":1939,"goal":"verify list_len_app_1939 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1940","id":91439,"instance":1940,"goal":"verify list_rev_rev_1940 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1941","id":91440,"instance":1941,"goal":"verify bool_and_comm_1941 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1942","id":91441,"instance":1942,"goal":"instance (x : string) . contract_1942 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1943","id":91442,"instance":1943,"goal":"lemma lemma_abs_1943 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1944","id":91443,"instance":1944,"goal":"theorem theorem_leq_1944 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1945","id":91444,"instance":1945,"goal":"verify add_comm_1945 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1946","id":91445,"instance":1946,"goal":"verify mul_comm_1946 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1947","id":91446,"instance":1947,"goal":"verify add_assoc_1947 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1948","id":91447,"instance":1948,"goal":"verify iff_self_1948 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1949","id":91448,"instance":1949,"goal":"verify if_true_1949 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1950","id":91449,"instance":1950,"goal":"verify list_app_nil_1950 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1951","id":91450,"instance":1951,"goal":"verify list_len_app_1951 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1952","id":91451,"instance":1952,"goal":"verify list_rev_rev_1952 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1953","id":91452,"instance":1953,"goal":"verify bool_and_comm_1953 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1954","id":91453,"instance":1954,"goal":"instance (x : string) . contract_1954 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1955","id":91454,"instance":1955,"goal":"lemma lemma_abs_1955 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1956","id":91455,"instance":1956,"goal":"theorem theorem_leq_1956 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1957","id":91456,"instance":1957,"goal":"verify add_comm_1957 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1958","id":91457,"instance":1958,"goal":"verify mul_comm_1958 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1959","id":91458,"instance":1959,"goal":"verify add_assoc_1959 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1960","id":91459,"instance":1960,"goal":"verify iff_self_1960 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1961","id":91460,"instance":1961,"goal":"verify if_true_1961 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1962","id":91461,"instance":1962,"goal":"verify list_app_nil_1962 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1963","id":91462,"instance":1963,"goal":"verify list_len_app_1963 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1964","id":91463,"instance":1964,"goal":"verify list_rev_rev_1964 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1965","id":91464,"instance":1965,"goal":"verify bool_and_comm_1965 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1966","id":91465,"instance":1966,"goal":"instance (x : string) . contract_1966 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1967","id":91466,"instance":1967,"goal":"lemma lemma_abs_1967 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1968","id":91467,"instance":1968,"goal":"theorem theorem_leq_1968 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1969","id":91468,"instance":1969,"goal":"verify add_comm_1969 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1970","id":91469,"instance":1970,"goal":"verify mul_comm_1970 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1971","id":91470,"instance":1971,"goal":"verify add_assoc_1971 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1972","id":91471,"instance":1972,"goal":"verify iff_self_1972 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1973","id":91472,"instance":1973,"goal":"verify if_true_1973 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1974","id":91473,"instance":1974,"goal":"verify list_app_nil_1974 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1975","id":91474,"instance":1975,"goal":"verify list_len_app_1975 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1976","id":91475,"instance":1976,"goal":"verify list_rev_rev_1976 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1977","id":91476,"instance":1977,"goal":"verify bool_and_comm_1977 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1978","id":91477,"instance":1978,"goal":"instance (x : string) . contract_1978 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1979","id":91478,"instance":1979,"goal":"lemma lemma_abs_1979 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1980","id":91479,"instance":1980,"goal":"theorem theorem_leq_1980 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1981","id":91480,"instance":1981,"goal":"verify add_comm_1981 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1982","id":91481,"instance":1982,"goal":"verify mul_comm_1982 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1983","id":91482,"instance":1983,"goal":"verify add_assoc_1983 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1984","id":91483,"instance":1984,"goal":"verify iff_self_1984 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1985","id":91484,"instance":1985,"goal":"verify if_true_1985 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1986","id":91485,"instance":1986,"goal":"verify list_app_nil_1986 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1987","id":91486,"instance":1987,"goal":"verify list_len_app_1987 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_1988","id":91487,"instance":1988,"goal":"verify list_rev_rev_1988 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_1989","id":91488,"instance":1989,"goal":"verify bool_and_comm_1989 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_1990","id":91489,"instance":1990,"goal":"instance (x : string) . contract_1990 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_1991","id":91490,"instance":1991,"goal":"lemma lemma_abs_1991 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_1992","id":91491,"instance":1992,"goal":"theorem theorem_leq_1992 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1993","id":91492,"instance":1993,"goal":"verify add_comm_1993 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1994","id":91493,"instance":1994,"goal":"verify mul_comm_1994 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1995","id":91494,"instance":1995,"goal":"verify add_assoc_1995 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_1996","id":91495,"instance":1996,"goal":"verify iff_self_1996 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_1997","id":91496,"instance":1997,"goal":"verify if_true_1997 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_1998","id":91497,"instance":1998,"goal":"verify list_app_nil_1998 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_1999","id":91498,"instance":1999,"goal":"verify list_len_app_1999 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2000","id":91499,"instance":2000,"goal":"verify list_rev_rev_2000 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2001","id":91500,"instance":2001,"goal":"verify bool_and_comm_2001 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2002","id":91501,"instance":2002,"goal":"instance (x : string) . contract_2002 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2003","id":91502,"instance":2003,"goal":"lemma lemma_abs_2003 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2004","id":91503,"instance":2004,"goal":"theorem theorem_leq_2004 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2005","id":91504,"instance":2005,"goal":"verify add_comm_2005 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2006","id":91505,"instance":2006,"goal":"verify mul_comm_2006 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2007","id":91506,"instance":2007,"goal":"verify add_assoc_2007 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2008","id":91507,"instance":2008,"goal":"verify iff_self_2008 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2009","id":91508,"instance":2009,"goal":"verify if_true_2009 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2010","id":91509,"instance":2010,"goal":"verify list_app_nil_2010 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2011","id":91510,"instance":2011,"goal":"verify list_len_app_2011 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2012","id":91511,"instance":2012,"goal":"verify list_rev_rev_2012 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2013","id":91512,"instance":2013,"goal":"verify bool_and_comm_2013 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2014","id":91513,"instance":2014,"goal":"instance (x : string) . contract_2014 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2015","id":91514,"instance":2015,"goal":"lemma lemma_abs_2015 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2016","id":91515,"instance":2016,"goal":"theorem theorem_leq_2016 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2017","id":91516,"instance":2017,"goal":"verify add_comm_2017 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2018","id":91517,"instance":2018,"goal":"verify mul_comm_2018 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2019","id":91518,"instance":2019,"goal":"verify add_assoc_2019 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2020","id":91519,"instance":2020,"goal":"verify iff_self_2020 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2021","id":91520,"instance":2021,"goal":"verify if_true_2021 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2022","id":91521,"instance":2022,"goal":"verify list_app_nil_2022 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2023","id":91522,"instance":2023,"goal":"verify list_len_app_2023 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2024","id":91523,"instance":2024,"goal":"verify list_rev_rev_2024 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2025","id":91524,"instance":2025,"goal":"verify bool_and_comm_2025 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2026","id":91525,"instance":2026,"goal":"instance (x : string) . contract_2026 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2027","id":91526,"instance":2027,"goal":"lemma lemma_abs_2027 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2028","id":91527,"instance":2028,"goal":"theorem theorem_leq_2028 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2029","id":91528,"instance":2029,"goal":"verify add_comm_2029 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2030","id":91529,"instance":2030,"goal":"verify mul_comm_2030 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2031","id":91530,"instance":2031,"goal":"verify add_assoc_2031 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2032","id":91531,"instance":2032,"goal":"verify iff_self_2032 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2033","id":91532,"instance":2033,"goal":"verify if_true_2033 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2034","id":91533,"instance":2034,"goal":"verify list_app_nil_2034 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2035","id":91534,"instance":2035,"goal":"verify list_len_app_2035 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2036","id":91535,"instance":2036,"goal":"verify list_rev_rev_2036 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2037","id":91536,"instance":2037,"goal":"verify bool_and_comm_2037 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2038","id":91537,"instance":2038,"goal":"instance (x : string) . contract_2038 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2039","id":91538,"instance":2039,"goal":"lemma lemma_abs_2039 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2040","id":91539,"instance":2040,"goal":"theorem theorem_leq_2040 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2041","id":91540,"instance":2041,"goal":"verify add_comm_2041 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2042","id":91541,"instance":2042,"goal":"verify mul_comm_2042 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2043","id":91542,"instance":2043,"goal":"verify add_assoc_2043 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2044","id":91543,"instance":2044,"goal":"verify iff_self_2044 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2045","id":91544,"instance":2045,"goal":"verify if_true_2045 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2046","id":91545,"instance":2046,"goal":"verify list_app_nil_2046 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2047","id":91546,"instance":2047,"goal":"verify list_len_app_2047 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2048","id":91547,"instance":2048,"goal":"verify list_rev_rev_2048 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2049","id":91548,"instance":2049,"goal":"verify bool_and_comm_2049 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2050","id":91549,"instance":2050,"goal":"instance (x : string) . contract_2050 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2051","id":91550,"instance":2051,"goal":"lemma lemma_abs_2051 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2052","id":91551,"instance":2052,"goal":"theorem theorem_leq_2052 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2053","id":91552,"instance":2053,"goal":"verify add_comm_2053 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2054","id":91553,"instance":2054,"goal":"verify mul_comm_2054 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2055","id":91554,"instance":2055,"goal":"verify add_assoc_2055 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2056","id":91555,"instance":2056,"goal":"verify iff_self_2056 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2057","id":91556,"instance":2057,"goal":"verify if_true_2057 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2058","id":91557,"instance":2058,"goal":"verify list_app_nil_2058 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2059","id":91558,"instance":2059,"goal":"verify list_len_app_2059 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2060","id":91559,"instance":2060,"goal":"verify list_rev_rev_2060 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2061","id":91560,"instance":2061,"goal":"verify bool_and_comm_2061 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2062","id":91561,"instance":2062,"goal":"instance (x : string) . contract_2062 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2063","id":91562,"instance":2063,"goal":"lemma lemma_abs_2063 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2064","id":91563,"instance":2064,"goal":"theorem theorem_leq_2064 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2065","id":91564,"instance":2065,"goal":"verify add_comm_2065 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2066","id":91565,"instance":2066,"goal":"verify mul_comm_2066 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2067","id":91566,"instance":2067,"goal":"verify add_assoc_2067 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2068","id":91567,"instance":2068,"goal":"verify iff_self_2068 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2069","id":91568,"instance":2069,"goal":"verify if_true_2069 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2070","id":91569,"instance":2070,"goal":"verify list_app_nil_2070 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2071","id":91570,"instance":2071,"goal":"verify list_len_app_2071 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2072","id":91571,"instance":2072,"goal":"verify list_rev_rev_2072 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2073","id":91572,"instance":2073,"goal":"verify bool_and_comm_2073 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2074","id":91573,"instance":2074,"goal":"instance (x : string) . contract_2074 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2075","id":91574,"instance":2075,"goal":"lemma lemma_abs_2075 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2076","id":91575,"instance":2076,"goal":"theorem theorem_leq_2076 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2077","id":91576,"instance":2077,"goal":"verify add_comm_2077 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2078","id":91577,"instance":2078,"goal":"verify mul_comm_2078 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2079","id":91578,"instance":2079,"goal":"verify add_assoc_2079 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2080","id":91579,"instance":2080,"goal":"verify iff_self_2080 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2081","id":91580,"instance":2081,"goal":"verify if_true_2081 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2082","id":91581,"instance":2082,"goal":"verify list_app_nil_2082 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2083","id":91582,"instance":2083,"goal":"verify list_len_app_2083 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2084","id":91583,"instance":2084,"goal":"verify list_rev_rev_2084 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2085","id":91584,"instance":2085,"goal":"verify bool_and_comm_2085 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2086","id":91585,"instance":2086,"goal":"instance (x : string) . contract_2086 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2087","id":91586,"instance":2087,"goal":"lemma lemma_abs_2087 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2088","id":91587,"instance":2088,"goal":"theorem theorem_leq_2088 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2089","id":91588,"instance":2089,"goal":"verify add_comm_2089 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2090","id":91589,"instance":2090,"goal":"verify mul_comm_2090 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2091","id":91590,"instance":2091,"goal":"verify add_assoc_2091 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2092","id":91591,"instance":2092,"goal":"verify iff_self_2092 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2093","id":91592,"instance":2093,"goal":"verify if_true_2093 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2094","id":91593,"instance":2094,"goal":"verify list_app_nil_2094 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2095","id":91594,"instance":2095,"goal":"verify list_len_app_2095 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2096","id":91595,"instance":2096,"goal":"verify list_rev_rev_2096 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2097","id":91596,"instance":2097,"goal":"verify bool_and_comm_2097 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2098","id":91597,"instance":2098,"goal":"instance (x : string) . contract_2098 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2099","id":91598,"instance":2099,"goal":"lemma lemma_abs_2099 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2100","id":91599,"instance":2100,"goal":"theorem theorem_leq_2100 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2101","id":91600,"instance":2101,"goal":"verify add_comm_2101 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2102","id":91601,"instance":2102,"goal":"verify mul_comm_2102 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2103","id":91602,"instance":2103,"goal":"verify add_assoc_2103 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2104","id":91603,"instance":2104,"goal":"verify iff_self_2104 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2105","id":91604,"instance":2105,"goal":"verify if_true_2105 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2106","id":91605,"instance":2106,"goal":"verify list_app_nil_2106 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2107","id":91606,"instance":2107,"goal":"verify list_len_app_2107 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2108","id":91607,"instance":2108,"goal":"verify list_rev_rev_2108 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2109","id":91608,"instance":2109,"goal":"verify bool_and_comm_2109 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2110","id":91609,"instance":2110,"goal":"instance (x : string) . contract_2110 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2111","id":91610,"instance":2111,"goal":"lemma lemma_abs_2111 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2112","id":91611,"instance":2112,"goal":"theorem theorem_leq_2112 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2113","id":91612,"instance":2113,"goal":"verify add_comm_2113 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2114","id":91613,"instance":2114,"goal":"verify mul_comm_2114 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2115","id":91614,"instance":2115,"goal":"verify add_assoc_2115 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2116","id":91615,"instance":2116,"goal":"verify iff_self_2116 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2117","id":91616,"instance":2117,"goal":"verify if_true_2117 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2118","id":91617,"instance":2118,"goal":"verify list_app_nil_2118 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2119","id":91618,"instance":2119,"goal":"verify list_len_app_2119 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2120","id":91619,"instance":2120,"goal":"verify list_rev_rev_2120 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2121","id":91620,"instance":2121,"goal":"verify bool_and_comm_2121 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2122","id":91621,"instance":2122,"goal":"instance (x : string) . contract_2122 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2123","id":91622,"instance":2123,"goal":"lemma lemma_abs_2123 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2124","id":91623,"instance":2124,"goal":"theorem theorem_leq_2124 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2125","id":91624,"instance":2125,"goal":"verify add_comm_2125 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2126","id":91625,"instance":2126,"goal":"verify mul_comm_2126 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2127","id":91626,"instance":2127,"goal":"verify add_assoc_2127 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2128","id":91627,"instance":2128,"goal":"verify iff_self_2128 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2129","id":91628,"instance":2129,"goal":"verify if_true_2129 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2130","id":91629,"instance":2130,"goal":"verify list_app_nil_2130 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2131","id":91630,"instance":2131,"goal":"verify list_len_app_2131 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2132","id":91631,"instance":2132,"goal":"verify list_rev_rev_2132 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2133","id":91632,"instance":2133,"goal":"verify bool_and_comm_2133 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2134","id":91633,"instance":2134,"goal":"instance (x : string) . contract_2134 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2135","id":91634,"instance":2135,"goal":"lemma lemma_abs_2135 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2136","id":91635,"instance":2136,"goal":"theorem theorem_leq_2136 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2137","id":91636,"instance":2137,"goal":"verify add_comm_2137 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2138","id":91637,"instance":2138,"goal":"verify mul_comm_2138 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2139","id":91638,"instance":2139,"goal":"verify add_assoc_2139 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2140","id":91639,"instance":2140,"goal":"verify iff_self_2140 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2141","id":91640,"instance":2141,"goal":"verify if_true_2141 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2142","id":91641,"instance":2142,"goal":"verify list_app_nil_2142 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2143","id":91642,"instance":2143,"goal":"verify list_len_app_2143 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2144","id":91643,"instance":2144,"goal":"verify list_rev_rev_2144 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2145","id":91644,"instance":2145,"goal":"verify bool_and_comm_2145 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2146","id":91645,"instance":2146,"goal":"instance (x : string) . contract_2146 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2147","id":91646,"instance":2147,"goal":"lemma lemma_abs_2147 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2148","id":91647,"instance":2148,"goal":"theorem theorem_leq_2148 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2149","id":91648,"instance":2149,"goal":"verify add_comm_2149 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2150","id":91649,"instance":2150,"goal":"verify mul_comm_2150 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2151","id":91650,"instance":2151,"goal":"verify add_assoc_2151 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2152","id":91651,"instance":2152,"goal":"verify iff_self_2152 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2153","id":91652,"instance":2153,"goal":"verify if_true_2153 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2154","id":91653,"instance":2154,"goal":"verify list_app_nil_2154 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2155","id":91654,"instance":2155,"goal":"verify list_len_app_2155 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2156","id":91655,"instance":2156,"goal":"verify list_rev_rev_2156 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2157","id":91656,"instance":2157,"goal":"verify bool_and_comm_2157 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2158","id":91657,"instance":2158,"goal":"instance (x : string) . contract_2158 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2159","id":91658,"instance":2159,"goal":"lemma lemma_abs_2159 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2160","id":91659,"instance":2160,"goal":"theorem theorem_leq_2160 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2161","id":91660,"instance":2161,"goal":"verify add_comm_2161 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2162","id":91661,"instance":2162,"goal":"verify mul_comm_2162 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2163","id":91662,"instance":2163,"goal":"verify add_assoc_2163 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2164","id":91663,"instance":2164,"goal":"verify iff_self_2164 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2165","id":91664,"instance":2165,"goal":"verify if_true_2165 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2166","id":91665,"instance":2166,"goal":"verify list_app_nil_2166 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2167","id":91666,"instance":2167,"goal":"verify list_len_app_2167 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2168","id":91667,"instance":2168,"goal":"verify list_rev_rev_2168 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2169","id":91668,"instance":2169,"goal":"verify bool_and_comm_2169 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2170","id":91669,"instance":2170,"goal":"instance (x : string) . contract_2170 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2171","id":91670,"instance":2171,"goal":"lemma lemma_abs_2171 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2172","id":91671,"instance":2172,"goal":"theorem theorem_leq_2172 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2173","id":91672,"instance":2173,"goal":"verify add_comm_2173 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2174","id":91673,"instance":2174,"goal":"verify mul_comm_2174 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2175","id":91674,"instance":2175,"goal":"verify add_assoc_2175 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2176","id":91675,"instance":2176,"goal":"verify iff_self_2176 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2177","id":91676,"instance":2177,"goal":"verify if_true_2177 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2178","id":91677,"instance":2178,"goal":"verify list_app_nil_2178 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2179","id":91678,"instance":2179,"goal":"verify list_len_app_2179 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2180","id":91679,"instance":2180,"goal":"verify list_rev_rev_2180 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2181","id":91680,"instance":2181,"goal":"verify bool_and_comm_2181 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2182","id":91681,"instance":2182,"goal":"instance (x : string) . contract_2182 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2183","id":91682,"instance":2183,"goal":"lemma lemma_abs_2183 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2184","id":91683,"instance":2184,"goal":"theorem theorem_leq_2184 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2185","id":91684,"instance":2185,"goal":"verify add_comm_2185 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2186","id":91685,"instance":2186,"goal":"verify mul_comm_2186 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2187","id":91686,"instance":2187,"goal":"verify add_assoc_2187 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2188","id":91687,"instance":2188,"goal":"verify iff_self_2188 (p : bool) = p = p","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"if_true","theorem":"if_true_2189","id":91688,"instance":2189,"goal":"verify if_true_2189 (x : int list) (y : int list) = (if true then x else y) = x","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_app_nil","theorem":"list_app_nil_2190","id":91689,"instance":2190,"goal":"verify list_app_nil_2190 (l : int list) = l @ [] = l","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_len_app","theorem":"list_len_app_2191","id":91690,"instance":2191,"goal":"verify list_len_app_2191 (a : int list) (b : int list) = List.length (a @ b) = List.length a + List.length b","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"list_rev_rev","theorem":"list_rev_rev_2192","id":91691,"instance":2192,"goal":"verify list_rev_rev_2192 (l : int list) = List.rev (List.rev l) = l","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"bool_and_comm","theorem":"bool_and_comm_2193","id":91692,"instance":2193,"goal":"verify bool_and_comm_2193 (p : bool) (q : bool) = (p && q) = (q && p)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"contract","theorem":"contract_2194","id":91693,"instance":2194,"goal":"instance (x : string) . contract_2194 x","type_family":"string","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"lemma_abs","theorem":"lemma_abs_2195","id":91694,"instance":2195,"goal":"lemma lemma_abs_2195 x = abs x >= 0","type_family":"int list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"theorem_leq","theorem":"theorem_leq_2196","id":91695,"instance":2196,"goal":"theorem theorem_leq_2196 (x : real list) = x <= x + 1","type_family":"real list","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2197","id":91696,"instance":2197,"goal":"verify add_comm_2197 (x : int) (y : int) = x + y = y + x","type_family":"int","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2198","id":91697,"instance":2198,"goal":"verify mul_comm_2198 (x : real) (y : real) = x * y = y * x","type_family":"real","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2199","id":91698,"instance":2199,"goal":"verify add_assoc_2199 (x : bool) (y : bool) (z : bool) = (x + y) + z = x + (y + z)","type_family":"bool","context":["verify"]} -{"prover":"Imandra","tactic_proof":"verify","source":"imandra_synthetic/scaled","variant":"iff_self","theorem":"iff_self_2200","id":91699,"instance":2200,"goal":"verify iff_self_2200 (p : bool) = p = p","type_family":"string","context":["verify"]} +{"id":89500,"prover":"Imandra","theorem":"add_comm","goal":"theorem add_comm x y = x + y = y + x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89501,"prover":"Imandra","theorem":"add_assoc","goal":"theorem add_assoc x y z = (x + y) + z = x + (y + z)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89502,"prover":"Imandra","theorem":"mul_comm","goal":"theorem mul_comm x y = x * y = y * x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89503,"prover":"Imandra","theorem":"mul_assoc","goal":"theorem mul_assoc x y z = (x * y) * z = x * (y * z)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89504,"prover":"Imandra","theorem":"distrib","goal":"theorem distrib x y z = x * (y + z) = x * y + x * z","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89505,"prover":"Imandra","theorem":"add_zero","goal":"theorem add_zero x = x + 0 = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89506,"prover":"Imandra","theorem":"mul_one","goal":"theorem mul_one x = x * 1 = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89507,"prover":"Imandra","theorem":"mul_zero","goal":"theorem mul_zero x = x * 0 = 0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89508,"prover":"Imandra","theorem":"abs_nonneg","goal":"theorem abs_nonneg x = abs x >= 0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89509,"prover":"Imandra","theorem":"abs_triangle","goal":"theorem abs_triangle x y = abs (x + y) <= abs x + abs y","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89510,"prover":"Imandra","theorem":"max_comm","goal":"theorem max_comm x y = max x y = max y x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89511,"prover":"Imandra","theorem":"min_comm","goal":"theorem min_comm x y = min x y = min y x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89512,"prover":"Imandra","theorem":"max_assoc","goal":"theorem max_assoc x y z = max x (max y z) = max (max x y) z","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89513,"prover":"Imandra","theorem":"min_max_absorb","goal":"theorem min_max_absorb x y = min x (max x y) = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89514,"prover":"Imandra","theorem":"div_mod_id","goal":"theorem div_mod_id a b = b > 0 ==> a = b * (a / b) + (a mod b)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/arithmetic"} +{"id":89515,"prover":"Imandra","theorem":"append_nil","goal":"theorem append_nil l = l @ [] = l","context":["auto"],"tactic_proof":"[@@induct l]","source":"imandra_synthetic/lists"} +{"id":89516,"prover":"Imandra","theorem":"append_assoc","goal":"theorem append_assoc l1 l2 l3 = (l1 @ l2) @ l3 = l1 @ (l2 @ l3)","context":["auto"],"tactic_proof":"[@@induct l1]","source":"imandra_synthetic/lists"} +{"id":89517,"prover":"Imandra","theorem":"length_append","goal":"theorem length_append l1 l2 = List.length (l1 @ l2) = List.length l1 + List.length l2","context":["auto"],"tactic_proof":"[@@induct l1]","source":"imandra_synthetic/lists"} +{"id":89518,"prover":"Imandra","theorem":"rev_rev","goal":"theorem rev_rev l = List.rev (List.rev l) = l","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/lists"} +{"id":89519,"prover":"Imandra","theorem":"map_id","goal":"theorem map_id l = List.map (fun x -> x) l = l","context":["auto"],"tactic_proof":"[@@induct l]","source":"imandra_synthetic/lists"} +{"id":89520,"prover":"Imandra","theorem":"map_compose","goal":"theorem map_compose f g l = List.map f (List.map g l) = List.map (fun x -> f (g x)) l","context":["auto"],"tactic_proof":"[@@induct l]","source":"imandra_synthetic/lists"} +{"id":89521,"prover":"Imandra","theorem":"length_map","goal":"theorem length_map f l = List.length (List.map f l) = List.length l","context":["auto"],"tactic_proof":"[@@induct l]","source":"imandra_synthetic/lists"} +{"id":89522,"prover":"Imandra","theorem":"length_rev","goal":"theorem length_rev l = List.length (List.rev l) = List.length l","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/lists"} +{"id":89523,"prover":"Imandra","theorem":"mem_append","goal":"theorem mem_append x l1 l2 = List.mem x (l1 @ l2) = (List.mem x l1 || List.mem x l2)","context":["auto"],"tactic_proof":"[@@induct l1]","source":"imandra_synthetic/lists"} +{"id":89524,"prover":"Imandra","theorem":"filter_append","goal":"theorem filter_append p l1 l2 = List.filter p (l1 @ l2) = List.filter p l1 @ List.filter p l2","context":["auto"],"tactic_proof":"[@@induct l1]","source":"imandra_synthetic/lists"} +{"id":89525,"prover":"Imandra","theorem":"flat_map_assoc","goal":"theorem flat_map_assoc f g l = List.concat_map g (List.concat_map f l) = List.concat_map (fun x -> List.concat_map g (f x)) l","context":["auto"],"tactic_proof":"[@@induct l]","source":"imandra_synthetic/lists"} +{"id":89526,"prover":"Imandra","theorem":"nth_map","goal":"theorem nth_map f l i = i < List.length l ==> List.nth (List.map f l) i = f (List.nth l i)","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/lists"} +{"id":89527,"prover":"Imandra","theorem":"for_all_append","goal":"theorem for_all_append p l1 l2 = List.for_all p (l1 @ l2) = (List.for_all p l1 && List.for_all p l2)","context":["auto"],"tactic_proof":"[@@induct l1]","source":"imandra_synthetic/lists"} +{"id":89528,"prover":"Imandra","theorem":"sorted_insert","goal":"theorem sorted_insert x l = sorted l ==> sorted (insert x l)","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89529,"prover":"Imandra","theorem":"sorted_sort","goal":"theorem sorted_sort l = sorted (sort l)","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89530,"prover":"Imandra","theorem":"perm_sort","goal":"theorem perm_sort l = is_perm l (sort l)","context":["auto"],"tactic_proof":"[@@induct l] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89531,"prover":"Imandra","theorem":"bsearch_correct","goal":"theorem bsearch_correct a v = sorted a ==> (match bsearch a v with Some i -> a.(i) = v | None -> not (Array.exists (fun x -> x = v) a))","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/verification"} +{"id":89532,"prover":"Imandra","theorem":"stack_push_pop","goal":"theorem stack_push_pop x s = pop (push x s) = (x, s)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/verification"} +{"id":89533,"prover":"Imandra","theorem":"queue_fifo","goal":"theorem queue_fifo x q = not (is_empty q) ==> fst (dequeue (enqueue x q)) = front q","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/verification"} +{"id":89534,"prover":"Imandra","theorem":"rbt_invariant","goal":"theorem rbt_invariant x t = is_rbt t ==> is_rbt (rbt_insert x t)","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89535,"prover":"Imandra","theorem":"bst_invariant","goal":"theorem bst_invariant x t = is_bst t ==> is_bst (bst_insert x t)","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89536,"prover":"Imandra","theorem":"heap_insert_min","goal":"theorem heap_insert_min x h = is_heap h ==> find_min (heap_insert x h) <= x","context":["auto"],"tactic_proof":"[@@induct h] [@@auto]","source":"imandra_synthetic/verification"} +{"id":89537,"prover":"Imandra","theorem":"margin_nonneg","goal":"theorem margin_nonneg acct = valid_account acct ==> margin_requirement acct >= 0.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/finance"} +{"id":89538,"prover":"Imandra","theorem":"order_fill_balance","goal":"theorem order_fill_balance order book = valid_order order ==> balance_after (fill order book) >= 0.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/finance"} +{"id":89539,"prover":"Imandra","theorem":"matching_symmetric","goal":"theorem matching_symmetric buy sell = can_match buy sell ==> can_match sell buy","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/finance"} +{"id":89540,"prover":"Imandra","theorem":"settlement_complete","goal":"theorem settlement_complete trades = all_valid trades ==> is_settled (settle trades)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/finance"} +{"id":89541,"prover":"Imandra","theorem":"risk_monotone","goal":"theorem risk_monotone p1 p2 = portfolio_subset p1 p2 ==> risk p1 <= risk p2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/finance"} +{"id":89542,"prover":"Imandra","theorem":"cx_list_rev_sorted","goal":"verify (fun l -> sorted l ==> sorted (List.rev l))","context":["auto"],"tactic_proof":"CX: l = [1; 3; 2] -- List.rev [1;3;2] = [2;3;1] which is not sorted","source":"imandra_synthetic/counterexamples"} +{"id":89543,"prover":"Imandra","theorem":"cx_append_comm","goal":"verify (fun l1 l2 -> l1 @ l2 = l2 @ l1)","context":["auto"],"tactic_proof":"CX: l1 = [1], l2 = [2] -- [1;2] <> [2;1]","source":"imandra_synthetic/counterexamples"} +{"id":89544,"prover":"Imandra","theorem":"cx_map_injective","goal":"verify (fun f l1 l2 -> List.map f l1 = List.map f l2 ==> l1 = l2)","context":["auto"],"tactic_proof":"CX: f = (fun _ -> 0), l1 = [1], l2 = [2]","source":"imandra_synthetic/counterexamples"} +{"id":89545,"prover":"Imandra","theorem":"cx_sort_idempotent","goal":"verify (fun l -> sort (sort l) = sort l)","context":["auto"],"tactic_proof":"Verified: true (sort is idempotent)","source":"imandra_synthetic/counterexamples"} +{"id":89546,"prover":"Imandra","theorem":"cx_filter_length","goal":"verify (fun p l -> List.length (List.filter p l) <= List.length l)","context":["auto"],"tactic_proof":"Verified: true","source":"imandra_synthetic/counterexamples"} +{"id":89547,"prover":"Imandra","theorem":"cx_subset_reflexive","goal":"verify (fun s -> subset s s)","context":["auto"],"tactic_proof":"Verified: true","source":"imandra_synthetic/counterexamples"} +{"id":89548,"prover":"Imandra","theorem":"cx_division_positive","goal":"verify (fun x y -> x > 0 && y > 0 ==> x / y >= 0)","context":["auto"],"tactic_proof":"Verified: true","source":"imandra_synthetic/counterexamples"} +{"id":89549,"prover":"Imandra","theorem":"cx_int_overflow","goal":"verify (fun (x:int) -> x + 1 > x)","context":["auto"],"tactic_proof":"CX: x = max_int -- overflow wraps to min_int","source":"imandra_synthetic/counterexamples"} +{"id":89550,"prover":"Imandra","theorem":"cx_float_assoc","goal":"verify (fun (a:float) b c -> (a +. b) +. c = a +. (b +. c))","context":["auto"],"tactic_proof":"CX: a=1e30, b= -1e30, c=1.0 -- floating point not associative","source":"imandra_synthetic/counterexamples"} +{"id":89551,"prover":"Imandra","theorem":"bst_insert_preserves","goal":"theorem bst_insert_preserves x t = is_bst t ==> is_bst (bst_insert x t)","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/trees_imandra"} +{"id":89552,"prover":"Imandra","theorem":"bst_member_after_insert","goal":"theorem bst_member_after_insert x t = bst_member x (bst_insert x t)","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/trees_imandra"} +{"id":89553,"prover":"Imandra","theorem":"tree_mirror_involutive","goal":"theorem tree_mirror_involutive t = mirror (mirror t) = t","context":["auto"],"tactic_proof":"[@@induct t]","source":"imandra_synthetic/trees_imandra"} +{"id":89554,"prover":"Imandra","theorem":"tree_size_positive","goal":"theorem tree_size_positive t = tree_size t >= 0","context":["auto"],"tactic_proof":"[@@induct t]","source":"imandra_synthetic/trees_imandra"} +{"id":89555,"prover":"Imandra","theorem":"tree_flatten_length","goal":"theorem tree_flatten_length t = List.length (flatten t) = tree_size t","context":["auto"],"tactic_proof":"[@@induct t]","source":"imandra_synthetic/trees_imandra"} +{"id":89556,"prover":"Imandra","theorem":"heap_extract_min_sorted","goal":"theorem heap_extract_min_sorted h = is_heap h ==> sorted (heapsort h)","context":["auto"],"tactic_proof":"[@@induct h] [@@auto]","source":"imandra_synthetic/trees_imandra"} +{"id":89557,"prover":"Imandra","theorem":"avl_balanced","goal":"theorem avl_balanced t = is_avl t ==> abs (height (left t) - height (right t)) <= 1","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/trees_imandra"} +{"id":89558,"prover":"Imandra","theorem":"rbt_black_height","goal":"theorem rbt_black_height t = is_rbt t ==> black_height (left t) = black_height (right t)","context":["auto"],"tactic_proof":"[@@induct t] [@@auto]","source":"imandra_synthetic/trees_imandra"} +{"id":89559,"prover":"Imandra","theorem":"abs_diff_symmetric","goal":"theorem abs_diff_symmetric x y = abs (x - y) = abs (y - x)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89560,"prover":"Imandra","theorem":"sqr_nonneg","goal":"theorem sqr_nonneg x = x * x >= 0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89561,"prover":"Imandra","theorem":"ge_trans","goal":"theorem ge_trans x y z = x >= y ==> y >= z ==> x >= z","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89562,"prover":"Imandra","theorem":"eq_refl_trans","goal":"theorem eq_refl_trans x y = x = y ==> y = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89563,"prover":"Imandra","theorem":"neg_neg","goal":"theorem neg_neg x = -(-x) = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89564,"prover":"Imandra","theorem":"sub_self_zero","goal":"theorem sub_self_zero x = x - x = 0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89565,"prover":"Imandra","theorem":"mul_distrib_sub","goal":"theorem mul_distrib_sub x y z = x * (y - z) = x*y - x*z","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89566,"prover":"Imandra","theorem":"pow_zero","goal":"theorem pow_zero x = x > 0 ==> x ** 0 = 1","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89567,"prover":"Imandra","theorem":"pow_succ","goal":"theorem pow_succ x n = n >= 0 ==> x ** (n+1) = x * (x ** n)","context":["auto"],"tactic_proof":"[@@induct n] [@@auto]","source":"imandra_synthetic/numerics"} +{"id":89568,"prover":"Imandra","theorem":"gcd_self","goal":"theorem gcd_self n = n >= 0 ==> gcd n n = n","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89569,"prover":"Imandra","theorem":"gcd_zero","goal":"theorem gcd_zero n = n >= 0 ==> gcd n 0 = n","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89570,"prover":"Imandra","theorem":"divides_refl","goal":"theorem divides_refl n = n <> 0 ==> divides n n","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/numerics"} +{"id":89571,"prover":"Imandra","theorem":"record_set_get","goal":"theorem record_set_get r v = (set_field r v).field = v","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89572,"prover":"Imandra","theorem":"record_get_set_idempotent","goal":"theorem record_get_set_idempotent r = set_field r (r.field) = r","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89573,"prover":"Imandra","theorem":"record_set_comm","goal":"theorem record_set_comm r v1 v2 = set_f2 (set_f1 r v1) v2 = set_f1 (set_f2 r v2) v1","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89574,"prover":"Imandra","theorem":"record_eq_fields","goal":"theorem record_eq_fields r1 r2 = r1.a = r2.a ==> r1.b = r2.b ==> r1 = r2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89575,"prover":"Imandra","theorem":"option_some_inj","goal":"theorem option_some_inj x y = Some x = Some y ==> x = y","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89576,"prover":"Imandra","theorem":"option_none_not_some","goal":"theorem option_none_not_some x = None <> Some x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89577,"prover":"Imandra","theorem":"tuple_fst_snd","goal":"theorem tuple_fst_snd p = (fst p, snd p) = p","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/records"} +{"id":89578,"prover":"Imandra","theorem":"fsm_deterministic","goal":"theorem fsm_deterministic s e = forall s1 s2. transition s e s1 ==> transition s e s2 ==> s1 = s2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89579,"prover":"Imandra","theorem":"fsm_reachable_trans","goal":"theorem fsm_reachable_trans s1 s2 s3 = reachable s1 s2 ==> reachable s2 s3 ==> reachable s1 s3","context":["induct","auto"],"tactic_proof":"[@@induct] [@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89580,"prover":"Imandra","theorem":"fsm_initial_reachable","goal":"theorem fsm_initial_reachable s = s = initial_state ==> reachable s s","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89581,"prover":"Imandra","theorem":"fsm_no_deadlock","goal":"theorem fsm_no_deadlock s = reachable initial_state s ==> exists e s'. transition s e s'","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89582,"prover":"Imandra","theorem":"fsm_invariant_preserve","goal":"theorem fsm_invariant_preserve s e s' = inv s ==> transition s e s' ==> inv s'","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89583,"prover":"Imandra","theorem":"safety_property","goal":"theorem safety_property s = reachable initial_state s ==> not (is_error s)","context":["induct","auto"],"tactic_proof":"[@@induct] [@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89584,"prover":"Imandra","theorem":"liveness_property","goal":"theorem liveness_property s = reachable initial_state s ==> eventually goal_state s","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/state_machines"} +{"id":89585,"prover":"Imandra","theorem":"hash_deterministic","goal":"theorem hash_deterministic m1 m2 = m1 = m2 ==> hash m1 = hash m2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89586,"prover":"Imandra","theorem":"signature_verify","goal":"theorem signature_verify sk pk m = keypair sk pk ==> verify pk m (sign sk m) = true","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89587,"prover":"Imandra","theorem":"encrypt_decrypt","goal":"theorem encrypt_decrypt k m = decrypt k (encrypt k m) = m","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89588,"prover":"Imandra","theorem":"merkle_root_unique","goal":"theorem merkle_root_unique l1 l2 = merkle_root l1 = merkle_root l2 ==> l1 = l2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89589,"prover":"Imandra","theorem":"xor_involutive","goal":"theorem xor_involutive x y = xor (xor x y) y = x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89590,"prover":"Imandra","theorem":"nonce_unique","goal":"theorem nonce_unique n1 n2 = generate_nonce n1 = generate_nonce n2 ==> n1 = n2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/crypto"} +{"id":89591,"prover":"Imandra","theorem":"block_prev_chain","goal":"theorem block_prev_chain b = valid_chain b ==> prev_hash (head b) = hash (tail b)","context":["auto"],"tactic_proof":"[@@induct b] [@@auto]","source":"imandra_synthetic/blockchain"} +{"id":89592,"prover":"Imandra","theorem":"tx_balance_conservation","goal":"theorem tx_balance_conservation tx = valid_tx tx ==> sum_inputs tx = sum_outputs tx + fee tx","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/blockchain"} +{"id":89593,"prover":"Imandra","theorem":"utxo_no_double_spend","goal":"theorem utxo_no_double_spend tx1 tx2 utxo = valid_tx tx1 ==> valid_tx tx2 ==> disjoint (inputs tx1) (inputs tx2)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/blockchain"} +{"id":89594,"prover":"Imandra","theorem":"consensus_majority","goal":"theorem consensus_majority votes = count_yes votes > length votes / 2 ==> accepted votes","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/blockchain"} +{"id":89595,"prover":"Imandra","theorem":"longest_chain","goal":"theorem longest_chain c1 c2 = valid_chain c1 ==> valid_chain c2 ==> height c1 >= height c2 ==> canonical c1","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/blockchain"} +{"id":89596,"prover":"Imandra","theorem":"group_identity_unique","goal":"theorem group_identity_unique e1 e2 = is_identity e1 ==> is_identity e2 ==> e1 = e2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89597,"prover":"Imandra","theorem":"group_inverse_unique","goal":"theorem group_inverse_unique a b c = op a b = identity ==> op a c = identity ==> b = c","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89598,"prover":"Imandra","theorem":"monoid_left_id","goal":"theorem monoid_left_id a = op identity a = a","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89599,"prover":"Imandra","theorem":"monoid_right_id","goal":"theorem monoid_right_id a = op a identity = a","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89600,"prover":"Imandra","theorem":"ring_distrib","goal":"theorem ring_distrib a b c = mul a (add b c) = add (mul a b) (mul a c)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89601,"prover":"Imandra","theorem":"ring_zero_mul","goal":"theorem ring_zero_mul a = mul zero a = zero","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89602,"prover":"Imandra","theorem":"semilat_idempotent","goal":"theorem semilat_idempotent a = meet a a = a","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/algebra"} +{"id":89603,"prover":"Imandra","theorem":"mutex_exclusive","goal":"theorem mutex_exclusive s = acquired s ==> not (available s)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89604,"prover":"Imandra","theorem":"lock_release","goal":"theorem lock_release s = acquire (release (acquire s)) = acquire s","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89605,"prover":"Imandra","theorem":"queue_ordering","goal":"theorem queue_ordering q msgs = push q msgs ==> fifo_order (drain q)","context":["auto"],"tactic_proof":"[@@induct msgs] [@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89606,"prover":"Imandra","theorem":"actor_isolation","goal":"theorem actor_isolation a b m = send a b m ==> state a = state a","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89607,"prover":"Imandra","theorem":"atomic_read_write","goal":"theorem atomic_read_write x v = read (write x v) = v","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89608,"prover":"Imandra","theorem":"cas_success","goal":"theorem cas_success x old new = read x = old ==> cas x old new = (true, write x new)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89609,"prover":"Imandra","theorem":"deadlock_free","goal":"theorem deadlock_free tree = acyclic tree ==> not (deadlocked tree)","context":["auto"],"tactic_proof":"[@@induct tree] [@@auto]","source":"imandra_synthetic/concurrency"} +{"id":89610,"prover":"Imandra","theorem":"acid_atomicity","goal":"theorem acid_atomicity tx db = commit tx db = Some db' \\/ commit tx db = None","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89611,"prover":"Imandra","theorem":"rollback_inverse","goal":"theorem rollback_inverse tx db = rollback tx (apply tx db) = db","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89612,"prover":"Imandra","theorem":"isolation_serializable","goal":"theorem isolation_serializable t1 t2 db = serializable [t1; t2] db = serializable [t2; t1] db","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89613,"prover":"Imandra","theorem":"index_consistent","goal":"theorem index_consistent idx tbl = valid_index idx tbl ==> forall k. lookup idx k = table_lookup tbl k","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89614,"prover":"Imandra","theorem":"foreign_key","goal":"theorem foreign_key p c = valid_db (p, c) ==> forall r in c. exists parent in p. fk_match r parent","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89615,"prover":"Imandra","theorem":"normalization_bcnf","goal":"theorem normalization_bcnf r = bcnf r ==> third_nf r","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/databases"} +{"id":89616,"prover":"Imandra","theorem":"tcp_in_order","goal":"theorem tcp_in_order conn pkts = established conn ==> tcp_deliver conn pkts = sort_by_seq pkts","context":["auto"],"tactic_proof":"[@@induct pkts] [@@auto]","source":"imandra_synthetic/networking"} +{"id":89617,"prover":"Imandra","theorem":"packet_checksum","goal":"theorem packet_checksum p = valid_packet p ==> verify_checksum p = true","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/networking"} +{"id":89618,"prover":"Imandra","theorem":"routing_loop_free","goal":"theorem routing_loop_free graph = valid_routing graph ==> acyclic (next_hop graph)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/networking"} +{"id":89619,"prover":"Imandra","theorem":"tls_handshake_secrecy","goal":"theorem tls_handshake_secrecy c s = secure_handshake c s ==> secret_key_safe c s","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/networking"} +{"id":89620,"prover":"Imandra","theorem":"retry_idempotent","goal":"theorem retry_idempotent req = idempotent req ==> retry req = send req","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/networking"} +{"id":89621,"prover":"Imandra","theorem":"auth_token_valid","goal":"theorem auth_token_valid t = verify_token t = Some user ==> user = decode t","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89622,"prover":"Imandra","theorem":"rbac_permission","goal":"theorem rbac_permission u r p = has_role u r && role_has_perm r p ==> authorized u p","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89623,"prover":"Imandra","theorem":"password_hash_oneway","goal":"theorem password_hash_oneway p1 p2 = hash_pw p1 = hash_pw p2 ==> p1 = p2 \\/ collision p1 p2","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89624,"prover":"Imandra","theorem":"input_sanitize","goal":"theorem input_sanitize s = safe (sanitize s)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89625,"prover":"Imandra","theorem":"session_expiry","goal":"theorem session_expiry s t = expired s t ==> not (valid_session s t)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89626,"prover":"Imandra","theorem":"audit_log_append","goal":"theorem audit_log_append log e = append log e = log ++ [e]","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/security"} +{"id":89627,"prover":"Imandra","theorem":"gradient_descent_converge","goal":"theorem gd_converge f lr x0 = convex f ==> lr > 0 ==> lr <= 1 / lipschitz f ==> converges (gd f lr x0)","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} +{"id":89628,"prover":"Imandra","theorem":"softmax_sum_one","goal":"theorem softmax_sum_one xs = List.length xs > 0 ==> List.fold_left (+.) 0.0 (softmax xs) = 1.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} +{"id":89629,"prover":"Imandra","theorem":"relu_nonneg","goal":"theorem relu_nonneg x = relu x >= 0.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} +{"id":89630,"prover":"Imandra","theorem":"sigmoid_bounded","goal":"theorem sigmoid_bounded x = 0.0 <= sigmoid x && sigmoid x <= 1.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} +{"id":89631,"prover":"Imandra","theorem":"loss_nonneg","goal":"theorem loss_nonneg y y_hat = mse y y_hat >= 0.0","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} +{"id":89632,"prover":"Imandra","theorem":"backprop_chain","goal":"theorem backprop_chain f g x = gradient (compose f g) x = gradient f (g x) *. gradient g x","context":["auto"],"tactic_proof":"[@@auto]","source":"imandra_synthetic/ml_correctness"} diff --git a/training_data/proof_states_minlog.jsonl b/training_data/proof_states_minlog.jsonl index ef48a5b6..8e26368b 100644 --- a/training_data/proof_states_minlog.jsonl +++ b/training_data/proof_states_minlog.jsonl @@ -1,2200 +1,148 @@ -{"prover":"Minlog","tactic_proof":"defn simp ind ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1","id":88500,"instance":1,"goal":"all w b = b","context":["defn","simp","ind","ng"]} -{"prover":"Minlog","tactic_proof":"defn defn rw intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2","id":88501,"instance":2,"goal":"all n + 0 = n","context":["defn","defn","rw","intro"]} -{"prover":"Minlog","tactic_proof":"intro simplify elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_3","id":88502,"instance":3,"goal":"all c g, c + g = g + c","context":["intro","simplify","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ng intro elim simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_4","id":88503,"instance":4,"goal":"all u d v, (u + d) + v = u + (d + v)","context":["ng","intro","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"use simp cases simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_5","id":88504,"instance":5,"goal":"all h, h * 0 = 0","context":["use","simp","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"rw cases cases ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_6","id":88505,"instance":6,"goal":"all h, h * 1 = h","context":["rw","cases","cases","ind"]} -{"prover":"Minlog","tactic_proof":"intro simplify elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_7","id":88506,"instance":7,"goal":"all b x w, (b * x) * w = b * (x * w)","context":["intro","simplify","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ng ind ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_8","id":88507,"instance":8,"goal":"all a e, a * e = e * a","context":["ng","ind","ind","rw"]} -{"prover":"Minlog","tactic_proof":"cases defn cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_9","id":88508,"instance":9,"goal":"all f p w, f * (p + w) = f * p + f * w","context":["cases","defn","cases","intro"]} -{"prover":"Minlog","tactic_proof":"rw defn rw elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_10","id":88509,"instance":10,"goal":"all n, n <= n","context":["rw","defn","rw","elim"]} -{"prover":"Minlog","tactic_proof":"rw cases use rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_11","id":88510,"instance":11,"goal":"all w b x, w <= b -> b <= x -> w <= x","context":["rw","cases","use","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_12","id":88511,"instance":12,"goal":"all m, not m < m","context":["ng","defn","simp","elim"]} -{"prover":"Minlog","tactic_proof":"intro elim ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_13","id":88512,"instance":13,"goal":"all g h, Even g -> Even h -> Even (g + h)","context":["intro","elim","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify ind intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_14","id":88513,"instance":14,"goal":"all a, a | a","context":["simplify","ind","intro","cases"]} -{"prover":"Minlog","tactic_proof":"defn simp cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_15","id":88514,"instance":15,"goal":"all x, Fact x > 0","context":["defn","simp","cases","rw"]} -{"prover":"Minlog","tactic_proof":"simplify ng elim defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_16","id":88515,"instance":16,"goal":"all g d = d","context":["simplify","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"use ind elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_17","id":88516,"instance":17,"goal":"all e + 0 = e","context":["use","ind","elim","defn"]} -{"prover":"Minlog","tactic_proof":"rw elim simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_18","id":88517,"instance":18,"goal":"all f b, f + b = b + f","context":["rw","elim","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"defn cases use ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_19","id":88518,"instance":19,"goal":"all f p z, (f + p) + z = f + (p + z)","context":["defn","cases","use","ind"]} -{"prover":"Minlog","tactic_proof":"rw simplify defn use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_20","id":88519,"instance":20,"goal":"all f, f * 0 = 0","context":["rw","simplify","defn","use"]} -{"prover":"Minlog","tactic_proof":"intro rw simp simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_21","id":88520,"instance":21,"goal":"all x, x * 1 = x","context":["intro","rw","simp","simp"]} -{"prover":"Minlog","tactic_proof":"use assume simplify ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_22","id":88521,"instance":22,"goal":"all y b q, (y * b) * q = y * (b * q)","context":["use","assume","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"simplify defn defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_23","id":88522,"instance":23,"goal":"all u b, u * b = b * u","context":["simplify","defn","defn","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume defn ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_24","id":88523,"instance":24,"goal":"all p e x, p * (e + x) = p * e + p * x","context":["ind","assume","defn","ind"]} -{"prover":"Minlog","tactic_proof":"defn cases rw cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_25","id":88524,"instance":25,"goal":"all m, m <= m","context":["defn","cases","rw","cases"]} -{"prover":"Minlog","tactic_proof":"ng defn intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_26","id":88525,"instance":26,"goal":"all n g z, n <= g -> g <= z -> n <= z","context":["ng","defn","intro","assume"]} -{"prover":"Minlog","tactic_proof":"ng defn simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_27","id":88526,"instance":27,"goal":"all h, not h < h","context":["ng","defn","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"defn defn elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_28","id":88527,"instance":28,"goal":"all w f, Even w -> Even f -> Even (w + f)","context":["defn","defn","elim","assume"]} -{"prover":"Minlog","tactic_proof":"intro defn defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_29","id":88528,"instance":29,"goal":"all v, v | v","context":["intro","defn","defn","use"]} -{"prover":"Minlog","tactic_proof":"simp simplify ind elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_30","id":88529,"instance":30,"goal":"all y, Fact y > 0","context":["simp","simplify","ind","elim"]} -{"prover":"Minlog","tactic_proof":"defn elim cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_31","id":88530,"instance":31,"goal":"all e x = x","context":["defn","elim","cases","intro"]} -{"prover":"Minlog","tactic_proof":"assume cases ind rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_32","id":88531,"instance":32,"goal":"all x + 0 = x","context":["assume","cases","ind","rw"]} -{"prover":"Minlog","tactic_proof":"elim rw elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_33","id":88532,"instance":33,"goal":"all m u, m + u = u + m","context":["elim","rw","elim","rw"]} -{"prover":"Minlog","tactic_proof":"simplify intro ind simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_34","id":88533,"instance":34,"goal":"all d e h, (d + e) + h = d + (e + h)","context":["simplify","intro","ind","simp"]} -{"prover":"Minlog","tactic_proof":"simplify use ng simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_35","id":88534,"instance":35,"goal":"all f, f * 0 = 0","context":["simplify","use","ng","simp"]} -{"prover":"Minlog","tactic_proof":"defn simp rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_36","id":88535,"instance":36,"goal":"all h, h * 1 = h","context":["defn","simp","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases use use elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_37","id":88536,"instance":37,"goal":"all c a f, (c * a) * f = c * (a * f)","context":["cases","use","use","elim"]} -{"prover":"Minlog","tactic_proof":"ind ng use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_38","id":88537,"instance":38,"goal":"all u a, u * a = a * u","context":["ind","ng","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw simp elim ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_39","id":88538,"instance":39,"goal":"all b g v, b * (g + v) = b * g + b * v","context":["rw","simp","elim","ind"]} -{"prover":"Minlog","tactic_proof":"cases intro defn intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_40","id":88539,"instance":40,"goal":"all n, n <= n","context":["cases","intro","defn","intro"]} -{"prover":"Minlog","tactic_proof":"simplify use defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_41","id":88540,"instance":41,"goal":"all u g n, u <= g -> g <= n -> u <= n","context":["simplify","use","defn","elim"]} -{"prover":"Minlog","tactic_proof":"assume use rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_42","id":88541,"instance":42,"goal":"all d, not d < d","context":["assume","use","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases simp use ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_43","id":88542,"instance":43,"goal":"all n h, Even n -> Even h -> Even (n + h)","context":["cases","simp","use","ind"]} -{"prover":"Minlog","tactic_proof":"simp defn defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_44","id":88543,"instance":44,"goal":"all q, q | q","context":["simp","defn","defn","simp"]} -{"prover":"Minlog","tactic_proof":"assume elim ng elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_45","id":88544,"instance":45,"goal":"all g, Fact g > 0","context":["assume","elim","ng","elim"]} -{"prover":"Minlog","tactic_proof":"simp rw use simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_46","id":88545,"instance":46,"goal":"all w x = x","context":["simp","rw","use","simp"]} -{"prover":"Minlog","tactic_proof":"use simp simp assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_47","id":88546,"instance":47,"goal":"all p + 0 = p","context":["use","simp","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume elim intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_48","id":88547,"instance":48,"goal":"all h x, h + x = x + h","context":["assume","elim","intro","ind"]} -{"prover":"Minlog","tactic_proof":"use ng simp ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_49","id":88548,"instance":49,"goal":"all v u p, (v + u) + p = v + (u + p)","context":["use","ng","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng elim cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_50","id":88549,"instance":50,"goal":"all z, z * 0 = 0","context":["ng","elim","cases","ng"]} -{"prover":"Minlog","tactic_proof":"cases simp assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_51","id":88550,"instance":51,"goal":"all v, v * 1 = v","context":["cases","simp","assume","intro"]} -{"prover":"Minlog","tactic_proof":"elim ind ng simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_52","id":88551,"instance":52,"goal":"all p g e, (p * g) * e = p * (g * e)","context":["elim","ind","ng","simp"]} -{"prover":"Minlog","tactic_proof":"use use simp simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_53","id":88552,"instance":53,"goal":"all x q, x * q = q * x","context":["use","use","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"simp use cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_54","id":88553,"instance":54,"goal":"all g y x, g * (y + x) = g * y + g * x","context":["simp","use","cases","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro assume simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_55","id":88554,"instance":55,"goal":"all d, d <= d","context":["intro","intro","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"ng ind ind intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_56","id":88555,"instance":56,"goal":"all w x e, w <= x -> x <= e -> w <= e","context":["ng","ind","ind","intro"]} -{"prover":"Minlog","tactic_proof":"intro rw elim ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_57","id":88556,"instance":57,"goal":"all v, not v < v","context":["intro","rw","elim","ind"]} -{"prover":"Minlog","tactic_proof":"defn intro elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_58","id":88557,"instance":58,"goal":"all c u, Even c -> Even u -> Even (c + u)","context":["defn","intro","elim","cases"]} -{"prover":"Minlog","tactic_proof":"rw use ind rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_59","id":88558,"instance":59,"goal":"all h, h | h","context":["rw","use","ind","rw"]} -{"prover":"Minlog","tactic_proof":"simplify ng cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_60","id":88559,"instance":60,"goal":"all u, Fact u > 0","context":["simplify","ng","cases","cases"]} -{"prover":"Minlog","tactic_proof":"use elim cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_61","id":88560,"instance":61,"goal":"all u w = w","context":["use","elim","cases","assume"]} -{"prover":"Minlog","tactic_proof":"rw use ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_62","id":88561,"instance":62,"goal":"all n + 0 = n","context":["rw","use","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume defn cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_63","id":88562,"instance":63,"goal":"all m g, m + g = g + m","context":["assume","defn","cases","cases"]} -{"prover":"Minlog","tactic_proof":"elim intro ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_64","id":88563,"instance":64,"goal":"all e u m, (e + u) + m = e + (u + m)","context":["elim","intro","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simp ind cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_65","id":88564,"instance":65,"goal":"all v, v * 0 = 0","context":["assume","simp","ind","cases"]} -{"prover":"Minlog","tactic_proof":"ng ng simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_66","id":88565,"instance":66,"goal":"all e, e * 1 = e","context":["ng","ng","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ng defn assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_67","id":88566,"instance":67,"goal":"all p m d, (p * m) * d = p * (m * d)","context":["ng","defn","assume","assume"]} -{"prover":"Minlog","tactic_proof":"intro ind ng simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_68","id":88567,"instance":68,"goal":"all g z, g * z = z * g","context":["intro","ind","ng","simp"]} -{"prover":"Minlog","tactic_proof":"ng ind simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_69","id":88568,"instance":69,"goal":"all b u a, b * (u + a) = b * u + b * a","context":["ng","ind","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"use elim cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_70","id":88569,"instance":70,"goal":"all v, v <= v","context":["use","elim","cases","use"]} -{"prover":"Minlog","tactic_proof":"cases elim rw defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_71","id":88570,"instance":71,"goal":"all p h v, p <= h -> h <= v -> p <= v","context":["cases","elim","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ng ind use elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_72","id":88571,"instance":72,"goal":"all y, not y < y","context":["ng","ind","use","elim"]} -{"prover":"Minlog","tactic_proof":"simp ng cases ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_73","id":88572,"instance":73,"goal":"all d w, Even d -> Even w -> Even (d + w)","context":["simp","ng","cases","ng"]} -{"prover":"Minlog","tactic_proof":"cases use simp intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_74","id":88573,"instance":74,"goal":"all w, w | w","context":["cases","use","simp","intro"]} -{"prover":"Minlog","tactic_proof":"assume assume use defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_75","id":88574,"instance":75,"goal":"all n, Fact n > 0","context":["assume","assume","use","defn"]} -{"prover":"Minlog","tactic_proof":"defn assume intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_76","id":88575,"instance":76,"goal":"all q e = e","context":["defn","assume","intro","assume"]} -{"prover":"Minlog","tactic_proof":"simplify ind assume cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_77","id":88576,"instance":77,"goal":"all w + 0 = w","context":["simplify","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"defn rw elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_78","id":88577,"instance":78,"goal":"all m z, m + z = z + m","context":["defn","rw","elim","use"]} -{"prover":"Minlog","tactic_proof":"cases ng simp ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_79","id":88578,"instance":79,"goal":"all d v c, (d + v) + c = d + (v + c)","context":["cases","ng","simp","ng"]} -{"prover":"Minlog","tactic_proof":"simplify defn intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_80","id":88579,"instance":80,"goal":"all v, v * 0 = 0","context":["simplify","defn","intro","ind"]} -{"prover":"Minlog","tactic_proof":"assume ng intro simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_81","id":88580,"instance":81,"goal":"all u, u * 1 = u","context":["assume","ng","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use ind assume cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_82","id":88581,"instance":82,"goal":"all y z f, (y * z) * f = y * (z * f)","context":["use","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"ng assume cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_83","id":88582,"instance":83,"goal":"all n z, n * z = z * n","context":["ng","assume","cases","ind"]} -{"prover":"Minlog","tactic_proof":"assume defn assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_84","id":88583,"instance":84,"goal":"all h y w, h * (y + w) = h * y + h * w","context":["assume","defn","assume","assume"]} -{"prover":"Minlog","tactic_proof":"intro simplify use defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_85","id":88584,"instance":85,"goal":"all b, b <= b","context":["intro","simplify","use","defn"]} -{"prover":"Minlog","tactic_proof":"rw defn rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_86","id":88585,"instance":86,"goal":"all b v g, b <= v -> v <= g -> b <= g","context":["rw","defn","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp ng elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_87","id":88586,"instance":87,"goal":"all q, not q < q","context":["simp","ng","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ng defn assume simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_88","id":88587,"instance":88,"goal":"all a f, Even a -> Even f -> Even (a + f)","context":["ng","defn","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"cases assume elim assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_89","id":88588,"instance":89,"goal":"all w, w | w","context":["cases","assume","elim","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_90","id":88589,"instance":90,"goal":"all n, Fact n > 0","context":["assume","simplify","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use ng use simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_91","id":88590,"instance":91,"goal":"all e p = p","context":["use","ng","use","simplify"]} -{"prover":"Minlog","tactic_proof":"assume defn use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_92","id":88591,"instance":92,"goal":"all p + 0 = p","context":["assume","defn","use","rw"]} -{"prover":"Minlog","tactic_proof":"elim cases use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_93","id":88592,"instance":93,"goal":"all u h, u + h = h + u","context":["elim","cases","use","intro"]} -{"prover":"Minlog","tactic_proof":"simplify cases intro cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_94","id":88593,"instance":94,"goal":"all a v n, (a + v) + n = a + (v + n)","context":["simplify","cases","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ng defn cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_95","id":88594,"instance":95,"goal":"all d, d * 0 = 0","context":["ng","defn","cases","use"]} -{"prover":"Minlog","tactic_proof":"ng cases simp ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_96","id":88595,"instance":96,"goal":"all m, m * 1 = m","context":["ng","cases","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng ind defn intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_97","id":88596,"instance":97,"goal":"all q m g, (q * m) * g = q * (m * g)","context":["ng","ind","defn","intro"]} -{"prover":"Minlog","tactic_proof":"ng elim simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_98","id":88597,"instance":98,"goal":"all e y, e * y = y * e","context":["ng","elim","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"assume elim ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_99","id":88598,"instance":99,"goal":"all x q h, x * (q + h) = x * q + x * h","context":["assume","elim","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simp use defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_100","id":88599,"instance":100,"goal":"all a, a <= a","context":["simp","use","defn","use"]} -{"prover":"Minlog","tactic_proof":"intro defn use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_101","id":88600,"instance":101,"goal":"all x y d, x <= y -> y <= d -> x <= d","context":["intro","defn","use","assume"]} -{"prover":"Minlog","tactic_proof":"elim defn assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_102","id":88601,"instance":102,"goal":"all v, not v < v","context":["elim","defn","assume","intro"]} -{"prover":"Minlog","tactic_proof":"intro simplify simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_103","id":88602,"instance":103,"goal":"all y b, Even y -> Even b -> Even (y + b)","context":["intro","simplify","simp","elim"]} -{"prover":"Minlog","tactic_proof":"cases ind ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_104","id":88603,"instance":104,"goal":"all c, c | c","context":["cases","ind","ng","defn"]} -{"prover":"Minlog","tactic_proof":"rw intro assume defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_105","id":88604,"instance":105,"goal":"all h, Fact h > 0","context":["rw","intro","assume","defn"]} -{"prover":"Minlog","tactic_proof":"cases simp use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_106","id":88605,"instance":106,"goal":"all y w = w","context":["cases","simp","use","rw"]} -{"prover":"Minlog","tactic_proof":"ind rw defn ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_107","id":88606,"instance":107,"goal":"all g + 0 = g","context":["ind","rw","defn","ind"]} -{"prover":"Minlog","tactic_proof":"assume defn elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_108","id":88607,"instance":108,"goal":"all w a, w + a = a + w","context":["assume","defn","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ind rw simp ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_109","id":88608,"instance":109,"goal":"all q h n, (q + h) + n = q + (h + n)","context":["ind","rw","simp","ng"]} -{"prover":"Minlog","tactic_proof":"assume defn rw simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_110","id":88609,"instance":110,"goal":"all u, u * 0 = 0","context":["assume","defn","rw","simp"]} -{"prover":"Minlog","tactic_proof":"elim defn intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_111","id":88610,"instance":111,"goal":"all b, b * 1 = b","context":["elim","defn","intro","cases"]} -{"prover":"Minlog","tactic_proof":"elim simplify simp ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_112","id":88611,"instance":112,"goal":"all f u v, (f * u) * v = f * (u * v)","context":["elim","simplify","simp","ng"]} -{"prover":"Minlog","tactic_proof":"simp elim simp cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_113","id":88612,"instance":113,"goal":"all w a, w * a = a * w","context":["simp","elim","simp","cases"]} -{"prover":"Minlog","tactic_proof":"intro assume elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_114","id":88613,"instance":114,"goal":"all n p g, n * (p + g) = n * p + n * g","context":["intro","assume","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ng ind rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_115","id":88614,"instance":115,"goal":"all n, n <= n","context":["ng","ind","rw","simp"]} -{"prover":"Minlog","tactic_proof":"rw simp simplify simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_116","id":88615,"instance":116,"goal":"all m f h, m <= f -> f <= h -> m <= h","context":["rw","simp","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"simp simp ng intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_117","id":88616,"instance":117,"goal":"all c, not c < c","context":["simp","simp","ng","intro"]} -{"prover":"Minlog","tactic_proof":"assume use intro rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_118","id":88617,"instance":118,"goal":"all u x, Even u -> Even x -> Even (u + x)","context":["assume","use","intro","rw"]} -{"prover":"Minlog","tactic_proof":"intro assume intro defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_119","id":88618,"instance":119,"goal":"all w, w | w","context":["intro","assume","intro","defn"]} -{"prover":"Minlog","tactic_proof":"simp ind use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_120","id":88619,"instance":120,"goal":"all x, Fact x > 0","context":["simp","ind","use","use"]} -{"prover":"Minlog","tactic_proof":"assume ng assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_121","id":88620,"instance":121,"goal":"all m n = n","context":["assume","ng","assume","intro"]} -{"prover":"Minlog","tactic_proof":"ind cases use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_122","id":88621,"instance":122,"goal":"all h + 0 = h","context":["ind","cases","use","simp"]} -{"prover":"Minlog","tactic_proof":"elim ind ind simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_123","id":88622,"instance":123,"goal":"all x h, x + h = h + x","context":["elim","ind","ind","simp"]} -{"prover":"Minlog","tactic_proof":"intro defn assume simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_124","id":88623,"instance":124,"goal":"all q c a, (q + c) + a = q + (c + a)","context":["intro","defn","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"defn intro simp assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_125","id":88624,"instance":125,"goal":"all u, u * 0 = 0","context":["defn","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"use rw defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_126","id":88625,"instance":126,"goal":"all e, e * 1 = e","context":["use","rw","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify use cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_127","id":88626,"instance":127,"goal":"all g c q, (g * c) * q = g * (c * q)","context":["simplify","use","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use assume elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_128","id":88627,"instance":128,"goal":"all z v, z * v = v * z","context":["use","assume","elim","ind"]} -{"prover":"Minlog","tactic_proof":"use rw cases intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_129","id":88628,"instance":129,"goal":"all u d a, u * (d + a) = u * d + u * a","context":["use","rw","cases","intro"]} -{"prover":"Minlog","tactic_proof":"assume simp intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_130","id":88629,"instance":130,"goal":"all p, p <= p","context":["assume","simp","intro","ng"]} -{"prover":"Minlog","tactic_proof":"simp ng simp defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_131","id":88630,"instance":131,"goal":"all a h f, a <= h -> h <= f -> a <= f","context":["simp","ng","simp","defn"]} -{"prover":"Minlog","tactic_proof":"ng cases elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_132","id":88631,"instance":132,"goal":"all a, not a < a","context":["ng","cases","elim","intro"]} -{"prover":"Minlog","tactic_proof":"elim intro simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_133","id":88632,"instance":133,"goal":"all v f, Even v -> Even f -> Even (v + f)","context":["elim","intro","simp","elim"]} -{"prover":"Minlog","tactic_proof":"defn simp ng rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_134","id":88633,"instance":134,"goal":"all c, c | c","context":["defn","simp","ng","rw"]} -{"prover":"Minlog","tactic_proof":"defn rw intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_135","id":88634,"instance":135,"goal":"all g, Fact g > 0","context":["defn","rw","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp elim use simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_136","id":88635,"instance":136,"goal":"all h y = y","context":["simp","elim","use","simplify"]} -{"prover":"Minlog","tactic_proof":"cases intro cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_137","id":88636,"instance":137,"goal":"all y + 0 = y","context":["cases","intro","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"rw rw ind use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_138","id":88637,"instance":138,"goal":"all p q, p + q = q + p","context":["rw","rw","ind","use"]} -{"prover":"Minlog","tactic_proof":"defn cases elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_139","id":88638,"instance":139,"goal":"all u w z, (u + w) + z = u + (w + z)","context":["defn","cases","elim","use"]} -{"prover":"Minlog","tactic_proof":"elim defn simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_140","id":88639,"instance":140,"goal":"all h, h * 0 = 0","context":["elim","defn","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ng simp rw defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_141","id":88640,"instance":141,"goal":"all z, z * 1 = z","context":["ng","simp","rw","defn"]} -{"prover":"Minlog","tactic_proof":"rw rw simp defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_142","id":88641,"instance":142,"goal":"all q x u, (q * x) * u = q * (x * u)","context":["rw","rw","simp","defn"]} -{"prover":"Minlog","tactic_proof":"elim ng assume simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_143","id":88642,"instance":143,"goal":"all b h, b * h = h * b","context":["elim","ng","assume","simp"]} -{"prover":"Minlog","tactic_proof":"simp simplify elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_144","id":88643,"instance":144,"goal":"all g e q, g * (e + q) = g * e + g * q","context":["simp","simplify","elim","ind"]} -{"prover":"Minlog","tactic_proof":"rw intro use use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_145","id":88644,"instance":145,"goal":"all h, h <= h","context":["rw","intro","use","use"]} -{"prover":"Minlog","tactic_proof":"use simp intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_146","id":88645,"instance":146,"goal":"all w h g, w <= h -> h <= g -> w <= g","context":["use","simp","intro","assume"]} -{"prover":"Minlog","tactic_proof":"cases elim ind use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_147","id":88646,"instance":147,"goal":"all g, not g < g","context":["cases","elim","ind","use"]} -{"prover":"Minlog","tactic_proof":"simp intro simp ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_148","id":88647,"instance":148,"goal":"all w y, Even w -> Even y -> Even (w + y)","context":["simp","intro","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng rw ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_149","id":88648,"instance":149,"goal":"all e, e | e","context":["ng","rw","ind","rw"]} -{"prover":"Minlog","tactic_proof":"rw assume intro use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_150","id":88649,"instance":150,"goal":"all y, Fact y > 0","context":["rw","assume","intro","use"]} -{"prover":"Minlog","tactic_proof":"ind simplify simp defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_151","id":88650,"instance":151,"goal":"all y g = g","context":["ind","simplify","simp","defn"]} -{"prover":"Minlog","tactic_proof":"defn use rw intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_152","id":88651,"instance":152,"goal":"all a + 0 = a","context":["defn","use","rw","intro"]} -{"prover":"Minlog","tactic_proof":"rw assume elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_153","id":88652,"instance":153,"goal":"all p f, p + f = f + p","context":["rw","assume","elim","intro"]} -{"prover":"Minlog","tactic_proof":"rw ind defn simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_154","id":88653,"instance":154,"goal":"all a u p, (a + u) + p = a + (u + p)","context":["rw","ind","defn","simp"]} -{"prover":"Minlog","tactic_proof":"use assume ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_155","id":88654,"instance":155,"goal":"all q, q * 0 = 0","context":["use","assume","ng","intro"]} -{"prover":"Minlog","tactic_proof":"defn assume rw simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_156","id":88655,"instance":156,"goal":"all m, m * 1 = m","context":["defn","assume","rw","simp"]} -{"prover":"Minlog","tactic_proof":"intro cases intro elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_157","id":88656,"instance":157,"goal":"all q u w, (q * u) * w = q * (u * w)","context":["intro","cases","intro","elim"]} -{"prover":"Minlog","tactic_proof":"ind assume use ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_158","id":88657,"instance":158,"goal":"all c x, c * x = x * c","context":["ind","assume","use","ind"]} -{"prover":"Minlog","tactic_proof":"ind simplify ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_159","id":88658,"instance":159,"goal":"all e b n, e * (b + n) = e * b + e * n","context":["ind","simplify","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simp ng rw ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_160","id":88659,"instance":160,"goal":"all d, d <= d","context":["simp","ng","rw","ng"]} -{"prover":"Minlog","tactic_proof":"use ng cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_161","id":88660,"instance":161,"goal":"all h c x, h <= c -> c <= x -> h <= x","context":["use","ng","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"simp intro simp cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_162","id":88661,"instance":162,"goal":"all g, not g < g","context":["simp","intro","simp","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_163","id":88662,"instance":163,"goal":"all q e, Even q -> Even e -> Even (q + e)","context":["ind","assume","elim","assume"]} -{"prover":"Minlog","tactic_proof":"rw ng intro use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_164","id":88663,"instance":164,"goal":"all n, n | n","context":["rw","ng","intro","use"]} -{"prover":"Minlog","tactic_proof":"rw intro cases use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_165","id":88664,"instance":165,"goal":"all n, Fact n > 0","context":["rw","intro","cases","use"]} -{"prover":"Minlog","tactic_proof":"defn use simp cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_166","id":88665,"instance":166,"goal":"all f g = g","context":["defn","use","simp","cases"]} -{"prover":"Minlog","tactic_proof":"rw cases ind assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_167","id":88666,"instance":167,"goal":"all g + 0 = g","context":["rw","cases","ind","assume"]} -{"prover":"Minlog","tactic_proof":"ind simplify use assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_168","id":88667,"instance":168,"goal":"all x z, x + z = z + x","context":["ind","simplify","use","assume"]} -{"prover":"Minlog","tactic_proof":"use assume simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_169","id":88668,"instance":169,"goal":"all b z m, (b + z) + m = b + (z + m)","context":["use","assume","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"cases simp use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_170","id":88669,"instance":170,"goal":"all d, d * 0 = 0","context":["cases","simp","use","rw"]} -{"prover":"Minlog","tactic_proof":"assume defn ind use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_171","id":88670,"instance":171,"goal":"all x, x * 1 = x","context":["assume","defn","ind","use"]} -{"prover":"Minlog","tactic_proof":"intro ng intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_172","id":88671,"instance":172,"goal":"all g n x, (g * n) * x = g * (n * x)","context":["intro","ng","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ng elim intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_173","id":88672,"instance":173,"goal":"all e y, e * y = y * e","context":["ng","elim","intro","cases"]} -{"prover":"Minlog","tactic_proof":"simp elim use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_174","id":88673,"instance":174,"goal":"all q v d, q * (v + d) = q * v + q * d","context":["simp","elim","use","rw"]} -{"prover":"Minlog","tactic_proof":"rw defn cases simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_175","id":88674,"instance":175,"goal":"all n, n <= n","context":["rw","defn","cases","simp"]} -{"prover":"Minlog","tactic_proof":"use elim elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_176","id":88675,"instance":176,"goal":"all p b d, p <= b -> b <= d -> p <= d","context":["use","elim","elim","ng"]} -{"prover":"Minlog","tactic_proof":"intro assume assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_177","id":88676,"instance":177,"goal":"all f, not f < f","context":["intro","assume","assume","assume"]} -{"prover":"Minlog","tactic_proof":"assume intro defn use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_178","id":88677,"instance":178,"goal":"all z d, Even z -> Even d -> Even (z + d)","context":["assume","intro","defn","use"]} -{"prover":"Minlog","tactic_proof":"ng ind elim simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_179","id":88678,"instance":179,"goal":"all a, a | a","context":["ng","ind","elim","simp"]} -{"prover":"Minlog","tactic_proof":"ng cases rw defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_180","id":88679,"instance":180,"goal":"all f, Fact f > 0","context":["ng","cases","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ind use ng simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_181","id":88680,"instance":181,"goal":"all c x = x","context":["ind","use","ng","simp"]} -{"prover":"Minlog","tactic_proof":"assume intro assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_182","id":88681,"instance":182,"goal":"all h + 0 = h","context":["assume","intro","assume","use"]} -{"prover":"Minlog","tactic_proof":"rw intro rw simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_183","id":88682,"instance":183,"goal":"all u b, u + b = b + u","context":["rw","intro","rw","simp"]} -{"prover":"Minlog","tactic_proof":"rw elim use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_184","id":88683,"instance":184,"goal":"all n y p, (n + y) + p = n + (y + p)","context":["rw","elim","use","rw"]} -{"prover":"Minlog","tactic_proof":"rw rw simp rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_185","id":88684,"instance":185,"goal":"all d, d * 0 = 0","context":["rw","rw","simp","rw"]} -{"prover":"Minlog","tactic_proof":"ng simplify defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_186","id":88685,"instance":186,"goal":"all x, x * 1 = x","context":["ng","simplify","defn","cases"]} -{"prover":"Minlog","tactic_proof":"ind simplify ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_187","id":88686,"instance":187,"goal":"all m f x, (m * f) * x = m * (f * x)","context":["ind","simplify","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume assume rw intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_188","id":88687,"instance":188,"goal":"all v x, v * x = x * v","context":["assume","assume","rw","intro"]} -{"prover":"Minlog","tactic_proof":"elim assume ng cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_189","id":88688,"instance":189,"goal":"all p y b, p * (y + b) = p * y + p * b","context":["elim","assume","ng","cases"]} -{"prover":"Minlog","tactic_proof":"cases assume assume assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_190","id":88689,"instance":190,"goal":"all z, z <= z","context":["cases","assume","assume","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_191","id":88690,"instance":191,"goal":"all d e m, d <= e -> e <= m -> d <= m","context":["assume","simplify","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ind ind elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_192","id":88691,"instance":192,"goal":"all h, not h < h","context":["ind","ind","elim","ind"]} -{"prover":"Minlog","tactic_proof":"assume defn simp simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_193","id":88692,"instance":193,"goal":"all v z, Even v -> Even z -> Even (v + z)","context":["assume","defn","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"simp ng simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_194","id":88693,"instance":194,"goal":"all h, h | h","context":["simp","ng","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"rw assume intro rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_195","id":88694,"instance":195,"goal":"all e, Fact e > 0","context":["rw","assume","intro","rw"]} -{"prover":"Minlog","tactic_proof":"cases ind simplify simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_196","id":88695,"instance":196,"goal":"all a x = x","context":["cases","ind","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"cases simp simp simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_197","id":88696,"instance":197,"goal":"all b + 0 = b","context":["cases","simp","simp","simp"]} -{"prover":"Minlog","tactic_proof":"elim rw ind simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_198","id":88697,"instance":198,"goal":"all b x, b + x = x + b","context":["elim","rw","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simplify use use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_199","id":88698,"instance":199,"goal":"all u f h, (u + f) + h = u + (f + h)","context":["assume","simplify","use","use"]} -{"prover":"Minlog","tactic_proof":"simp use elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_200","id":88699,"instance":200,"goal":"all f, f * 0 = 0","context":["simp","use","elim","ng"]} -{"prover":"Minlog","tactic_proof":"ng intro assume cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_201","id":88700,"instance":201,"goal":"all v, v * 1 = v","context":["ng","intro","assume","cases"]} -{"prover":"Minlog","tactic_proof":"elim cases elim use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_202","id":88701,"instance":202,"goal":"all v m a, (v * m) * a = v * (m * a)","context":["elim","cases","elim","use"]} -{"prover":"Minlog","tactic_proof":"assume elim use defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_203","id":88702,"instance":203,"goal":"all q c, q * c = c * q","context":["assume","elim","use","defn"]} -{"prover":"Minlog","tactic_proof":"simp defn intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_204","id":88703,"instance":204,"goal":"all m q x, m * (q + x) = m * q + m * x","context":["simp","defn","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"cases defn defn simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_205","id":88704,"instance":205,"goal":"all a, a <= a","context":["cases","defn","defn","simp"]} -{"prover":"Minlog","tactic_proof":"simplify assume ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_206","id":88705,"instance":206,"goal":"all z b m, z <= b -> b <= m -> z <= m","context":["simplify","assume","ind","elim"]} -{"prover":"Minlog","tactic_proof":"assume use rw simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_207","id":88706,"instance":207,"goal":"all q, not q < q","context":["assume","use","rw","simp"]} -{"prover":"Minlog","tactic_proof":"simplify ind ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_208","id":88707,"instance":208,"goal":"all c b, Even c -> Even b -> Even (c + b)","context":["simplify","ind","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify assume ng intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_209","id":88708,"instance":209,"goal":"all m, m | m","context":["simplify","assume","ng","intro"]} -{"prover":"Minlog","tactic_proof":"simp ng rw ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_210","id":88709,"instance":210,"goal":"all x, Fact x > 0","context":["simp","ng","rw","ng"]} -{"prover":"Minlog","tactic_proof":"use ind simplify ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_211","id":88710,"instance":211,"goal":"all f e = e","context":["use","ind","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"ng simplify cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_212","id":88711,"instance":212,"goal":"all p + 0 = p","context":["ng","simplify","cases","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn use ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_213","id":88712,"instance":213,"goal":"all g y, g + y = y + g","context":["ng","defn","use","ind"]} -{"prover":"Minlog","tactic_proof":"simp rw simp intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_214","id":88713,"instance":214,"goal":"all f x h, (f + x) + h = f + (x + h)","context":["simp","rw","simp","intro"]} -{"prover":"Minlog","tactic_proof":"intro elim ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_215","id":88714,"instance":215,"goal":"all g, g * 0 = 0","context":["intro","elim","ind","rw"]} -{"prover":"Minlog","tactic_proof":"cases ind intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_216","id":88715,"instance":216,"goal":"all a, a * 1 = a","context":["cases","ind","intro","use"]} -{"prover":"Minlog","tactic_proof":"ind defn elim simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_217","id":88716,"instance":217,"goal":"all g w m, (g * w) * m = g * (w * m)","context":["ind","defn","elim","simp"]} -{"prover":"Minlog","tactic_proof":"elim assume simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_218","id":88717,"instance":218,"goal":"all h e, h * e = e * h","context":["elim","assume","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"assume defn rw defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_219","id":88718,"instance":219,"goal":"all u a c, u * (a + c) = u * a + u * c","context":["assume","defn","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn assume cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_220","id":88719,"instance":220,"goal":"all c, c <= c","context":["ng","defn","assume","cases"]} -{"prover":"Minlog","tactic_proof":"intro defn assume cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_221","id":88720,"instance":221,"goal":"all e b q, e <= b -> b <= q -> e <= q","context":["intro","defn","assume","cases"]} -{"prover":"Minlog","tactic_proof":"simp ind defn assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_222","id":88721,"instance":222,"goal":"all v, not v < v","context":["simp","ind","defn","assume"]} -{"prover":"Minlog","tactic_proof":"ind intro assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_223","id":88722,"instance":223,"goal":"all c v, Even c -> Even v -> Even (c + v)","context":["ind","intro","assume","cases"]} -{"prover":"Minlog","tactic_proof":"ng assume assume simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_224","id":88723,"instance":224,"goal":"all d, d | d","context":["ng","assume","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"elim use simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_225","id":88724,"instance":225,"goal":"all q, Fact q > 0","context":["elim","use","simp","elim"]} -{"prover":"Minlog","tactic_proof":"rw cases intro ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_226","id":88725,"instance":226,"goal":"all w c = c","context":["rw","cases","intro","ind"]} -{"prover":"Minlog","tactic_proof":"cases use rw intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_227","id":88726,"instance":227,"goal":"all m + 0 = m","context":["cases","use","rw","intro"]} -{"prover":"Minlog","tactic_proof":"defn ng intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_228","id":88727,"instance":228,"goal":"all b q, b + q = q + b","context":["defn","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"defn simp elim intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_229","id":88728,"instance":229,"goal":"all e f w, (e + f) + w = e + (f + w)","context":["defn","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ind cases assume rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_230","id":88729,"instance":230,"goal":"all p, p * 0 = 0","context":["ind","cases","assume","rw"]} -{"prover":"Minlog","tactic_proof":"use simplify assume ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_231","id":88730,"instance":231,"goal":"all m, m * 1 = m","context":["use","simplify","assume","ng"]} -{"prover":"Minlog","tactic_proof":"defn elim assume defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_232","id":88731,"instance":232,"goal":"all w m p, (w * m) * p = w * (m * p)","context":["defn","elim","assume","defn"]} -{"prover":"Minlog","tactic_proof":"cases ind ind simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_233","id":88732,"instance":233,"goal":"all p e, p * e = e * p","context":["cases","ind","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"defn ng defn cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_234","id":88733,"instance":234,"goal":"all z n v, z * (n + v) = z * n + z * v","context":["defn","ng","defn","cases"]} -{"prover":"Minlog","tactic_proof":"assume defn use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_235","id":88734,"instance":235,"goal":"all y, y <= y","context":["assume","defn","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw ind elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_236","id":88735,"instance":236,"goal":"all p d y, p <= d -> d <= y -> p <= y","context":["rw","ind","elim","intro"]} -{"prover":"Minlog","tactic_proof":"defn ind elim use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_237","id":88736,"instance":237,"goal":"all d, not d < d","context":["defn","ind","elim","use"]} -{"prover":"Minlog","tactic_proof":"simp rw ind cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_238","id":88737,"instance":238,"goal":"all f e, Even f -> Even e -> Even (f + e)","context":["simp","rw","ind","cases"]} -{"prover":"Minlog","tactic_proof":"rw use cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_239","id":88738,"instance":239,"goal":"all z, z | z","context":["rw","use","cases","rw"]} -{"prover":"Minlog","tactic_proof":"simplify use defn ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_240","id":88739,"instance":240,"goal":"all y, Fact y > 0","context":["simplify","use","defn","ng"]} -{"prover":"Minlog","tactic_proof":"ind ind assume ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_241","id":88740,"instance":241,"goal":"all f g = g","context":["ind","ind","assume","ind"]} -{"prover":"Minlog","tactic_proof":"assume assume assume defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_242","id":88741,"instance":242,"goal":"all v + 0 = v","context":["assume","assume","assume","defn"]} -{"prover":"Minlog","tactic_proof":"intro cases ng simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_243","id":88742,"instance":243,"goal":"all f a, f + a = a + f","context":["intro","cases","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"defn intro defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_244","id":88743,"instance":244,"goal":"all d y x, (d + y) + x = d + (y + x)","context":["defn","intro","defn","cases"]} -{"prover":"Minlog","tactic_proof":"elim rw cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_245","id":88744,"instance":245,"goal":"all h, h * 0 = 0","context":["elim","rw","cases","intro"]} -{"prover":"Minlog","tactic_proof":"assume rw simp ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_246","id":88745,"instance":246,"goal":"all x, x * 1 = x","context":["assume","rw","simp","ind"]} -{"prover":"Minlog","tactic_proof":"use assume rw simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_247","id":88746,"instance":247,"goal":"all f u h, (f * u) * h = f * (u * h)","context":["use","assume","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"cases elim use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_248","id":88747,"instance":248,"goal":"all m q, m * q = q * m","context":["cases","elim","use","rw"]} -{"prover":"Minlog","tactic_proof":"cases simplify cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_249","id":88748,"instance":249,"goal":"all n p y, n * (p + y) = n * p + n * y","context":["cases","simplify","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simp defn intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_250","id":88749,"instance":250,"goal":"all p, p <= p","context":["simp","defn","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ng ind simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_251","id":88750,"instance":251,"goal":"all g e c, g <= e -> e <= c -> g <= c","context":["ng","ind","simp","elim"]} -{"prover":"Minlog","tactic_proof":"elim elim ng intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_252","id":88751,"instance":252,"goal":"all h, not h < h","context":["elim","elim","ng","intro"]} -{"prover":"Minlog","tactic_proof":"simplify elim use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_253","id":88752,"instance":253,"goal":"all x v, Even x -> Even v -> Even (x + v)","context":["simplify","elim","use","intro"]} -{"prover":"Minlog","tactic_proof":"defn ng simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_254","id":88753,"instance":254,"goal":"all f, f | f","context":["defn","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"simp ind defn simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_255","id":88754,"instance":255,"goal":"all g, Fact g > 0","context":["simp","ind","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"use simp cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_256","id":88755,"instance":256,"goal":"all x e = e","context":["use","simp","cases","assume"]} -{"prover":"Minlog","tactic_proof":"rw assume use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_257","id":88756,"instance":257,"goal":"all p + 0 = p","context":["rw","assume","use","simp"]} -{"prover":"Minlog","tactic_proof":"elim simp elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_258","id":88757,"instance":258,"goal":"all b e, b + e = e + b","context":["elim","simp","elim","use"]} -{"prover":"Minlog","tactic_proof":"simp ind ng intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_259","id":88758,"instance":259,"goal":"all u f d, (u + f) + d = u + (f + d)","context":["simp","ind","ng","intro"]} -{"prover":"Minlog","tactic_proof":"simplify use ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_260","id":88759,"instance":260,"goal":"all y, y * 0 = 0","context":["simplify","use","ind","cases"]} -{"prover":"Minlog","tactic_proof":"cases use elim defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_261","id":88760,"instance":261,"goal":"all y, y * 1 = y","context":["cases","use","elim","defn"]} -{"prover":"Minlog","tactic_proof":"assume rw assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_262","id":88761,"instance":262,"goal":"all y d p, (y * d) * p = y * (d * p)","context":["assume","rw","assume","ng"]} -{"prover":"Minlog","tactic_proof":"rw defn simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_263","id":88762,"instance":263,"goal":"all e z, e * z = z * e","context":["rw","defn","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"rw ind elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_264","id":88763,"instance":264,"goal":"all g e h, g * (e + h) = g * e + g * h","context":["rw","ind","elim","assume"]} -{"prover":"Minlog","tactic_proof":"intro assume ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_265","id":88764,"instance":265,"goal":"all q, q <= q","context":["intro","assume","ind","defn"]} -{"prover":"Minlog","tactic_proof":"defn assume ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_266","id":88765,"instance":266,"goal":"all b w f, b <= w -> w <= f -> b <= f","context":["defn","assume","ng","simp"]} -{"prover":"Minlog","tactic_proof":"assume intro use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_267","id":88766,"instance":267,"goal":"all n, not n < n","context":["assume","intro","use","ng"]} -{"prover":"Minlog","tactic_proof":"simplify defn assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_268","id":88767,"instance":268,"goal":"all u v, Even u -> Even v -> Even (u + v)","context":["simplify","defn","assume","rw"]} -{"prover":"Minlog","tactic_proof":"elim rw assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_269","id":88768,"instance":269,"goal":"all n, n | n","context":["elim","rw","assume","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_270","id":88769,"instance":270,"goal":"all m, Fact m > 0","context":["intro","cases","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"defn rw assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_271","id":88770,"instance":271,"goal":"all h f = f","context":["defn","rw","assume","defn"]} -{"prover":"Minlog","tactic_proof":"assume cases defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_272","id":88771,"instance":272,"goal":"all h + 0 = h","context":["assume","cases","defn","cases"]} -{"prover":"Minlog","tactic_proof":"elim cases ng use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_273","id":88772,"instance":273,"goal":"all b c, b + c = c + b","context":["elim","cases","ng","use"]} -{"prover":"Minlog","tactic_proof":"elim elim defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_274","id":88773,"instance":274,"goal":"all f q v, (f + q) + v = f + (q + v)","context":["elim","elim","defn","use"]} -{"prover":"Minlog","tactic_proof":"use elim ng simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_275","id":88774,"instance":275,"goal":"all w, w * 0 = 0","context":["use","elim","ng","simp"]} -{"prover":"Minlog","tactic_proof":"ng simp ng ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_276","id":88775,"instance":276,"goal":"all h, h * 1 = h","context":["ng","simp","ng","ng"]} -{"prover":"Minlog","tactic_proof":"ng cases elim ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_277","id":88776,"instance":277,"goal":"all x c h, (x * c) * h = x * (c * h)","context":["ng","cases","elim","ind"]} -{"prover":"Minlog","tactic_proof":"rw ind cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_278","id":88777,"instance":278,"goal":"all b e, b * e = e * b","context":["rw","ind","cases","simp"]} -{"prover":"Minlog","tactic_proof":"intro use simplify ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_279","id":88778,"instance":279,"goal":"all f h m, f * (h + m) = f * h + f * m","context":["intro","use","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ng defn intro cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_280","id":88779,"instance":280,"goal":"all b, b <= b","context":["ng","defn","intro","cases"]} -{"prover":"Minlog","tactic_proof":"rw ind use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_281","id":88780,"instance":281,"goal":"all m f z, m <= f -> f <= z -> m <= z","context":["rw","ind","use","intro"]} -{"prover":"Minlog","tactic_proof":"intro simp rw ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_282","id":88781,"instance":282,"goal":"all b, not b < b","context":["intro","simp","rw","ng"]} -{"prover":"Minlog","tactic_proof":"cases defn use ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_283","id":88782,"instance":283,"goal":"all d v, Even d -> Even v -> Even (d + v)","context":["cases","defn","use","ind"]} -{"prover":"Minlog","tactic_proof":"use simp use assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_284","id":88783,"instance":284,"goal":"all u, u | u","context":["use","simp","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw defn simp defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_285","id":88784,"instance":285,"goal":"all x, Fact x > 0","context":["rw","defn","simp","defn"]} -{"prover":"Minlog","tactic_proof":"intro assume use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_286","id":88785,"instance":286,"goal":"all f v = v","context":["intro","assume","use","assume"]} -{"prover":"Minlog","tactic_proof":"defn use rw elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_287","id":88786,"instance":287,"goal":"all w + 0 = w","context":["defn","use","rw","elim"]} -{"prover":"Minlog","tactic_proof":"use simplify intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_288","id":88787,"instance":288,"goal":"all m g, m + g = g + m","context":["use","simplify","intro","elim"]} -{"prover":"Minlog","tactic_proof":"rw simp assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_289","id":88788,"instance":289,"goal":"all g c n, (g + c) + n = g + (c + n)","context":["rw","simp","assume","intro"]} -{"prover":"Minlog","tactic_proof":"use simplify simplify ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_290","id":88789,"instance":290,"goal":"all p, p * 0 = 0","context":["use","simplify","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"ng cases use use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_291","id":88790,"instance":291,"goal":"all w, w * 1 = w","context":["ng","cases","use","use"]} -{"prover":"Minlog","tactic_proof":"ng cases elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_292","id":88791,"instance":292,"goal":"all a z y, (a * z) * y = a * (z * y)","context":["ng","cases","elim","elim"]} -{"prover":"Minlog","tactic_proof":"defn simp intro intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_293","id":88792,"instance":293,"goal":"all g q, g * q = q * g","context":["defn","simp","intro","intro"]} -{"prover":"Minlog","tactic_proof":"simplify ng defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_294","id":88793,"instance":294,"goal":"all m n v, m * (n + v) = m * n + m * v","context":["simplify","ng","defn","assume"]} -{"prover":"Minlog","tactic_proof":"cases simplify cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_295","id":88794,"instance":295,"goal":"all q, q <= q","context":["cases","simplify","cases","use"]} -{"prover":"Minlog","tactic_proof":"defn simplify elim cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_296","id":88795,"instance":296,"goal":"all c z q, c <= z -> z <= q -> c <= q","context":["defn","simplify","elim","cases"]} -{"prover":"Minlog","tactic_proof":"cases intro cases simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_297","id":88796,"instance":297,"goal":"all c, not c < c","context":["cases","intro","cases","simp"]} -{"prover":"Minlog","tactic_proof":"simp use elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_298","id":88797,"instance":298,"goal":"all c q, Even c -> Even q -> Even (c + q)","context":["simp","use","elim","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_299","id":88798,"instance":299,"goal":"all p, p | p","context":["elim","ng","use","rw"]} -{"prover":"Minlog","tactic_proof":"rw assume assume ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_300","id":88799,"instance":300,"goal":"all z, Fact z > 0","context":["rw","assume","assume","ind"]} -{"prover":"Minlog","tactic_proof":"simp simplify intro simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_301","id":88800,"instance":301,"goal":"all u b = b","context":["simp","simplify","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use intro elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_302","id":88801,"instance":302,"goal":"all m + 0 = m","context":["use","intro","elim","rw"]} -{"prover":"Minlog","tactic_proof":"cases intro defn elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_303","id":88802,"instance":303,"goal":"all m v, m + v = v + m","context":["cases","intro","defn","elim"]} -{"prover":"Minlog","tactic_proof":"rw simplify ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_304","id":88803,"instance":304,"goal":"all g m d, (g + m) + d = g + (m + d)","context":["rw","simplify","ng","defn"]} -{"prover":"Minlog","tactic_proof":"rw ind simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_305","id":88804,"instance":305,"goal":"all a, a * 0 = 0","context":["rw","ind","simp","elim"]} -{"prover":"Minlog","tactic_proof":"elim elim use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_306","id":88805,"instance":306,"goal":"all q, q * 1 = q","context":["elim","elim","use","use"]} -{"prover":"Minlog","tactic_proof":"use cases assume use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_307","id":88806,"instance":307,"goal":"all m u w, (m * u) * w = m * (u * w)","context":["use","cases","assume","use"]} -{"prover":"Minlog","tactic_proof":"use cases ind use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_308","id":88807,"instance":308,"goal":"all f n, f * n = n * f","context":["use","cases","ind","use"]} -{"prover":"Minlog","tactic_proof":"elim ng elim use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_309","id":88808,"instance":309,"goal":"all q n h, q * (n + h) = q * n + q * h","context":["elim","ng","elim","use"]} -{"prover":"Minlog","tactic_proof":"defn intro rw simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_310","id":88809,"instance":310,"goal":"all c, c <= c","context":["defn","intro","rw","simp"]} -{"prover":"Minlog","tactic_proof":"elim elim simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_311","id":88810,"instance":311,"goal":"all p z x, p <= z -> z <= x -> p <= x","context":["elim","elim","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simplify elim elim ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_312","id":88811,"instance":312,"goal":"all p, not p < p","context":["simplify","elim","elim","ind"]} -{"prover":"Minlog","tactic_proof":"rw defn assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_313","id":88812,"instance":313,"goal":"all w a, Even w -> Even a -> Even (w + a)","context":["rw","defn","assume","use"]} -{"prover":"Minlog","tactic_proof":"intro defn elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_314","id":88813,"instance":314,"goal":"all y, y | y","context":["intro","defn","elim","ng"]} -{"prover":"Minlog","tactic_proof":"intro assume use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_315","id":88814,"instance":315,"goal":"all q, Fact q > 0","context":["intro","assume","use","cases"]} -{"prover":"Minlog","tactic_proof":"simp simp elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_316","id":88815,"instance":316,"goal":"all b g = g","context":["simp","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"cases simplify simplify simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_317","id":88816,"instance":317,"goal":"all z + 0 = z","context":["cases","simplify","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"intro ng use intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_318","id":88817,"instance":318,"goal":"all f a, f + a = a + f","context":["intro","ng","use","intro"]} -{"prover":"Minlog","tactic_proof":"defn use rw use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_319","id":88818,"instance":319,"goal":"all y v w, (y + v) + w = y + (v + w)","context":["defn","use","rw","use"]} -{"prover":"Minlog","tactic_proof":"simp ng simp ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_320","id":88819,"instance":320,"goal":"all f, f * 0 = 0","context":["simp","ng","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng cases intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_321","id":88820,"instance":321,"goal":"all f, f * 1 = f","context":["ng","cases","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_322","id":88821,"instance":322,"goal":"all q v m, (q * v) * m = q * (v * m)","context":["ind","assume","use","intro"]} -{"prover":"Minlog","tactic_proof":"simplify rw ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_323","id":88822,"instance":323,"goal":"all z v, z * v = v * z","context":["simplify","rw","ind","rw"]} -{"prover":"Minlog","tactic_proof":"defn use use ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_324","id":88823,"instance":324,"goal":"all n d x, n * (d + x) = n * d + n * x","context":["defn","use","use","ind"]} -{"prover":"Minlog","tactic_proof":"intro simplify elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_325","id":88824,"instance":325,"goal":"all w, w <= w","context":["intro","simplify","elim","rw"]} -{"prover":"Minlog","tactic_proof":"assume elim defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_326","id":88825,"instance":326,"goal":"all g c m, g <= c -> c <= m -> g <= m","context":["assume","elim","defn","rw"]} -{"prover":"Minlog","tactic_proof":"ind rw ind simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_327","id":88826,"instance":327,"goal":"all e, not e < e","context":["ind","rw","ind","simp"]} -{"prover":"Minlog","tactic_proof":"assume assume intro cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_328","id":88827,"instance":328,"goal":"all b g, Even b -> Even g -> Even (b + g)","context":["assume","assume","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ng simplify elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_329","id":88828,"instance":329,"goal":"all f, f | f","context":["ng","simplify","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ind defn simplify rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_330","id":88829,"instance":330,"goal":"all h, Fact h > 0","context":["ind","defn","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"elim ng assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_331","id":88830,"instance":331,"goal":"all f y = y","context":["elim","ng","assume","elim"]} -{"prover":"Minlog","tactic_proof":"ng ind elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_332","id":88831,"instance":332,"goal":"all z + 0 = z","context":["ng","ind","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ind use ind use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_333","id":88832,"instance":333,"goal":"all d m, d + m = m + d","context":["ind","use","ind","use"]} -{"prover":"Minlog","tactic_proof":"assume elim ind cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_334","id":88833,"instance":334,"goal":"all p a z, (p + a) + z = p + (a + z)","context":["assume","elim","ind","cases"]} -{"prover":"Minlog","tactic_proof":"elim intro use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_335","id":88834,"instance":335,"goal":"all y, y * 0 = 0","context":["elim","intro","use","use"]} -{"prover":"Minlog","tactic_proof":"intro cases use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_336","id":88835,"instance":336,"goal":"all w, w * 1 = w","context":["intro","cases","use","rw"]} -{"prover":"Minlog","tactic_proof":"simp ng cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_337","id":88836,"instance":337,"goal":"all e f a, (e * f) * a = e * (f * a)","context":["simp","ng","cases","cases"]} -{"prover":"Minlog","tactic_proof":"ind cases cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_338","id":88837,"instance":338,"goal":"all f g, f * g = g * f","context":["ind","cases","cases","assume"]} -{"prover":"Minlog","tactic_proof":"defn ng simplify cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_339","id":88838,"instance":339,"goal":"all b z x, b * (z + x) = b * z + b * x","context":["defn","ng","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"assume defn intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_340","id":88839,"instance":340,"goal":"all m, m <= m","context":["assume","defn","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use use simp simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_341","id":88840,"instance":341,"goal":"all a c w, a <= c -> c <= w -> a <= w","context":["use","use","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"assume ind ng intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_342","id":88841,"instance":342,"goal":"all w, not w < w","context":["assume","ind","ng","intro"]} -{"prover":"Minlog","tactic_proof":"cases defn simplify rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_343","id":88842,"instance":343,"goal":"all p q, Even p -> Even q -> Even (p + q)","context":["cases","defn","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"assume simp simplify defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_344","id":88843,"instance":344,"goal":"all c, c | c","context":["assume","simp","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"ng elim defn ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_345","id":88844,"instance":345,"goal":"all a, Fact a > 0","context":["ng","elim","defn","ind"]} -{"prover":"Minlog","tactic_proof":"defn elim simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_346","id":88845,"instance":346,"goal":"all h x = x","context":["defn","elim","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng defn ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_347","id":88846,"instance":347,"goal":"all h + 0 = h","context":["elim","ng","defn","ind"]} -{"prover":"Minlog","tactic_proof":"cases ind intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_348","id":88847,"instance":348,"goal":"all y w, y + w = w + y","context":["cases","ind","intro","assume"]} -{"prover":"Minlog","tactic_proof":"assume defn use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_349","id":88848,"instance":349,"goal":"all z h y, (z + h) + y = z + (h + y)","context":["assume","defn","use","rw"]} -{"prover":"Minlog","tactic_proof":"rw ind elim rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_350","id":88849,"instance":350,"goal":"all v, v * 0 = 0","context":["rw","ind","elim","rw"]} -{"prover":"Minlog","tactic_proof":"rw ind cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_351","id":88850,"instance":351,"goal":"all h, h * 1 = h","context":["rw","ind","cases","intro"]} -{"prover":"Minlog","tactic_proof":"simp rw simplify elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_352","id":88851,"instance":352,"goal":"all z b c, (z * b) * c = z * (b * c)","context":["simp","rw","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"cases simp defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_353","id":88852,"instance":353,"goal":"all d x, d * x = x * d","context":["cases","simp","defn","elim"]} -{"prover":"Minlog","tactic_proof":"elim cases cases rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_354","id":88853,"instance":354,"goal":"all c n v, c * (n + v) = c * n + c * v","context":["elim","cases","cases","rw"]} -{"prover":"Minlog","tactic_proof":"cases use rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_355","id":88854,"instance":355,"goal":"all p, p <= p","context":["cases","use","rw","ng"]} -{"prover":"Minlog","tactic_proof":"cases cases use intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_356","id":88855,"instance":356,"goal":"all n q v, n <= q -> q <= v -> n <= v","context":["cases","cases","use","intro"]} -{"prover":"Minlog","tactic_proof":"cases ng ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_357","id":88856,"instance":357,"goal":"all f, not f < f","context":["cases","ng","ind","intro"]} -{"prover":"Minlog","tactic_proof":"elim simplify use use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_358","id":88857,"instance":358,"goal":"all m b, Even m -> Even b -> Even (m + b)","context":["elim","simplify","use","use"]} -{"prover":"Minlog","tactic_proof":"simp intro cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_359","id":88858,"instance":359,"goal":"all g, g | g","context":["simp","intro","cases","assume"]} -{"prover":"Minlog","tactic_proof":"assume intro cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_360","id":88859,"instance":360,"goal":"all g, Fact g > 0","context":["assume","intro","cases","assume"]} -{"prover":"Minlog","tactic_proof":"rw simplify ng rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_361","id":88860,"instance":361,"goal":"all b e = e","context":["rw","simplify","ng","rw"]} -{"prover":"Minlog","tactic_proof":"assume rw assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_362","id":88861,"instance":362,"goal":"all q + 0 = q","context":["assume","rw","assume","intro"]} -{"prover":"Minlog","tactic_proof":"simplify ind use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_363","id":88862,"instance":363,"goal":"all x c, x + c = c + x","context":["simplify","ind","use","cases"]} -{"prover":"Minlog","tactic_proof":"rw rw defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_364","id":88863,"instance":364,"goal":"all c n e, (c + n) + e = c + (n + e)","context":["rw","rw","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simp assume defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_365","id":88864,"instance":365,"goal":"all n, n * 0 = 0","context":["simp","assume","defn","rw"]} -{"prover":"Minlog","tactic_proof":"assume assume cases cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_366","id":88865,"instance":366,"goal":"all q, q * 1 = q","context":["assume","assume","cases","cases"]} -{"prover":"Minlog","tactic_proof":"elim intro elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_367","id":88866,"instance":367,"goal":"all m x p, (m * x) * p = m * (x * p)","context":["elim","intro","elim","cases"]} -{"prover":"Minlog","tactic_proof":"elim intro intro intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_368","id":88867,"instance":368,"goal":"all w z, w * z = z * w","context":["elim","intro","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ind simp simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_369","id":88868,"instance":369,"goal":"all z e x, z * (e + x) = z * e + z * x","context":["ind","simp","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simplify rw defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_370","id":88869,"instance":370,"goal":"all u, u <= u","context":["simplify","rw","defn","rw"]} -{"prover":"Minlog","tactic_proof":"rw elim elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_371","id":88870,"instance":371,"goal":"all f w x, f <= w -> w <= x -> f <= x","context":["rw","elim","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ng simp ng defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_372","id":88871,"instance":372,"goal":"all g, not g < g","context":["ng","simp","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro use rw defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_373","id":88872,"instance":373,"goal":"all y c, Even y -> Even c -> Even (y + c)","context":["intro","use","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ind cases ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_374","id":88873,"instance":374,"goal":"all g, g | g","context":["ind","cases","ng","ng"]} -{"prover":"Minlog","tactic_proof":"assume defn defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_375","id":88874,"instance":375,"goal":"all z, Fact z > 0","context":["assume","defn","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify intro ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_376","id":88875,"instance":376,"goal":"all v p = p","context":["simplify","intro","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simp intro elim simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_377","id":88876,"instance":377,"goal":"all q + 0 = q","context":["simp","intro","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"intro ind simp cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_378","id":88877,"instance":378,"goal":"all x g, x + g = g + x","context":["intro","ind","simp","cases"]} -{"prover":"Minlog","tactic_proof":"assume use elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_379","id":88878,"instance":379,"goal":"all p q e, (p + q) + e = p + (q + e)","context":["assume","use","elim","ind"]} -{"prover":"Minlog","tactic_proof":"cases cases ind intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_380","id":88879,"instance":380,"goal":"all g, g * 0 = 0","context":["cases","cases","ind","intro"]} -{"prover":"Minlog","tactic_proof":"ng elim use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_381","id":88880,"instance":381,"goal":"all p, p * 1 = p","context":["ng","elim","use","simp"]} -{"prover":"Minlog","tactic_proof":"assume simplify ind assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_382","id":88881,"instance":382,"goal":"all b g d, (b * g) * d = b * (g * d)","context":["assume","simplify","ind","assume"]} -{"prover":"Minlog","tactic_proof":"elim simplify ng simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_383","id":88882,"instance":383,"goal":"all y u, y * u = u * y","context":["elim","simplify","ng","simp"]} -{"prover":"Minlog","tactic_proof":"elim intro elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_384","id":88883,"instance":384,"goal":"all u x q, u * (x + q) = u * x + u * q","context":["elim","intro","elim","ng"]} -{"prover":"Minlog","tactic_proof":"cases intro intro rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_385","id":88884,"instance":385,"goal":"all a, a <= a","context":["cases","intro","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ind ind cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_386","id":88885,"instance":386,"goal":"all m y q, m <= y -> y <= q -> m <= q","context":["ind","ind","cases","intro"]} -{"prover":"Minlog","tactic_proof":"use ng elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_387","id":88886,"instance":387,"goal":"all d, not d < d","context":["use","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"simplify assume defn ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_388","id":88887,"instance":388,"goal":"all g v, Even g -> Even v -> Even (g + v)","context":["simplify","assume","defn","ind"]} -{"prover":"Minlog","tactic_proof":"ind ng ind cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_389","id":88888,"instance":389,"goal":"all x, x | x","context":["ind","ng","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simp defn rw simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_390","id":88889,"instance":390,"goal":"all y, Fact y > 0","context":["simp","defn","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"ind intro intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_391","id":88890,"instance":391,"goal":"all f b = b","context":["ind","intro","intro","assume"]} -{"prover":"Minlog","tactic_proof":"intro assume intro simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_392","id":88891,"instance":392,"goal":"all q + 0 = q","context":["intro","assume","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"rw assume simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_393","id":88892,"instance":393,"goal":"all a u, a + u = u + a","context":["rw","assume","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"elim defn simplify assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_394","id":88893,"instance":394,"goal":"all h f v, (h + f) + v = h + (f + v)","context":["elim","defn","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ng ind elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_395","id":88894,"instance":395,"goal":"all m, m * 0 = 0","context":["ng","ind","elim","rw"]} -{"prover":"Minlog","tactic_proof":"elim elim defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_396","id":88895,"instance":396,"goal":"all x, x * 1 = x","context":["elim","elim","defn","assume"]} -{"prover":"Minlog","tactic_proof":"intro defn simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_397","id":88896,"instance":397,"goal":"all p v x, (p * v) * x = p * (v * x)","context":["intro","defn","simp","ind"]} -{"prover":"Minlog","tactic_proof":"intro simplify use defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_398","id":88897,"instance":398,"goal":"all n a, n * a = a * n","context":["intro","simplify","use","defn"]} -{"prover":"Minlog","tactic_proof":"elim intro ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_399","id":88898,"instance":399,"goal":"all f q h, f * (q + h) = f * q + f * h","context":["elim","intro","ind","defn"]} -{"prover":"Minlog","tactic_proof":"rw ng assume elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_400","id":88899,"instance":400,"goal":"all q, q <= q","context":["rw","ng","assume","elim"]} -{"prover":"Minlog","tactic_proof":"ng elim ind use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_401","id":88900,"instance":401,"goal":"all f z n, f <= z -> z <= n -> f <= n","context":["ng","elim","ind","use"]} -{"prover":"Minlog","tactic_proof":"intro defn intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_402","id":88901,"instance":402,"goal":"all c, not c < c","context":["intro","defn","intro","simp"]} -{"prover":"Minlog","tactic_proof":"use intro simp elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_403","id":88902,"instance":403,"goal":"all c u, Even c -> Even u -> Even (c + u)","context":["use","intro","simp","elim"]} -{"prover":"Minlog","tactic_proof":"assume cases use elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_404","id":88903,"instance":404,"goal":"all n, n | n","context":["assume","cases","use","elim"]} -{"prover":"Minlog","tactic_proof":"elim use defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_405","id":88904,"instance":405,"goal":"all f, Fact f > 0","context":["elim","use","defn","simp"]} -{"prover":"Minlog","tactic_proof":"rw simplify ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_406","id":88905,"instance":406,"goal":"all c b = b","context":["rw","simplify","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ind elim elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_407","id":88906,"instance":407,"goal":"all c + 0 = c","context":["ind","elim","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ind assume ind ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_408","id":88907,"instance":408,"goal":"all m e, m + e = e + m","context":["ind","assume","ind","ng"]} -{"prover":"Minlog","tactic_proof":"intro defn intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_409","id":88908,"instance":409,"goal":"all e z h, (e + z) + h = e + (z + h)","context":["intro","defn","intro","cases"]} -{"prover":"Minlog","tactic_proof":"elim simplify elim ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_410","id":88909,"instance":410,"goal":"all e, e * 0 = 0","context":["elim","simplify","elim","ind"]} -{"prover":"Minlog","tactic_proof":"ng intro cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_411","id":88910,"instance":411,"goal":"all g, g * 1 = g","context":["ng","intro","cases","use"]} -{"prover":"Minlog","tactic_proof":"intro simplify defn simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_412","id":88911,"instance":412,"goal":"all d a g, (d * a) * g = d * (a * g)","context":["intro","simplify","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"rw ng defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_413","id":88912,"instance":413,"goal":"all c a, c * a = a * c","context":["rw","ng","defn","use"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_414","id":88913,"instance":414,"goal":"all h z p, h * (z + p) = h * z + h * p","context":["defn","simplify","rw","rw"]} -{"prover":"Minlog","tactic_proof":"simp cases cases rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_415","id":88914,"instance":415,"goal":"all v, v <= v","context":["simp","cases","cases","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn rw elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_416","id":88915,"instance":416,"goal":"all n f q, n <= f -> f <= q -> n <= q","context":["ng","defn","rw","elim"]} -{"prover":"Minlog","tactic_proof":"defn simp ng cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_417","id":88916,"instance":417,"goal":"all p, not p < p","context":["defn","simp","ng","cases"]} -{"prover":"Minlog","tactic_proof":"use intro assume elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_418","id":88917,"instance":418,"goal":"all a v, Even a -> Even v -> Even (a + v)","context":["use","intro","assume","elim"]} -{"prover":"Minlog","tactic_proof":"simplify intro rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_419","id":88918,"instance":419,"goal":"all f, f | f","context":["simplify","intro","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ng simp ind rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_420","id":88919,"instance":420,"goal":"all p, Fact p > 0","context":["ng","simp","ind","rw"]} -{"prover":"Minlog","tactic_proof":"elim assume cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_421","id":88920,"instance":421,"goal":"all u v = v","context":["elim","assume","cases","rw"]} -{"prover":"Minlog","tactic_proof":"use ng defn ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_422","id":88921,"instance":422,"goal":"all u + 0 = u","context":["use","ng","defn","ng"]} -{"prover":"Minlog","tactic_proof":"ng defn defn defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_423","id":88922,"instance":423,"goal":"all w x, w + x = x + w","context":["ng","defn","defn","defn"]} -{"prover":"Minlog","tactic_proof":"use simp assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_424","id":88923,"instance":424,"goal":"all y f m, (y + f) + m = y + (f + m)","context":["use","simp","assume","cases"]} -{"prover":"Minlog","tactic_proof":"intro assume elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_425","id":88924,"instance":425,"goal":"all b, b * 0 = 0","context":["intro","assume","elim","rw"]} -{"prover":"Minlog","tactic_proof":"simplify defn ng use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_426","id":88925,"instance":426,"goal":"all y, y * 1 = y","context":["simplify","defn","ng","use"]} -{"prover":"Minlog","tactic_proof":"ng ng assume simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_427","id":88926,"instance":427,"goal":"all q e w, (q * e) * w = q * (e * w)","context":["ng","ng","assume","simp"]} -{"prover":"Minlog","tactic_proof":"rw elim rw rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_428","id":88927,"instance":428,"goal":"all h p, h * p = p * h","context":["rw","elim","rw","rw"]} -{"prover":"Minlog","tactic_proof":"cases defn simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_429","id":88928,"instance":429,"goal":"all n y m, n * (y + m) = n * y + n * m","context":["cases","defn","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"rw use ng assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_430","id":88929,"instance":430,"goal":"all e, e <= e","context":["rw","use","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_431","id":88930,"instance":431,"goal":"all c x e, c <= x -> x <= e -> c <= e","context":["ind","ng","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"ind rw simplify simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_432","id":88931,"instance":432,"goal":"all b, not b < b","context":["ind","rw","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_433","id":88932,"instance":433,"goal":"all n h, Even n -> Even h -> Even (n + h)","context":["ind","ng","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"elim use ind simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_434","id":88933,"instance":434,"goal":"all q, q | q","context":["elim","use","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify intro simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_435","id":88934,"instance":435,"goal":"all q, Fact q > 0","context":["simplify","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng simplify rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_436","id":88935,"instance":436,"goal":"all w f = f","context":["elim","ng","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"assume elim rw elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_437","id":88936,"instance":437,"goal":"all z + 0 = z","context":["assume","elim","rw","elim"]} -{"prover":"Minlog","tactic_proof":"ng simp cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_438","id":88937,"instance":438,"goal":"all x u, x + u = u + x","context":["ng","simp","cases","elim"]} -{"prover":"Minlog","tactic_proof":"simplify ind defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_439","id":88938,"instance":439,"goal":"all x a g, (x + a) + g = x + (a + g)","context":["simplify","ind","defn","use"]} -{"prover":"Minlog","tactic_proof":"elim ind elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_440","id":88939,"instance":440,"goal":"all y, y * 0 = 0","context":["elim","ind","elim","rw"]} -{"prover":"Minlog","tactic_proof":"simp rw intro assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_441","id":88940,"instance":441,"goal":"all f, f * 1 = f","context":["simp","rw","intro","assume"]} -{"prover":"Minlog","tactic_proof":"elim simp cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_442","id":88941,"instance":442,"goal":"all u b c, (u * b) * c = u * (b * c)","context":["elim","simp","cases","intro"]} -{"prover":"Minlog","tactic_proof":"defn cases assume use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_443","id":88942,"instance":443,"goal":"all q z, q * z = z * q","context":["defn","cases","assume","use"]} -{"prover":"Minlog","tactic_proof":"intro ind rw elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_444","id":88943,"instance":444,"goal":"all c u h, c * (u + h) = c * u + c * h","context":["intro","ind","rw","elim"]} -{"prover":"Minlog","tactic_proof":"defn ng elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_445","id":88944,"instance":445,"goal":"all n, n <= n","context":["defn","ng","elim","cases"]} -{"prover":"Minlog","tactic_proof":"use defn simplify simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_446","id":88945,"instance":446,"goal":"all u b x, u <= b -> b <= x -> u <= x","context":["use","defn","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"use intro simplify intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_447","id":88946,"instance":447,"goal":"all w, not w < w","context":["use","intro","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases cases simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_448","id":88947,"instance":448,"goal":"all u b, Even u -> Even b -> Even (u + b)","context":["intro","cases","cases","simp"]} -{"prover":"Minlog","tactic_proof":"simp assume simplify ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_449","id":88948,"instance":449,"goal":"all h, h | h","context":["simp","assume","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"assume rw use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_450","id":88949,"instance":450,"goal":"all a, Fact a > 0","context":["assume","rw","use","cases"]} -{"prover":"Minlog","tactic_proof":"cases ng rw ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_451","id":88950,"instance":451,"goal":"all q e = e","context":["cases","ng","rw","ind"]} -{"prover":"Minlog","tactic_proof":"intro use use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_452","id":88951,"instance":452,"goal":"all h + 0 = h","context":["intro","use","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw defn assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_453","id":88952,"instance":453,"goal":"all u d, u + d = d + u","context":["rw","defn","assume","defn"]} -{"prover":"Minlog","tactic_proof":"ind intro elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_454","id":88953,"instance":454,"goal":"all e p q, (e + p) + q = e + (p + q)","context":["ind","intro","elim","defn"]} -{"prover":"Minlog","tactic_proof":"assume use defn simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_455","id":88954,"instance":455,"goal":"all g, g * 0 = 0","context":["assume","use","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"ind use assume simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_456","id":88955,"instance":456,"goal":"all z, z * 1 = z","context":["ind","use","assume","simp"]} -{"prover":"Minlog","tactic_proof":"defn rw assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_457","id":88956,"instance":457,"goal":"all v y u, (v * y) * u = v * (y * u)","context":["defn","rw","assume","rw"]} -{"prover":"Minlog","tactic_proof":"simplify simplify ind cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_458","id":88957,"instance":458,"goal":"all q d, q * d = d * q","context":["simplify","simplify","ind","cases"]} -{"prover":"Minlog","tactic_proof":"intro simp simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_459","id":88958,"instance":459,"goal":"all a b d, a * (b + d) = a * b + a * d","context":["intro","simp","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simp defn ind defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_460","id":88959,"instance":460,"goal":"all e, e <= e","context":["simp","defn","ind","defn"]} -{"prover":"Minlog","tactic_proof":"defn simplify elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_461","id":88960,"instance":461,"goal":"all b g p, b <= g -> g <= p -> b <= p","context":["defn","simplify","elim","intro"]} -{"prover":"Minlog","tactic_proof":"simp use defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_462","id":88961,"instance":462,"goal":"all f, not f < f","context":["simp","use","defn","assume"]} -{"prover":"Minlog","tactic_proof":"elim simp intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_463","id":88962,"instance":463,"goal":"all m g, Even m -> Even g -> Even (m + g)","context":["elim","simp","intro","ind"]} -{"prover":"Minlog","tactic_proof":"rw assume simp cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_464","id":88963,"instance":464,"goal":"all m, m | m","context":["rw","assume","simp","cases"]} -{"prover":"Minlog","tactic_proof":"rw use simplify use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_465","id":88964,"instance":465,"goal":"all w, Fact w > 0","context":["rw","use","simplify","use"]} -{"prover":"Minlog","tactic_proof":"defn rw simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_466","id":88965,"instance":466,"goal":"all n g = g","context":["defn","rw","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_467","id":88966,"instance":467,"goal":"all f + 0 = f","context":["intro","cases","ng","intro"]} -{"prover":"Minlog","tactic_proof":"intro intro elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_468","id":88967,"instance":468,"goal":"all a e, a + e = e + a","context":["intro","intro","elim","elim"]} -{"prover":"Minlog","tactic_proof":"cases ind ng ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_469","id":88968,"instance":469,"goal":"all h b f, (h + b) + f = h + (b + f)","context":["cases","ind","ng","ind"]} -{"prover":"Minlog","tactic_proof":"defn assume ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_470","id":88969,"instance":470,"goal":"all f, f * 0 = 0","context":["defn","assume","ind","intro"]} -{"prover":"Minlog","tactic_proof":"ng cases intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_471","id":88970,"instance":471,"goal":"all q, q * 1 = q","context":["ng","cases","intro","assume"]} -{"prover":"Minlog","tactic_proof":"intro use simp intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_472","id":88971,"instance":472,"goal":"all x y h, (x * y) * h = x * (y * h)","context":["intro","use","simp","intro"]} -{"prover":"Minlog","tactic_proof":"ng cases ind intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_473","id":88972,"instance":473,"goal":"all g h, g * h = h * g","context":["ng","cases","ind","intro"]} -{"prover":"Minlog","tactic_proof":"simp intro ind elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_474","id":88973,"instance":474,"goal":"all d b w, d * (b + w) = d * b + d * w","context":["simp","intro","ind","elim"]} -{"prover":"Minlog","tactic_proof":"assume cases cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_475","id":88974,"instance":475,"goal":"all z, z <= z","context":["assume","cases","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simplify defn elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_476","id":88975,"instance":476,"goal":"all n x g, n <= x -> x <= g -> n <= g","context":["simplify","defn","elim","cases"]} -{"prover":"Minlog","tactic_proof":"elim simp intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_477","id":88976,"instance":477,"goal":"all g, not g < g","context":["elim","simp","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use cases assume defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_478","id":88977,"instance":478,"goal":"all c d, Even c -> Even d -> Even (c + d)","context":["use","cases","assume","defn"]} -{"prover":"Minlog","tactic_proof":"simplify use assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_479","id":88978,"instance":479,"goal":"all m, m | m","context":["simplify","use","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ind cases cases simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_480","id":88979,"instance":480,"goal":"all g, Fact g > 0","context":["ind","cases","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify elim assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_481","id":88980,"instance":481,"goal":"all h a = a","context":["simplify","elim","assume","elim"]} -{"prover":"Minlog","tactic_proof":"ng cases rw use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_482","id":88981,"instance":482,"goal":"all n + 0 = n","context":["ng","cases","rw","use"]} -{"prover":"Minlog","tactic_proof":"cases cases intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_483","id":88982,"instance":483,"goal":"all m p, m + p = p + m","context":["cases","cases","intro","ind"]} -{"prover":"Minlog","tactic_proof":"elim cases use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_484","id":88983,"instance":484,"goal":"all h c q, (h + c) + q = h + (c + q)","context":["elim","cases","use","elim"]} -{"prover":"Minlog","tactic_proof":"intro assume cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_485","id":88984,"instance":485,"goal":"all z, z * 0 = 0","context":["intro","assume","cases","rw"]} -{"prover":"Minlog","tactic_proof":"use use intro defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_486","id":88985,"instance":486,"goal":"all n, n * 1 = n","context":["use","use","intro","defn"]} -{"prover":"Minlog","tactic_proof":"simplify cases simplify defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_487","id":88986,"instance":487,"goal":"all e x z, (e * x) * z = e * (x * z)","context":["simplify","cases","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"elim defn elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_488","id":88987,"instance":488,"goal":"all m q, m * q = q * m","context":["elim","defn","elim","defn"]} -{"prover":"Minlog","tactic_proof":"assume defn intro simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_489","id":88988,"instance":489,"goal":"all p c z, p * (c + z) = p * c + p * z","context":["assume","defn","intro","simp"]} -{"prover":"Minlog","tactic_proof":"cases use elim rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_490","id":88989,"instance":490,"goal":"all x, x <= x","context":["cases","use","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ng simp ng defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_491","id":88990,"instance":491,"goal":"all u d a, u <= d -> d <= a -> u <= a","context":["ng","simp","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simplify ind assume use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_492","id":88991,"instance":492,"goal":"all h, not h < h","context":["simplify","ind","assume","use"]} -{"prover":"Minlog","tactic_proof":"defn cases use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_493","id":88992,"instance":493,"goal":"all y b, Even y -> Even b -> Even (y + b)","context":["defn","cases","use","intro"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_494","id":88993,"instance":494,"goal":"all c, c | c","context":["intro","simplify","intro","use"]} -{"prover":"Minlog","tactic_proof":"simplify cases elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_495","id":88994,"instance":495,"goal":"all u, Fact u > 0","context":["simplify","cases","elim","simp"]} -{"prover":"Minlog","tactic_proof":"defn rw ng simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_496","id":88995,"instance":496,"goal":"all h x = x","context":["defn","rw","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"ng rw cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_497","id":88996,"instance":497,"goal":"all d + 0 = d","context":["ng","rw","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simp elim cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_498","id":88997,"instance":498,"goal":"all q g, q + g = g + q","context":["simp","elim","cases","use"]} -{"prover":"Minlog","tactic_proof":"simp simp assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_499","id":88998,"instance":499,"goal":"all w m p, (w + m) + p = w + (m + p)","context":["simp","simp","assume","elim"]} -{"prover":"Minlog","tactic_proof":"use intro simplify simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_500","id":88999,"instance":500,"goal":"all f, f * 0 = 0","context":["use","intro","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"ng cases elim simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_501","id":89000,"instance":501,"goal":"all p, p * 1 = p","context":["ng","cases","elim","simp"]} -{"prover":"Minlog","tactic_proof":"simp simp defn ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_502","id":89001,"instance":502,"goal":"all q f e, (q * f) * e = q * (f * e)","context":["simp","simp","defn","ind"]} -{"prover":"Minlog","tactic_proof":"simplify simp elim simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_503","id":89002,"instance":503,"goal":"all g b, g * b = b * g","context":["simplify","simp","elim","simp"]} -{"prover":"Minlog","tactic_proof":"assume simp simplify elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_504","id":89003,"instance":504,"goal":"all e n m, e * (n + m) = e * n + e * m","context":["assume","simp","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"elim simp simp simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_505","id":89004,"instance":505,"goal":"all d, d <= d","context":["elim","simp","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"intro defn ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_506","id":89005,"instance":506,"goal":"all c m w, c <= m -> m <= w -> c <= w","context":["intro","defn","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ind use ind ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_507","id":89006,"instance":507,"goal":"all p, not p < p","context":["ind","use","ind","ng"]} -{"prover":"Minlog","tactic_proof":"ind use cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_508","id":89007,"instance":508,"goal":"all h c, Even h -> Even c -> Even (h + c)","context":["ind","use","cases","cases"]} -{"prover":"Minlog","tactic_proof":"rw rw elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_509","id":89008,"instance":509,"goal":"all h, h | h","context":["rw","rw","elim","use"]} -{"prover":"Minlog","tactic_proof":"ng use rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_510","id":89009,"instance":510,"goal":"all y, Fact y > 0","context":["ng","use","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp intro ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_511","id":89010,"instance":511,"goal":"all x a = a","context":["simp","intro","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro ind simplify rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_512","id":89011,"instance":512,"goal":"all z + 0 = z","context":["intro","ind","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simp rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_513","id":89012,"instance":513,"goal":"all b n, b + n = n + b","context":["simplify","simplify","simp","rw"]} -{"prover":"Minlog","tactic_proof":"ind use ind use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_514","id":89013,"instance":514,"goal":"all e p g, (e + p) + g = e + (p + g)","context":["ind","use","ind","use"]} -{"prover":"Minlog","tactic_proof":"elim use rw simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_515","id":89014,"instance":515,"goal":"all b, b * 0 = 0","context":["elim","use","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"simp intro cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_516","id":89015,"instance":516,"goal":"all w, w * 1 = w","context":["simp","intro","cases","assume"]} -{"prover":"Minlog","tactic_proof":"cases ng rw simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_517","id":89016,"instance":517,"goal":"all d q v, (d * q) * v = d * (q * v)","context":["cases","ng","rw","simp"]} -{"prover":"Minlog","tactic_proof":"cases simplify ind assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_518","id":89017,"instance":518,"goal":"all a n, a * n = n * a","context":["cases","simplify","ind","assume"]} -{"prover":"Minlog","tactic_proof":"cases ng use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_519","id":89018,"instance":519,"goal":"all f e a, f * (e + a) = f * e + f * a","context":["cases","ng","use","defn"]} -{"prover":"Minlog","tactic_proof":"elim defn defn simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_520","id":89019,"instance":520,"goal":"all e, e <= e","context":["elim","defn","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify ng use defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_521","id":89020,"instance":521,"goal":"all a e n, a <= e -> e <= n -> a <= n","context":["simplify","ng","use","defn"]} -{"prover":"Minlog","tactic_proof":"simp rw ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_522","id":89021,"instance":522,"goal":"all e, not e < e","context":["simp","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"cases elim simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_523","id":89022,"instance":523,"goal":"all d h, Even d -> Even h -> Even (d + h)","context":["cases","elim","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"simplify elim simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_524","id":89023,"instance":524,"goal":"all v, v | v","context":["simplify","elim","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"rw cases use simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_525","id":89024,"instance":525,"goal":"all g, Fact g > 0","context":["rw","cases","use","simplify"]} -{"prover":"Minlog","tactic_proof":"ind intro intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_526","id":89025,"instance":526,"goal":"all f g = g","context":["ind","intro","intro","cases"]} -{"prover":"Minlog","tactic_proof":"rw simp defn rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_527","id":89026,"instance":527,"goal":"all a + 0 = a","context":["rw","simp","defn","rw"]} -{"prover":"Minlog","tactic_proof":"assume elim intro intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_528","id":89027,"instance":528,"goal":"all c f, c + f = f + c","context":["assume","elim","intro","intro"]} -{"prover":"Minlog","tactic_proof":"simp simp use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_529","id":89028,"instance":529,"goal":"all w v x, (w + v) + x = w + (v + x)","context":["simp","simp","use","assume"]} -{"prover":"Minlog","tactic_proof":"use cases assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_530","id":89029,"instance":530,"goal":"all f, f * 0 = 0","context":["use","cases","assume","ng"]} -{"prover":"Minlog","tactic_proof":"assume rw use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_531","id":89030,"instance":531,"goal":"all z, z * 1 = z","context":["assume","rw","use","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify use simp rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_532","id":89031,"instance":532,"goal":"all a g n, (a * g) * n = a * (g * n)","context":["simplify","use","simp","rw"]} -{"prover":"Minlog","tactic_proof":"assume rw defn use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_533","id":89032,"instance":533,"goal":"all p d, p * d = d * p","context":["assume","rw","defn","use"]} -{"prover":"Minlog","tactic_proof":"cases simp ind rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_534","id":89033,"instance":534,"goal":"all z a w, z * (a + w) = z * a + z * w","context":["cases","simp","ind","rw"]} -{"prover":"Minlog","tactic_proof":"elim simp simp rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_535","id":89034,"instance":535,"goal":"all u, u <= u","context":["elim","simp","simp","rw"]} -{"prover":"Minlog","tactic_proof":"rw elim defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_536","id":89035,"instance":536,"goal":"all c g e, c <= g -> g <= e -> c <= e","context":["rw","elim","defn","use"]} -{"prover":"Minlog","tactic_proof":"defn elim ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_537","id":89036,"instance":537,"goal":"all h, not h < h","context":["defn","elim","ind","ind"]} -{"prover":"Minlog","tactic_proof":"assume elim simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_538","id":89037,"instance":538,"goal":"all m p, Even m -> Even p -> Even (m + p)","context":["assume","elim","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ng rw assume rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_539","id":89038,"instance":539,"goal":"all m, m | m","context":["ng","rw","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ind intro simp assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_540","id":89039,"instance":540,"goal":"all e, Fact e > 0","context":["ind","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"ng simp simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_541","id":89040,"instance":541,"goal":"all c e = e","context":["ng","simp","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"rw elim ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_542","id":89041,"instance":542,"goal":"all q + 0 = q","context":["rw","elim","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro use ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_543","id":89042,"instance":543,"goal":"all p x, p + x = x + p","context":["intro","use","ind","defn"]} -{"prover":"Minlog","tactic_proof":"ind ind elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_544","id":89043,"instance":544,"goal":"all b n v, (b + n) + v = b + (n + v)","context":["ind","ind","elim","simp"]} -{"prover":"Minlog","tactic_proof":"use ng rw cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_545","id":89044,"instance":545,"goal":"all m, m * 0 = 0","context":["use","ng","rw","cases"]} -{"prover":"Minlog","tactic_proof":"ind cases rw elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_546","id":89045,"instance":546,"goal":"all a, a * 1 = a","context":["ind","cases","rw","elim"]} -{"prover":"Minlog","tactic_proof":"elim use defn ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_547","id":89046,"instance":547,"goal":"all b n z, (b * n) * z = b * (n * z)","context":["elim","use","defn","ng"]} -{"prover":"Minlog","tactic_proof":"defn use rw intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_548","id":89047,"instance":548,"goal":"all m n, m * n = n * m","context":["defn","use","rw","intro"]} -{"prover":"Minlog","tactic_proof":"ind assume intro simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_549","id":89048,"instance":549,"goal":"all u a e, u * (a + e) = u * a + u * e","context":["ind","assume","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp ind intro simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_550","id":89049,"instance":550,"goal":"all n, n <= n","context":["simp","ind","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simplify cases simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_551","id":89050,"instance":551,"goal":"all a u m, a <= u -> u <= m -> a <= m","context":["simplify","cases","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"assume cases simp simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_552","id":89051,"instance":552,"goal":"all y, not y < y","context":["assume","cases","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"elim defn intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_553","id":89052,"instance":553,"goal":"all c m, Even c -> Even m -> Even (c + m)","context":["elim","defn","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use intro ind use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_554","id":89053,"instance":554,"goal":"all y, y | y","context":["use","intro","ind","use"]} -{"prover":"Minlog","tactic_proof":"cases simp elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_555","id":89054,"instance":555,"goal":"all p, Fact p > 0","context":["cases","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_556","id":89055,"instance":556,"goal":"all b y = y","context":["intro","cases","cases","use"]} -{"prover":"Minlog","tactic_proof":"elim defn ind simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_557","id":89056,"instance":557,"goal":"all x + 0 = x","context":["elim","defn","ind","simp"]} -{"prover":"Minlog","tactic_proof":"intro elim elim rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_558","id":89057,"instance":558,"goal":"all e z, e + z = z + e","context":["intro","elim","elim","rw"]} -{"prover":"Minlog","tactic_proof":"simp intro elim rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_559","id":89058,"instance":559,"goal":"all h g f, (h + g) + f = h + (g + f)","context":["simp","intro","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ind defn simplify intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_560","id":89059,"instance":560,"goal":"all y, y * 0 = 0","context":["ind","defn","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"ng simplify cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_561","id":89060,"instance":561,"goal":"all x, x * 1 = x","context":["ng","simplify","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"elim defn ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_562","id":89061,"instance":562,"goal":"all d m e, (d * m) * e = d * (m * e)","context":["elim","defn","ng","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_563","id":89062,"instance":563,"goal":"all f e, f * e = e * f","context":["elim","ng","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng intro elim intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_564","id":89063,"instance":564,"goal":"all h v a, h * (v + a) = h * v + h * a","context":["ng","intro","elim","intro"]} -{"prover":"Minlog","tactic_proof":"elim intro simp assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_565","id":89064,"instance":565,"goal":"all u, u <= u","context":["elim","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"ng simp cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_566","id":89065,"instance":566,"goal":"all d z b, d <= z -> z <= b -> d <= b","context":["ng","simp","cases","elim"]} -{"prover":"Minlog","tactic_proof":"elim ind use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_567","id":89066,"instance":567,"goal":"all c, not c < c","context":["elim","ind","use","cases"]} -{"prover":"Minlog","tactic_proof":"rw ng simplify defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_568","id":89067,"instance":568,"goal":"all d x, Even d -> Even x -> Even (d + x)","context":["rw","ng","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"cases rw simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_569","id":89068,"instance":569,"goal":"all m, m | m","context":["cases","rw","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"defn cases elim defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_570","id":89069,"instance":570,"goal":"all d, Fact d > 0","context":["defn","cases","elim","defn"]} -{"prover":"Minlog","tactic_proof":"rw rw elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_571","id":89070,"instance":571,"goal":"all w b = b","context":["rw","rw","elim","intro"]} -{"prover":"Minlog","tactic_proof":"simplify assume use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_572","id":89071,"instance":572,"goal":"all u + 0 = u","context":["simplify","assume","use","assume"]} -{"prover":"Minlog","tactic_proof":"use elim ind assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_573","id":89072,"instance":573,"goal":"all p m, p + m = m + p","context":["use","elim","ind","assume"]} -{"prover":"Minlog","tactic_proof":"use defn simp simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_574","id":89073,"instance":574,"goal":"all v c x, (v + c) + x = v + (c + x)","context":["use","defn","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"ind simplify use defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_575","id":89074,"instance":575,"goal":"all p, p * 0 = 0","context":["ind","simplify","use","defn"]} -{"prover":"Minlog","tactic_proof":"simp use intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_576","id":89075,"instance":576,"goal":"all d, d * 1 = d","context":["simp","use","intro","defn"]} -{"prover":"Minlog","tactic_proof":"cases rw ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_577","id":89076,"instance":577,"goal":"all x y a, (x * y) * a = x * (y * a)","context":["cases","rw","ng","simp"]} -{"prover":"Minlog","tactic_proof":"defn simp defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_578","id":89077,"instance":578,"goal":"all d x, d * x = x * d","context":["defn","simp","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify ind intro intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_579","id":89078,"instance":579,"goal":"all a h b, a * (h + b) = a * h + a * b","context":["simplify","ind","intro","intro"]} -{"prover":"Minlog","tactic_proof":"use defn ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_580","id":89079,"instance":580,"goal":"all n, n <= n","context":["use","defn","ng","defn"]} -{"prover":"Minlog","tactic_proof":"ind ng use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_581","id":89080,"instance":581,"goal":"all w v h, w <= v -> v <= h -> w <= h","context":["ind","ng","use","cases"]} -{"prover":"Minlog","tactic_proof":"use ind cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_582","id":89081,"instance":582,"goal":"all z, not z < z","context":["use","ind","cases","intro"]} -{"prover":"Minlog","tactic_proof":"intro ng cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_583","id":89082,"instance":583,"goal":"all m d, Even m -> Even d -> Even (m + d)","context":["intro","ng","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use ng defn ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_584","id":89083,"instance":584,"goal":"all f, f | f","context":["use","ng","defn","ng"]} -{"prover":"Minlog","tactic_proof":"use intro intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_585","id":89084,"instance":585,"goal":"all u, Fact u > 0","context":["use","intro","intro","use"]} -{"prover":"Minlog","tactic_proof":"intro ind intro defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_586","id":89085,"instance":586,"goal":"all n h = h","context":["intro","ind","intro","defn"]} -{"prover":"Minlog","tactic_proof":"assume simp elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_587","id":89086,"instance":587,"goal":"all e + 0 = e","context":["assume","simp","elim","rw"]} -{"prover":"Minlog","tactic_proof":"use simplify assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_588","id":89087,"instance":588,"goal":"all c q, c + q = q + c","context":["use","simplify","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ind intro simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_589","id":89088,"instance":589,"goal":"all w y n, (w + y) + n = w + (y + n)","context":["ind","intro","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"simp defn rw intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_590","id":89089,"instance":590,"goal":"all n, n * 0 = 0","context":["simp","defn","rw","intro"]} -{"prover":"Minlog","tactic_proof":"intro defn elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_591","id":89090,"instance":591,"goal":"all g, g * 1 = g","context":["intro","defn","elim","elim"]} -{"prover":"Minlog","tactic_proof":"intro rw rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_592","id":89091,"instance":592,"goal":"all x h v, (x * h) * v = x * (h * v)","context":["intro","rw","rw","simp"]} -{"prover":"Minlog","tactic_proof":"use ind assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_593","id":89092,"instance":593,"goal":"all n p, n * p = p * n","context":["use","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"ind intro intro intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_594","id":89093,"instance":594,"goal":"all v m x, v * (m + x) = v * m + v * x","context":["ind","intro","intro","intro"]} -{"prover":"Minlog","tactic_proof":"simp simp defn intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_595","id":89094,"instance":595,"goal":"all e, e <= e","context":["simp","simp","defn","intro"]} -{"prover":"Minlog","tactic_proof":"ng ind use use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_596","id":89095,"instance":596,"goal":"all c y b, c <= y -> y <= b -> c <= b","context":["ng","ind","use","use"]} -{"prover":"Minlog","tactic_proof":"simplify simp rw simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_597","id":89096,"instance":597,"goal":"all x, not x < x","context":["simplify","simp","rw","simp"]} -{"prover":"Minlog","tactic_proof":"assume ind assume defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_598","id":89097,"instance":598,"goal":"all w e, Even w -> Even e -> Even (w + e)","context":["assume","ind","assume","defn"]} -{"prover":"Minlog","tactic_proof":"rw simp simplify simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_599","id":89098,"instance":599,"goal":"all z, z | z","context":["rw","simp","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"defn assume ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_600","id":89099,"instance":600,"goal":"all v, Fact v > 0","context":["defn","assume","ng","cases"]} -{"prover":"Minlog","tactic_proof":"assume use cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_601","id":89100,"instance":601,"goal":"all u p = p","context":["assume","use","cases","ng"]} -{"prover":"Minlog","tactic_proof":"ind elim intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_602","id":89101,"instance":602,"goal":"all x + 0 = x","context":["ind","elim","intro","defn"]} -{"prover":"Minlog","tactic_proof":"rw simp simp assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_603","id":89102,"instance":603,"goal":"all y n, y + n = n + y","context":["rw","simp","simp","assume"]} -{"prover":"Minlog","tactic_proof":"ng defn simplify simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_604","id":89103,"instance":604,"goal":"all a n x, (a + n) + x = a + (n + x)","context":["ng","defn","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"ind elim ind simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_605","id":89104,"instance":605,"goal":"all p, p * 0 = 0","context":["ind","elim","ind","simp"]} -{"prover":"Minlog","tactic_proof":"cases simp defn assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_606","id":89105,"instance":606,"goal":"all h, h * 1 = h","context":["cases","simp","defn","assume"]} -{"prover":"Minlog","tactic_proof":"use intro rw use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_607","id":89106,"instance":607,"goal":"all p z c, (p * z) * c = p * (z * c)","context":["use","intro","rw","use"]} -{"prover":"Minlog","tactic_proof":"use elim assume ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_608","id":89107,"instance":608,"goal":"all x b, x * b = b * x","context":["use","elim","assume","ng"]} -{"prover":"Minlog","tactic_proof":"use ind cases intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_609","id":89108,"instance":609,"goal":"all a v y, a * (v + y) = a * v + a * y","context":["use","ind","cases","intro"]} -{"prover":"Minlog","tactic_proof":"use ind elim assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_610","id":89109,"instance":610,"goal":"all e, e <= e","context":["use","ind","elim","assume"]} -{"prover":"Minlog","tactic_proof":"cases cases ind ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_611","id":89110,"instance":611,"goal":"all p c a, p <= c -> c <= a -> p <= a","context":["cases","cases","ind","ind"]} -{"prover":"Minlog","tactic_proof":"assume simp intro assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_612","id":89111,"instance":612,"goal":"all f, not f < f","context":["assume","simp","intro","assume"]} -{"prover":"Minlog","tactic_proof":"cases intro simplify rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_613","id":89112,"instance":613,"goal":"all v q, Even v -> Even q -> Even (v + q)","context":["cases","intro","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"ind ind simplify defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_614","id":89113,"instance":614,"goal":"all a, a | a","context":["ind","ind","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"assume ng intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_615","id":89114,"instance":615,"goal":"all w, Fact w > 0","context":["assume","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"cases defn rw rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_616","id":89115,"instance":616,"goal":"all x y = y","context":["cases","defn","rw","rw"]} -{"prover":"Minlog","tactic_proof":"elim ind simp ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_617","id":89116,"instance":617,"goal":"all q + 0 = q","context":["elim","ind","simp","ind"]} -{"prover":"Minlog","tactic_proof":"rw ng rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_618","id":89117,"instance":618,"goal":"all h g, h + g = g + h","context":["rw","ng","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ng ng cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_619","id":89118,"instance":619,"goal":"all a q w, (a + q) + w = a + (q + w)","context":["ng","ng","cases","elim"]} -{"prover":"Minlog","tactic_proof":"simp rw rw ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_620","id":89119,"instance":620,"goal":"all f, f * 0 = 0","context":["simp","rw","rw","ind"]} -{"prover":"Minlog","tactic_proof":"elim assume simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_621","id":89120,"instance":621,"goal":"all u, u * 1 = u","context":["elim","assume","simp","use"]} -{"prover":"Minlog","tactic_proof":"ng intro rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_622","id":89121,"instance":622,"goal":"all y b x, (y * b) * x = y * (b * x)","context":["ng","intro","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"use ind simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_623","id":89122,"instance":623,"goal":"all w d, w * d = d * w","context":["use","ind","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"simp defn assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_624","id":89123,"instance":624,"goal":"all f g e, f * (g + e) = f * g + f * e","context":["simp","defn","assume","cases"]} -{"prover":"Minlog","tactic_proof":"cases simplify rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_625","id":89124,"instance":625,"goal":"all x, x <= x","context":["cases","simplify","rw","simp"]} -{"prover":"Minlog","tactic_proof":"cases elim assume assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_626","id":89125,"instance":626,"goal":"all p w v, p <= w -> w <= v -> p <= v","context":["cases","elim","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ng assume defn ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_627","id":89126,"instance":627,"goal":"all u, not u < u","context":["ng","assume","defn","ind"]} -{"prover":"Minlog","tactic_proof":"ng cases use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_628","id":89127,"instance":628,"goal":"all v x, Even v -> Even x -> Even (v + x)","context":["ng","cases","use","defn"]} -{"prover":"Minlog","tactic_proof":"simp assume ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_629","id":89128,"instance":629,"goal":"all u, u | u","context":["simp","assume","ind","ind"]} -{"prover":"Minlog","tactic_proof":"use use cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_630","id":89129,"instance":630,"goal":"all n, Fact n > 0","context":["use","use","cases","simp"]} -{"prover":"Minlog","tactic_proof":"defn cases cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_631","id":89130,"instance":631,"goal":"all y f = f","context":["defn","cases","cases","defn"]} -{"prover":"Minlog","tactic_proof":"simplify defn simp rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_632","id":89131,"instance":632,"goal":"all q + 0 = q","context":["simplify","defn","simp","rw"]} -{"prover":"Minlog","tactic_proof":"cases ng intro simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_633","id":89132,"instance":633,"goal":"all a v, a + v = v + a","context":["cases","ng","intro","simp"]} -{"prover":"Minlog","tactic_proof":"intro elim elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_634","id":89133,"instance":634,"goal":"all v h b, (v + h) + b = v + (h + b)","context":["intro","elim","elim","assume"]} -{"prover":"Minlog","tactic_proof":"ng use simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_635","id":89134,"instance":635,"goal":"all z, z * 0 = 0","context":["ng","use","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ng intro cases defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_636","id":89135,"instance":636,"goal":"all g, g * 1 = g","context":["ng","intro","cases","defn"]} -{"prover":"Minlog","tactic_proof":"defn elim use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_637","id":89136,"instance":637,"goal":"all c u e, (c * u) * e = c * (u * e)","context":["defn","elim","use","use"]} -{"prover":"Minlog","tactic_proof":"elim cases defn ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_638","id":89137,"instance":638,"goal":"all w h, w * h = h * w","context":["elim","cases","defn","ng"]} -{"prover":"Minlog","tactic_proof":"elim simp simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_639","id":89138,"instance":639,"goal":"all a n p, a * (n + p) = a * n + a * p","context":["elim","simp","simplify","use"]} -{"prover":"Minlog","tactic_proof":"intro rw simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_640","id":89139,"instance":640,"goal":"all w, w <= w","context":["intro","rw","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"assume simplify simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_641","id":89140,"instance":641,"goal":"all q z b, q <= z -> z <= b -> q <= b","context":["assume","simplify","simp","ng"]} -{"prover":"Minlog","tactic_proof":"assume simplify use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_642","id":89141,"instance":642,"goal":"all d, not d < d","context":["assume","simplify","use","intro"]} -{"prover":"Minlog","tactic_proof":"intro defn elim ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_643","id":89142,"instance":643,"goal":"all e n, Even e -> Even n -> Even (e + n)","context":["intro","defn","elim","ng"]} -{"prover":"Minlog","tactic_proof":"rw intro elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_644","id":89143,"instance":644,"goal":"all q, q | q","context":["rw","intro","elim","elim"]} -{"prover":"Minlog","tactic_proof":"simp use rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_645","id":89144,"instance":645,"goal":"all p, Fact p > 0","context":["simp","use","rw","simp"]} -{"prover":"Minlog","tactic_proof":"cases ng ind elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_646","id":89145,"instance":646,"goal":"all v n = n","context":["cases","ng","ind","elim"]} -{"prover":"Minlog","tactic_proof":"ind simplify ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_647","id":89146,"instance":647,"goal":"all v + 0 = v","context":["ind","simplify","ng","ind"]} -{"prover":"Minlog","tactic_proof":"elim ng rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_648","id":89147,"instance":648,"goal":"all e b, e + b = b + e","context":["elim","ng","rw","assume"]} -{"prover":"Minlog","tactic_proof":"elim assume ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_649","id":89148,"instance":649,"goal":"all u d e, (u + d) + e = u + (d + e)","context":["elim","assume","ind","ind"]} -{"prover":"Minlog","tactic_proof":"intro ng assume ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_650","id":89149,"instance":650,"goal":"all a, a * 0 = 0","context":["intro","ng","assume","ind"]} -{"prover":"Minlog","tactic_proof":"assume elim ng ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_651","id":89150,"instance":651,"goal":"all w, w * 1 = w","context":["assume","elim","ng","ng"]} -{"prover":"Minlog","tactic_proof":"intro use ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_652","id":89151,"instance":652,"goal":"all b n h, (b * n) * h = b * (n * h)","context":["intro","use","ind","rw"]} -{"prover":"Minlog","tactic_proof":"defn assume simplify ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_653","id":89152,"instance":653,"goal":"all m v, m * v = v * m","context":["defn","assume","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"rw ind elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_654","id":89153,"instance":654,"goal":"all f z h, f * (z + h) = f * z + f * h","context":["rw","ind","elim","use"]} -{"prover":"Minlog","tactic_proof":"ind simplify rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_655","id":89154,"instance":655,"goal":"all e, e <= e","context":["ind","simplify","rw","simp"]} -{"prover":"Minlog","tactic_proof":"cases ind use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_656","id":89155,"instance":656,"goal":"all a n d, a <= n -> n <= d -> a <= d","context":["cases","ind","use","cases"]} -{"prover":"Minlog","tactic_proof":"assume use rw cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_657","id":89156,"instance":657,"goal":"all m, not m < m","context":["assume","use","rw","cases"]} -{"prover":"Minlog","tactic_proof":"elim ng simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_658","id":89157,"instance":658,"goal":"all g w, Even g -> Even w -> Even (g + w)","context":["elim","ng","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"assume simp simp simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_659","id":89158,"instance":659,"goal":"all y, y | y","context":["assume","simp","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"assume use simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_660","id":89159,"instance":660,"goal":"all u, Fact u > 0","context":["assume","use","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ind simplify defn ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_661","id":89160,"instance":661,"goal":"all m x = x","context":["ind","simplify","defn","ng"]} -{"prover":"Minlog","tactic_proof":"assume ng cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_662","id":89161,"instance":662,"goal":"all a + 0 = a","context":["assume","ng","cases","ind"]} -{"prover":"Minlog","tactic_proof":"cases use intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_663","id":89162,"instance":663,"goal":"all m h, m + h = h + m","context":["cases","use","intro","ind"]} -{"prover":"Minlog","tactic_proof":"simplify assume rw cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_664","id":89163,"instance":664,"goal":"all e p q, (e + p) + q = e + (p + q)","context":["simplify","assume","rw","cases"]} -{"prover":"Minlog","tactic_proof":"ng rw cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_665","id":89164,"instance":665,"goal":"all n, n * 0 = 0","context":["ng","rw","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simp cases use use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_666","id":89165,"instance":666,"goal":"all b, b * 1 = b","context":["simp","cases","use","use"]} -{"prover":"Minlog","tactic_proof":"simplify use elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_667","id":89166,"instance":667,"goal":"all v x p, (v * x) * p = v * (x * p)","context":["simplify","use","elim","ind"]} -{"prover":"Minlog","tactic_proof":"intro assume intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_668","id":89167,"instance":668,"goal":"all y q, y * q = q * y","context":["intro","assume","intro","elim"]} -{"prover":"Minlog","tactic_proof":"elim intro ind ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_669","id":89168,"instance":669,"goal":"all x c w, x * (c + w) = x * c + x * w","context":["elim","intro","ind","ng"]} -{"prover":"Minlog","tactic_proof":"use assume elim simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_670","id":89169,"instance":670,"goal":"all g, g <= g","context":["use","assume","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify simplify ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_671","id":89170,"instance":671,"goal":"all a m q, a <= m -> m <= q -> a <= q","context":["simplify","simplify","ind","rw"]} -{"prover":"Minlog","tactic_proof":"ind elim assume rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_672","id":89171,"instance":672,"goal":"all p, not p < p","context":["ind","elim","assume","rw"]} -{"prover":"Minlog","tactic_proof":"cases simp ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_673","id":89172,"instance":673,"goal":"all x g, Even x -> Even g -> Even (x + g)","context":["cases","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"ind ng intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_674","id":89173,"instance":674,"goal":"all f, f | f","context":["ind","ng","intro","assume"]} -{"prover":"Minlog","tactic_proof":"ng intro assume simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_675","id":89174,"instance":675,"goal":"all d, Fact d > 0","context":["ng","intro","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"ind elim assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_676","id":89175,"instance":676,"goal":"all c p = p","context":["ind","elim","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ind assume use simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_677","id":89176,"instance":677,"goal":"all p + 0 = p","context":["ind","assume","use","simplify"]} -{"prover":"Minlog","tactic_proof":"use ng simplify ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_678","id":89177,"instance":678,"goal":"all u n, u + n = n + u","context":["use","ng","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"simp simplify assume simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_679","id":89178,"instance":679,"goal":"all z a f, (z + a) + f = z + (a + f)","context":["simp","simplify","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify cases ind ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_680","id":89179,"instance":680,"goal":"all c, c * 0 = 0","context":["simplify","cases","ind","ind"]} -{"prover":"Minlog","tactic_proof":"assume ind ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_681","id":89180,"instance":681,"goal":"all q, q * 1 = q","context":["assume","ind","ng","cases"]} -{"prover":"Minlog","tactic_proof":"rw use intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_682","id":89181,"instance":682,"goal":"all y v d, (y * v) * d = y * (v * d)","context":["rw","use","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ng simplify ind simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_683","id":89182,"instance":683,"goal":"all c p, c * p = p * c","context":["ng","simplify","ind","simp"]} -{"prover":"Minlog","tactic_proof":"cases rw defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_684","id":89183,"instance":684,"goal":"all y z q, y * (z + q) = y * z + y * q","context":["cases","rw","defn","cases"]} -{"prover":"Minlog","tactic_proof":"use assume cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_685","id":89184,"instance":685,"goal":"all d, d <= d","context":["use","assume","cases","ind"]} -{"prover":"Minlog","tactic_proof":"ind cases intro cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_686","id":89185,"instance":686,"goal":"all c v z, c <= v -> v <= z -> c <= z","context":["ind","cases","intro","cases"]} -{"prover":"Minlog","tactic_proof":"assume ng intro ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_687","id":89186,"instance":687,"goal":"all b, not b < b","context":["assume","ng","intro","ng"]} -{"prover":"Minlog","tactic_proof":"use elim ng elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_688","id":89187,"instance":688,"goal":"all g z, Even g -> Even z -> Even (g + z)","context":["use","elim","ng","elim"]} -{"prover":"Minlog","tactic_proof":"rw rw intro rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_689","id":89188,"instance":689,"goal":"all h, h | h","context":["rw","rw","intro","rw"]} -{"prover":"Minlog","tactic_proof":"assume simplify ng cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_690","id":89189,"instance":690,"goal":"all a, Fact a > 0","context":["assume","simplify","ng","cases"]} -{"prover":"Minlog","tactic_proof":"ng assume simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_691","id":89190,"instance":691,"goal":"all f c = c","context":["ng","assume","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"intro rw use simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_692","id":89191,"instance":692,"goal":"all q + 0 = q","context":["intro","rw","use","simplify"]} -{"prover":"Minlog","tactic_proof":"cases use use elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_693","id":89192,"instance":693,"goal":"all n w, n + w = w + n","context":["cases","use","use","elim"]} -{"prover":"Minlog","tactic_proof":"intro ind cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_694","id":89193,"instance":694,"goal":"all p b y, (p + b) + y = p + (b + y)","context":["intro","ind","cases","defn"]} -{"prover":"Minlog","tactic_proof":"intro ng elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_695","id":89194,"instance":695,"goal":"all a, a * 0 = 0","context":["intro","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn ng rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_696","id":89195,"instance":696,"goal":"all m, m * 1 = m","context":["ng","defn","ng","rw"]} -{"prover":"Minlog","tactic_proof":"assume ind assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_697","id":89196,"instance":697,"goal":"all w n y, (w * n) * y = w * (n * y)","context":["assume","ind","assume","defn"]} -{"prover":"Minlog","tactic_proof":"intro simplify assume use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_698","id":89197,"instance":698,"goal":"all a d, a * d = d * a","context":["intro","simplify","assume","use"]} -{"prover":"Minlog","tactic_proof":"elim ng assume defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_699","id":89198,"instance":699,"goal":"all v z e, v * (z + e) = v * z + v * e","context":["elim","ng","assume","defn"]} -{"prover":"Minlog","tactic_proof":"ind simp cases simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_700","id":89199,"instance":700,"goal":"all m, m <= m","context":["ind","simp","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"ng ind ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_701","id":89200,"instance":701,"goal":"all a b n, a <= b -> b <= n -> a <= n","context":["ng","ind","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simp simplify ng ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_702","id":89201,"instance":702,"goal":"all c, not c < c","context":["simp","simplify","ng","ind"]} -{"prover":"Minlog","tactic_proof":"cases rw rw elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_703","id":89202,"instance":703,"goal":"all f q, Even f -> Even q -> Even (f + q)","context":["cases","rw","rw","elim"]} -{"prover":"Minlog","tactic_proof":"intro defn simp ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_704","id":89203,"instance":704,"goal":"all m, m | m","context":["intro","defn","simp","ind"]} -{"prover":"Minlog","tactic_proof":"intro ng use ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_705","id":89204,"instance":705,"goal":"all a, Fact a > 0","context":["intro","ng","use","ng"]} -{"prover":"Minlog","tactic_proof":"simp ng ng rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_706","id":89205,"instance":706,"goal":"all p x = x","context":["simp","ng","ng","rw"]} -{"prover":"Minlog","tactic_proof":"intro cases simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_707","id":89206,"instance":707,"goal":"all m + 0 = m","context":["intro","cases","simplify","use"]} -{"prover":"Minlog","tactic_proof":"simplify intro ind ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_708","id":89207,"instance":708,"goal":"all q b, q + b = b + q","context":["simplify","intro","ind","ind"]} -{"prover":"Minlog","tactic_proof":"cases simplify simplify cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_709","id":89208,"instance":709,"goal":"all w g q, (w + g) + q = w + (g + q)","context":["cases","simplify","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"use defn cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_710","id":89209,"instance":710,"goal":"all u, u * 0 = 0","context":["use","defn","cases","assume"]} -{"prover":"Minlog","tactic_proof":"rw use use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_711","id":89210,"instance":711,"goal":"all w, w * 1 = w","context":["rw","use","use","ng"]} -{"prover":"Minlog","tactic_proof":"defn use defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_712","id":89211,"instance":712,"goal":"all z h e, (z * h) * e = z * (h * e)","context":["defn","use","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify cases simp rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_713","id":89212,"instance":713,"goal":"all x d, x * d = d * x","context":["simplify","cases","simp","rw"]} -{"prover":"Minlog","tactic_proof":"use ng cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_714","id":89213,"instance":714,"goal":"all u d b, u * (d + b) = u * d + u * b","context":["use","ng","cases","ng"]} -{"prover":"Minlog","tactic_proof":"ind rw simp intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_715","id":89214,"instance":715,"goal":"all m, m <= m","context":["ind","rw","simp","intro"]} -{"prover":"Minlog","tactic_proof":"ind cases defn simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_716","id":89215,"instance":716,"goal":"all v e n, v <= e -> e <= n -> v <= n","context":["ind","cases","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify assume rw rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_717","id":89216,"instance":717,"goal":"all g, not g < g","context":["simplify","assume","rw","rw"]} -{"prover":"Minlog","tactic_proof":"assume ind simplify rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_718","id":89217,"instance":718,"goal":"all q m, Even q -> Even m -> Even (q + m)","context":["assume","ind","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"rw assume ng ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_719","id":89218,"instance":719,"goal":"all a, a | a","context":["rw","assume","ng","ind"]} -{"prover":"Minlog","tactic_proof":"use use use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_720","id":89219,"instance":720,"goal":"all q, Fact q > 0","context":["use","use","use","cases"]} -{"prover":"Minlog","tactic_proof":"simp cases simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_721","id":89220,"instance":721,"goal":"all u c = c","context":["simp","cases","simplify","use"]} -{"prover":"Minlog","tactic_proof":"use ind simp simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_722","id":89221,"instance":722,"goal":"all b + 0 = b","context":["use","ind","simp","simp"]} -{"prover":"Minlog","tactic_proof":"rw ng simplify ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_723","id":89222,"instance":723,"goal":"all w y, w + y = y + w","context":["rw","ng","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ng defn assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_724","id":89223,"instance":724,"goal":"all g y m, (g + y) + m = g + (y + m)","context":["ng","defn","assume","intro"]} -{"prover":"Minlog","tactic_proof":"defn assume cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_725","id":89224,"instance":725,"goal":"all x, x * 0 = 0","context":["defn","assume","cases","elim"]} -{"prover":"Minlog","tactic_proof":"intro simplify rw use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_726","id":89225,"instance":726,"goal":"all y, y * 1 = y","context":["intro","simplify","rw","use"]} -{"prover":"Minlog","tactic_proof":"intro intro simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_727","id":89226,"instance":727,"goal":"all m a u, (m * a) * u = m * (a * u)","context":["intro","intro","simp","use"]} -{"prover":"Minlog","tactic_proof":"simplify ng elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_728","id":89227,"instance":728,"goal":"all a h, a * h = h * a","context":["simplify","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"defn elim assume simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_729","id":89228,"instance":729,"goal":"all v n a, v * (n + a) = v * n + v * a","context":["defn","elim","assume","simp"]} -{"prover":"Minlog","tactic_proof":"defn use rw use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_730","id":89229,"instance":730,"goal":"all u, u <= u","context":["defn","use","rw","use"]} -{"prover":"Minlog","tactic_proof":"ng simplify simplify elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_731","id":89230,"instance":731,"goal":"all m h f, m <= h -> h <= f -> m <= f","context":["ng","simplify","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"simp ng defn ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_732","id":89231,"instance":732,"goal":"all y, not y < y","context":["simp","ng","defn","ng"]} -{"prover":"Minlog","tactic_proof":"assume assume rw cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_733","id":89232,"instance":733,"goal":"all q h, Even q -> Even h -> Even (q + h)","context":["assume","assume","rw","cases"]} -{"prover":"Minlog","tactic_proof":"intro ind defn rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_734","id":89233,"instance":734,"goal":"all m, m | m","context":["intro","ind","defn","rw"]} -{"prover":"Minlog","tactic_proof":"elim simp defn intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_735","id":89234,"instance":735,"goal":"all f, Fact f > 0","context":["elim","simp","defn","intro"]} -{"prover":"Minlog","tactic_proof":"simp assume cases rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_736","id":89235,"instance":736,"goal":"all h a = a","context":["simp","assume","cases","rw"]} -{"prover":"Minlog","tactic_proof":"rw ng use simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_737","id":89236,"instance":737,"goal":"all u + 0 = u","context":["rw","ng","use","simplify"]} -{"prover":"Minlog","tactic_proof":"rw elim cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_738","id":89237,"instance":738,"goal":"all v z, v + z = z + v","context":["rw","elim","cases","simp"]} -{"prover":"Minlog","tactic_proof":"simplify assume simp use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_739","id":89238,"instance":739,"goal":"all w v x, (w + v) + x = w + (v + x)","context":["simplify","assume","simp","use"]} -{"prover":"Minlog","tactic_proof":"rw defn simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_740","id":89239,"instance":740,"goal":"all g, g * 0 = 0","context":["rw","defn","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ind defn defn ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_741","id":89240,"instance":741,"goal":"all p, p * 1 = p","context":["ind","defn","defn","ng"]} -{"prover":"Minlog","tactic_proof":"intro simp ind simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_742","id":89241,"instance":742,"goal":"all y z w, (y * z) * w = y * (z * w)","context":["intro","simp","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify elim intro assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_743","id":89242,"instance":743,"goal":"all f x, f * x = x * f","context":["simplify","elim","intro","assume"]} -{"prover":"Minlog","tactic_proof":"simp simplify use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_744","id":89243,"instance":744,"goal":"all d b x, d * (b + x) = d * b + d * x","context":["simp","simplify","use","ng"]} -{"prover":"Minlog","tactic_proof":"simp rw simp simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_745","id":89244,"instance":745,"goal":"all z, z <= z","context":["simp","rw","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"use elim use cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_746","id":89245,"instance":746,"goal":"all v b g, v <= b -> b <= g -> v <= g","context":["use","elim","use","cases"]} -{"prover":"Minlog","tactic_proof":"simp rw elim ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_747","id":89246,"instance":747,"goal":"all v, not v < v","context":["simp","rw","elim","ng"]} -{"prover":"Minlog","tactic_proof":"elim rw ind simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_748","id":89247,"instance":748,"goal":"all m w, Even m -> Even w -> Even (m + w)","context":["elim","rw","ind","simp"]} -{"prover":"Minlog","tactic_proof":"cases rw intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_749","id":89248,"instance":749,"goal":"all p, p | p","context":["cases","rw","intro","assume"]} -{"prover":"Minlog","tactic_proof":"elim rw cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_750","id":89249,"instance":750,"goal":"all y, Fact y > 0","context":["elim","rw","cases","simp"]} -{"prover":"Minlog","tactic_proof":"use rw ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_751","id":89250,"instance":751,"goal":"all n m = m","context":["use","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simplify simp cases use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_752","id":89251,"instance":752,"goal":"all c + 0 = c","context":["simplify","simp","cases","use"]} -{"prover":"Minlog","tactic_proof":"rw ng intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_753","id":89252,"instance":753,"goal":"all c v, c + v = v + c","context":["rw","ng","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simplify defn ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_754","id":89253,"instance":754,"goal":"all y a f, (y + a) + f = y + (a + f)","context":["assume","simplify","defn","ng"]} -{"prover":"Minlog","tactic_proof":"defn ind simp assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_755","id":89254,"instance":755,"goal":"all z, z * 0 = 0","context":["defn","ind","simp","assume"]} -{"prover":"Minlog","tactic_proof":"ind ng simp use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_756","id":89255,"instance":756,"goal":"all g, g * 1 = g","context":["ind","ng","simp","use"]} -{"prover":"Minlog","tactic_proof":"rw elim ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_757","id":89256,"instance":757,"goal":"all b p d, (b * p) * d = b * (p * d)","context":["rw","elim","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ind simplify elim ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_758","id":89257,"instance":758,"goal":"all z u, z * u = u * z","context":["ind","simplify","elim","ng"]} -{"prover":"Minlog","tactic_proof":"rw cases intro use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_759","id":89258,"instance":759,"goal":"all f x p, f * (x + p) = f * x + f * p","context":["rw","cases","intro","use"]} -{"prover":"Minlog","tactic_proof":"simplify ind intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_760","id":89259,"instance":760,"goal":"all b, b <= b","context":["simplify","ind","intro","use"]} -{"prover":"Minlog","tactic_proof":"ng simp defn ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_761","id":89260,"instance":761,"goal":"all m h d, m <= h -> h <= d -> m <= d","context":["ng","simp","defn","ng"]} -{"prover":"Minlog","tactic_proof":"cases use simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_762","id":89261,"instance":762,"goal":"all z, not z < z","context":["cases","use","simp","ind"]} -{"prover":"Minlog","tactic_proof":"defn elim ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_763","id":89262,"instance":763,"goal":"all b f, Even b -> Even f -> Even (b + f)","context":["defn","elim","ng","assume"]} -{"prover":"Minlog","tactic_proof":"cases use intro elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_764","id":89263,"instance":764,"goal":"all b, b | b","context":["cases","use","intro","elim"]} -{"prover":"Minlog","tactic_proof":"cases elim defn rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_765","id":89264,"instance":765,"goal":"all x, Fact x > 0","context":["cases","elim","defn","rw"]} -{"prover":"Minlog","tactic_proof":"intro defn simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_766","id":89265,"instance":766,"goal":"all h c = c","context":["intro","defn","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"elim ind rw elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_767","id":89266,"instance":767,"goal":"all a + 0 = a","context":["elim","ind","rw","elim"]} -{"prover":"Minlog","tactic_proof":"cases ng intro elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_768","id":89267,"instance":768,"goal":"all a q, a + q = q + a","context":["cases","ng","intro","elim"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_769","id":89268,"instance":769,"goal":"all v q m, (v + q) + m = v + (q + m)","context":["intro","simplify","intro","elim"]} -{"prover":"Minlog","tactic_proof":"assume elim intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_770","id":89269,"instance":770,"goal":"all q, q * 0 = 0","context":["assume","elim","intro","elim"]} -{"prover":"Minlog","tactic_proof":"ng ng ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_771","id":89270,"instance":771,"goal":"all p, p * 1 = p","context":["ng","ng","ng","ind"]} -{"prover":"Minlog","tactic_proof":"cases ind elim ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_772","id":89271,"instance":772,"goal":"all p e y, (p * e) * y = p * (e * y)","context":["cases","ind","elim","ng"]} -{"prover":"Minlog","tactic_proof":"simplify rw intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_773","id":89272,"instance":773,"goal":"all g f, g * f = f * g","context":["simplify","rw","intro","use"]} -{"prover":"Minlog","tactic_proof":"ng simplify simplify simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_774","id":89273,"instance":774,"goal":"all u p h, u * (p + h) = u * p + u * h","context":["ng","simplify","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"ng simp defn assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_775","id":89274,"instance":775,"goal":"all g, g <= g","context":["ng","simp","defn","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro assume elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_776","id":89275,"instance":776,"goal":"all b d z, b <= d -> d <= z -> b <= z","context":["intro","intro","assume","elim"]} -{"prover":"Minlog","tactic_proof":"rw intro use assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_777","id":89276,"instance":777,"goal":"all f, not f < f","context":["rw","intro","use","assume"]} -{"prover":"Minlog","tactic_proof":"use simplify intro simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_778","id":89277,"instance":778,"goal":"all n q, Even n -> Even q -> Even (n + q)","context":["use","simplify","intro","simp"]} -{"prover":"Minlog","tactic_proof":"use use cases rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_779","id":89278,"instance":779,"goal":"all h, h | h","context":["use","use","cases","rw"]} -{"prover":"Minlog","tactic_proof":"rw elim rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_780","id":89279,"instance":780,"goal":"all h, Fact h > 0","context":["rw","elim","rw","rw"]} -{"prover":"Minlog","tactic_proof":"simplify ind ind assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_781","id":89280,"instance":781,"goal":"all z a = a","context":["simplify","ind","ind","assume"]} -{"prover":"Minlog","tactic_proof":"intro simp intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_782","id":89281,"instance":782,"goal":"all f + 0 = f","context":["intro","simp","intro","use"]} -{"prover":"Minlog","tactic_proof":"assume simplify ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_783","id":89282,"instance":783,"goal":"all z h, z + h = h + z","context":["assume","simplify","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ng simplify cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_784","id":89283,"instance":784,"goal":"all u n z, (u + n) + z = u + (n + z)","context":["ng","simplify","cases","assume"]} -{"prover":"Minlog","tactic_proof":"ng simplify intro assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_785","id":89284,"instance":785,"goal":"all c, c * 0 = 0","context":["ng","simplify","intro","assume"]} -{"prover":"Minlog","tactic_proof":"assume cases cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_786","id":89285,"instance":786,"goal":"all g, g * 1 = g","context":["assume","cases","cases","cases"]} -{"prover":"Minlog","tactic_proof":"simplify assume cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_787","id":89286,"instance":787,"goal":"all b m z, (b * m) * z = b * (m * z)","context":["simplify","assume","cases","defn"]} -{"prover":"Minlog","tactic_proof":"use use ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_788","id":89287,"instance":788,"goal":"all v d, v * d = d * v","context":["use","use","ng","simp"]} -{"prover":"Minlog","tactic_proof":"ind ind elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_789","id":89288,"instance":789,"goal":"all w h c, w * (h + c) = w * h + w * c","context":["ind","ind","elim","elim"]} -{"prover":"Minlog","tactic_proof":"elim simp defn ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_790","id":89289,"instance":790,"goal":"all d, d <= d","context":["elim","simp","defn","ng"]} -{"prover":"Minlog","tactic_proof":"simp intro simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_791","id":89290,"instance":791,"goal":"all e u q, e <= u -> u <= q -> e <= q","context":["simp","intro","simplify","use"]} -{"prover":"Minlog","tactic_proof":"rw use ng elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_792","id":89291,"instance":792,"goal":"all e, not e < e","context":["rw","use","ng","elim"]} -{"prover":"Minlog","tactic_proof":"ind defn use simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_793","id":89292,"instance":793,"goal":"all c q, Even c -> Even q -> Even (c + q)","context":["ind","defn","use","simplify"]} -{"prover":"Minlog","tactic_proof":"ind ng intro simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_794","id":89293,"instance":794,"goal":"all u, u | u","context":["ind","ng","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify ind ind intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_795","id":89294,"instance":795,"goal":"all u, Fact u > 0","context":["simplify","ind","ind","intro"]} -{"prover":"Minlog","tactic_proof":"ind elim elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_796","id":89295,"instance":796,"goal":"all a u = u","context":["ind","elim","elim","ind"]} -{"prover":"Minlog","tactic_proof":"use assume simplify assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_797","id":89296,"instance":797,"goal":"all z + 0 = z","context":["use","assume","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"use simp cases ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_798","id":89297,"instance":798,"goal":"all d n, d + n = n + d","context":["use","simp","cases","ng"]} -{"prover":"Minlog","tactic_proof":"use simplify use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_799","id":89298,"instance":799,"goal":"all b u y, (b + u) + y = b + (u + y)","context":["use","simplify","use","defn"]} -{"prover":"Minlog","tactic_proof":"rw use defn intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_800","id":89299,"instance":800,"goal":"all a, a * 0 = 0","context":["rw","use","defn","intro"]} -{"prover":"Minlog","tactic_proof":"assume use use ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_801","id":89300,"instance":801,"goal":"all x, x * 1 = x","context":["assume","use","use","ind"]} -{"prover":"Minlog","tactic_proof":"rw simplify simp rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_802","id":89301,"instance":802,"goal":"all e g d, (e * g) * d = e * (g * d)","context":["rw","simplify","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simp intro assume simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_803","id":89302,"instance":803,"goal":"all e y, e * y = y * e","context":["simp","intro","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"ng simplify ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_804","id":89303,"instance":804,"goal":"all h x d, h * (x + d) = h * x + h * d","context":["ng","simplify","ind","ind"]} -{"prover":"Minlog","tactic_proof":"cases simplify ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_805","id":89304,"instance":805,"goal":"all c, c <= c","context":["cases","simplify","ng","assume"]} -{"prover":"Minlog","tactic_proof":"assume defn rw ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_806","id":89305,"instance":806,"goal":"all u g a, u <= g -> g <= a -> u <= a","context":["assume","defn","rw","ind"]} -{"prover":"Minlog","tactic_proof":"ng defn use simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_807","id":89306,"instance":807,"goal":"all w, not w < w","context":["ng","defn","use","simplify"]} -{"prover":"Minlog","tactic_proof":"elim simp intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_808","id":89307,"instance":808,"goal":"all h f, Even h -> Even f -> Even (h + f)","context":["elim","simp","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"elim ind defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_809","id":89308,"instance":809,"goal":"all m, m | m","context":["elim","ind","defn","rw"]} -{"prover":"Minlog","tactic_proof":"cases ind ind elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_810","id":89309,"instance":810,"goal":"all m, Fact m > 0","context":["cases","ind","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simp ng simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_811","id":89310,"instance":811,"goal":"all e u = u","context":["simp","ng","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ind ind ind ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_812","id":89311,"instance":812,"goal":"all u + 0 = u","context":["ind","ind","ind","ng"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_813","id":89312,"instance":813,"goal":"all w y, w + y = y + w","context":["intro","simplify","intro","ng"]} -{"prover":"Minlog","tactic_proof":"intro rw ng elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_814","id":89313,"instance":814,"goal":"all n p q, (n + p) + q = n + (p + q)","context":["intro","rw","ng","elim"]} -{"prover":"Minlog","tactic_proof":"cases rw rw defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_815","id":89314,"instance":815,"goal":"all w, w * 0 = 0","context":["cases","rw","rw","defn"]} -{"prover":"Minlog","tactic_proof":"intro use ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_816","id":89315,"instance":816,"goal":"all p, p * 1 = p","context":["intro","use","ind","rw"]} -{"prover":"Minlog","tactic_proof":"assume ind use rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_817","id":89316,"instance":817,"goal":"all y v e, (y * v) * e = y * (v * e)","context":["assume","ind","use","rw"]} -{"prover":"Minlog","tactic_proof":"ind cases simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_818","id":89317,"instance":818,"goal":"all c v, c * v = v * c","context":["ind","cases","simp","use"]} -{"prover":"Minlog","tactic_proof":"intro ind intro simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_819","id":89318,"instance":819,"goal":"all b f e, b * (f + e) = b * f + b * e","context":["intro","ind","intro","simp"]} -{"prover":"Minlog","tactic_proof":"assume ind defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_820","id":89319,"instance":820,"goal":"all a, a <= a","context":["assume","ind","defn","use"]} -{"prover":"Minlog","tactic_proof":"intro cases simplify ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_821","id":89320,"instance":821,"goal":"all z e f, z <= e -> e <= f -> z <= f","context":["intro","cases","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ng assume simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_822","id":89321,"instance":822,"goal":"all b, not b < b","context":["ng","assume","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"elim rw assume simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_823","id":89322,"instance":823,"goal":"all n a, Even n -> Even a -> Even (n + a)","context":["elim","rw","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"cases rw simplify use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_824","id":89323,"instance":824,"goal":"all e, e | e","context":["cases","rw","simplify","use"]} -{"prover":"Minlog","tactic_proof":"defn cases intro defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_825","id":89324,"instance":825,"goal":"all n, Fact n > 0","context":["defn","cases","intro","defn"]} -{"prover":"Minlog","tactic_proof":"rw elim use ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_826","id":89325,"instance":826,"goal":"all n g = g","context":["rw","elim","use","ng"]} -{"prover":"Minlog","tactic_proof":"elim ng use ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_827","id":89326,"instance":827,"goal":"all v + 0 = v","context":["elim","ng","use","ind"]} -{"prover":"Minlog","tactic_proof":"defn elim assume use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_828","id":89327,"instance":828,"goal":"all v u, v + u = u + v","context":["defn","elim","assume","use"]} -{"prover":"Minlog","tactic_proof":"simp simplify rw ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_829","id":89328,"instance":829,"goal":"all e g z, (e + g) + z = e + (g + z)","context":["simp","simplify","rw","ind"]} -{"prover":"Minlog","tactic_proof":"elim use intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_830","id":89329,"instance":830,"goal":"all x, x * 0 = 0","context":["elim","use","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ind simp simplify cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_831","id":89330,"instance":831,"goal":"all z, z * 1 = z","context":["ind","simp","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"simp ind simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_832","id":89331,"instance":832,"goal":"all e a n, (e * a) * n = e * (a * n)","context":["simp","ind","simp","ind"]} -{"prover":"Minlog","tactic_proof":"use ind use ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_833","id":89332,"instance":833,"goal":"all h u, h * u = u * h","context":["use","ind","use","ng"]} -{"prover":"Minlog","tactic_proof":"intro ind ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_834","id":89333,"instance":834,"goal":"all m h y, m * (h + y) = m * h + m * y","context":["intro","ind","ng","simp"]} -{"prover":"Minlog","tactic_proof":"simp rw assume simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_835","id":89334,"instance":835,"goal":"all p, p <= p","context":["simp","rw","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"intro cases cases intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_836","id":89335,"instance":836,"goal":"all w z m, w <= z -> z <= m -> w <= m","context":["intro","cases","cases","intro"]} -{"prover":"Minlog","tactic_proof":"use rw use ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_837","id":89336,"instance":837,"goal":"all n, not n < n","context":["use","rw","use","ng"]} -{"prover":"Minlog","tactic_proof":"elim rw elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_838","id":89337,"instance":838,"goal":"all f v, Even f -> Even v -> Even (f + v)","context":["elim","rw","elim","use"]} -{"prover":"Minlog","tactic_proof":"ng cases rw use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_839","id":89338,"instance":839,"goal":"all q, q | q","context":["ng","cases","rw","use"]} -{"prover":"Minlog","tactic_proof":"use use use intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_840","id":89339,"instance":840,"goal":"all g, Fact g > 0","context":["use","use","use","intro"]} -{"prover":"Minlog","tactic_proof":"rw simplify use use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_841","id":89340,"instance":841,"goal":"all e w = w","context":["rw","simplify","use","use"]} -{"prover":"Minlog","tactic_proof":"assume ng ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_842","id":89341,"instance":842,"goal":"all h + 0 = h","context":["assume","ng","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume rw assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_843","id":89342,"instance":843,"goal":"all y d, y + d = d + y","context":["assume","rw","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ng intro elim ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_844","id":89343,"instance":844,"goal":"all p z h, (p + z) + h = p + (z + h)","context":["ng","intro","elim","ng"]} -{"prover":"Minlog","tactic_proof":"assume intro elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_845","id":89344,"instance":845,"goal":"all a, a * 0 = 0","context":["assume","intro","elim","cases"]} -{"prover":"Minlog","tactic_proof":"assume ind ind cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_846","id":89345,"instance":846,"goal":"all m, m * 1 = m","context":["assume","ind","ind","cases"]} -{"prover":"Minlog","tactic_proof":"use cases ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_847","id":89346,"instance":847,"goal":"all a d h, (a * d) * h = a * (d * h)","context":["use","cases","ind","intro"]} -{"prover":"Minlog","tactic_proof":"rw ind cases rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_848","id":89347,"instance":848,"goal":"all x n, x * n = n * x","context":["rw","ind","cases","rw"]} -{"prover":"Minlog","tactic_proof":"elim defn cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_849","id":89348,"instance":849,"goal":"all v d g, v * (d + g) = v * d + v * g","context":["elim","defn","cases","rw"]} -{"prover":"Minlog","tactic_proof":"elim simplify defn simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_850","id":89349,"instance":850,"goal":"all u, u <= u","context":["elim","simplify","defn","simp"]} -{"prover":"Minlog","tactic_proof":"defn cases ind defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_851","id":89350,"instance":851,"goal":"all n m x, n <= m -> m <= x -> n <= x","context":["defn","cases","ind","defn"]} -{"prover":"Minlog","tactic_proof":"cases elim assume simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_852","id":89351,"instance":852,"goal":"all y, not y < y","context":["cases","elim","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify simp intro simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_853","id":89352,"instance":853,"goal":"all v f, Even v -> Even f -> Even (v + f)","context":["simplify","simp","intro","simp"]} -{"prover":"Minlog","tactic_proof":"use use assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_854","id":89353,"instance":854,"goal":"all g, g | g","context":["use","use","assume","ng"]} -{"prover":"Minlog","tactic_proof":"simp rw assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_855","id":89354,"instance":855,"goal":"all y, Fact y > 0","context":["simp","rw","assume","intro"]} -{"prover":"Minlog","tactic_proof":"defn simp assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_856","id":89355,"instance":856,"goal":"all z h = h","context":["defn","simp","assume","cases"]} -{"prover":"Minlog","tactic_proof":"defn rw elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_857","id":89356,"instance":857,"goal":"all u + 0 = u","context":["defn","rw","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ind use rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_858","id":89357,"instance":858,"goal":"all e z, e + z = z + e","context":["ind","use","rw","simp"]} -{"prover":"Minlog","tactic_proof":"ind cases intro simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_859","id":89358,"instance":859,"goal":"all y v w, (y + v) + w = y + (v + w)","context":["ind","cases","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use simp ng rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_860","id":89359,"instance":860,"goal":"all m, m * 0 = 0","context":["use","simp","ng","rw"]} -{"prover":"Minlog","tactic_proof":"intro intro defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_861","id":89360,"instance":861,"goal":"all x, x * 1 = x","context":["intro","intro","defn","rw"]} -{"prover":"Minlog","tactic_proof":"simp rw use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_862","id":89361,"instance":862,"goal":"all x v m, (x * v) * m = x * (v * m)","context":["simp","rw","use","intro"]} -{"prover":"Minlog","tactic_proof":"cases simplify ind simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_863","id":89362,"instance":863,"goal":"all x g, x * g = g * x","context":["cases","simplify","ind","simp"]} -{"prover":"Minlog","tactic_proof":"cases simplify use rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_864","id":89363,"instance":864,"goal":"all g u y, g * (u + y) = g * u + g * y","context":["cases","simplify","use","rw"]} -{"prover":"Minlog","tactic_proof":"simplify intro ng simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_865","id":89364,"instance":865,"goal":"all c, c <= c","context":["simplify","intro","ng","simp"]} -{"prover":"Minlog","tactic_proof":"ng ind assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_866","id":89365,"instance":866,"goal":"all b g f, b <= g -> g <= f -> b <= f","context":["ng","ind","assume","intro"]} -{"prover":"Minlog","tactic_proof":"ng ind elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_867","id":89366,"instance":867,"goal":"all z, not z < z","context":["ng","ind","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"simp rw ng defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_868","id":89367,"instance":868,"goal":"all n y, Even n -> Even y -> Even (n + y)","context":["simp","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"use ng intro rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_869","id":89368,"instance":869,"goal":"all b, b | b","context":["use","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"cases cases defn ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_870","id":89369,"instance":870,"goal":"all n, Fact n > 0","context":["cases","cases","defn","ng"]} -{"prover":"Minlog","tactic_proof":"intro cases simplify cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_871","id":89370,"instance":871,"goal":"all x g = g","context":["intro","cases","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"ind simp intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_872","id":89371,"instance":872,"goal":"all g + 0 = g","context":["ind","simp","intro","use"]} -{"prover":"Minlog","tactic_proof":"use ng simp simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_873","id":89372,"instance":873,"goal":"all m c, m + c = c + m","context":["use","ng","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ng ng simp defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_874","id":89373,"instance":874,"goal":"all b g z, (b + g) + z = b + (g + z)","context":["ng","ng","simp","defn"]} -{"prover":"Minlog","tactic_proof":"simplify simp use intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_875","id":89374,"instance":875,"goal":"all q, q * 0 = 0","context":["simplify","simp","use","intro"]} -{"prover":"Minlog","tactic_proof":"defn simp elim ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_876","id":89375,"instance":876,"goal":"all v, v * 1 = v","context":["defn","simp","elim","ng"]} -{"prover":"Minlog","tactic_proof":"ng assume simplify simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_877","id":89376,"instance":877,"goal":"all q g n, (q * g) * n = q * (g * n)","context":["ng","assume","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ng simp defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_878","id":89377,"instance":878,"goal":"all u e, u * e = e * u","context":["cases","ng","simp","defn"]} -{"prover":"Minlog","tactic_proof":"cases use rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_879","id":89378,"instance":879,"goal":"all e g x, e * (g + x) = e * g + e * x","context":["cases","use","rw","assume"]} -{"prover":"Minlog","tactic_proof":"defn ng intro ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_880","id":89379,"instance":880,"goal":"all a, a <= a","context":["defn","ng","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ng ng ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_881","id":89380,"instance":881,"goal":"all x w c, x <= w -> w <= c -> x <= c","context":["ng","ng","ng","cases"]} -{"prover":"Minlog","tactic_proof":"use ind rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_882","id":89381,"instance":882,"goal":"all v, not v < v","context":["use","ind","rw","assume"]} -{"prover":"Minlog","tactic_proof":"intro assume rw defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_883","id":89382,"instance":883,"goal":"all e n, Even e -> Even n -> Even (e + n)","context":["intro","assume","rw","defn"]} -{"prover":"Minlog","tactic_proof":"rw cases elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_884","id":89383,"instance":884,"goal":"all q, q | q","context":["rw","cases","elim","use"]} -{"prover":"Minlog","tactic_proof":"use cases ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_885","id":89384,"instance":885,"goal":"all h, Fact h > 0","context":["use","cases","ng","cases"]} -{"prover":"Minlog","tactic_proof":"elim ind elim ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_886","id":89385,"instance":886,"goal":"all v u = u","context":["elim","ind","elim","ng"]} -{"prover":"Minlog","tactic_proof":"intro simp defn intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_887","id":89386,"instance":887,"goal":"all z + 0 = z","context":["intro","simp","defn","intro"]} -{"prover":"Minlog","tactic_proof":"cases rw ind elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_888","id":89387,"instance":888,"goal":"all z w, z + w = w + z","context":["cases","rw","ind","elim"]} -{"prover":"Minlog","tactic_proof":"ind simp assume simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_889","id":89388,"instance":889,"goal":"all h c w, (h + c) + w = h + (c + w)","context":["ind","simp","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_890","id":89389,"instance":890,"goal":"all n, n * 0 = 0","context":["intro","simplify","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ind elim intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_891","id":89390,"instance":891,"goal":"all f, f * 1 = f","context":["ind","elim","intro","ind"]} -{"prover":"Minlog","tactic_proof":"intro intro cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_892","id":89391,"instance":892,"goal":"all b f y, (b * f) * y = b * (f * y)","context":["intro","intro","cases","intro"]} -{"prover":"Minlog","tactic_proof":"elim use simplify use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_893","id":89392,"instance":893,"goal":"all z c, z * c = c * z","context":["elim","use","simplify","use"]} -{"prover":"Minlog","tactic_proof":"use simplify use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_894","id":89393,"instance":894,"goal":"all v c m, v * (c + m) = v * c + v * m","context":["use","simplify","use","assume"]} -{"prover":"Minlog","tactic_proof":"intro simp simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_895","id":89394,"instance":895,"goal":"all m, m <= m","context":["intro","simp","simp","ind"]} -{"prover":"Minlog","tactic_proof":"assume ind ind simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_896","id":89395,"instance":896,"goal":"all p z w, p <= z -> z <= w -> p <= w","context":["assume","ind","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"cases cases intro intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_897","id":89396,"instance":897,"goal":"all x, not x < x","context":["cases","cases","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ng simp use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_898","id":89397,"instance":898,"goal":"all y d, Even y -> Even d -> Even (y + d)","context":["ng","simp","use","simp"]} -{"prover":"Minlog","tactic_proof":"use assume simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_899","id":89398,"instance":899,"goal":"all g, g | g","context":["use","assume","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"elim ind ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_900","id":89399,"instance":900,"goal":"all y, Fact y > 0","context":["elim","ind","ng","ind"]} -{"prover":"Minlog","tactic_proof":"rw intro cases elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_901","id":89400,"instance":901,"goal":"all d y = y","context":["rw","intro","cases","elim"]} -{"prover":"Minlog","tactic_proof":"simp cases use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_902","id":89401,"instance":902,"goal":"all p + 0 = p","context":["simp","cases","use","elim"]} -{"prover":"Minlog","tactic_proof":"simplify defn cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_903","id":89402,"instance":903,"goal":"all e u, e + u = u + e","context":["simplify","defn","cases","cases"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_904","id":89403,"instance":904,"goal":"all y f b, (y + f) + b = y + (f + b)","context":["ind","ng","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"rw simp ind simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_905","id":89404,"instance":905,"goal":"all u, u * 0 = 0","context":["rw","simp","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ng ng simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_906","id":89405,"instance":906,"goal":"all u, u * 1 = u","context":["cases","ng","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"rw simp rw ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_907","id":89406,"instance":907,"goal":"all w u p, (w * u) * p = w * (u * p)","context":["rw","simp","rw","ind"]} -{"prover":"Minlog","tactic_proof":"simplify assume rw defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_908","id":89407,"instance":908,"goal":"all m x, m * x = x * m","context":["simplify","assume","rw","defn"]} -{"prover":"Minlog","tactic_proof":"assume ng intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_909","id":89408,"instance":909,"goal":"all v p x, v * (p + x) = v * p + v * x","context":["assume","ng","intro","ng"]} -{"prover":"Minlog","tactic_proof":"simplify assume intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_910","id":89409,"instance":910,"goal":"all p, p <= p","context":["simplify","assume","intro","rw"]} -{"prover":"Minlog","tactic_proof":"simp cases ind rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_911","id":89410,"instance":911,"goal":"all d b f, d <= b -> b <= f -> d <= f","context":["simp","cases","ind","rw"]} -{"prover":"Minlog","tactic_proof":"rw simplify intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_912","id":89411,"instance":912,"goal":"all c, not c < c","context":["rw","simplify","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ind simp simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_913","id":89412,"instance":913,"goal":"all h q, Even h -> Even q -> Even (h + q)","context":["ind","simp","simp","rw"]} -{"prover":"Minlog","tactic_proof":"ind ng intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_914","id":89413,"instance":914,"goal":"all u, u | u","context":["ind","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ng cases assume use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_915","id":89414,"instance":915,"goal":"all v, Fact v > 0","context":["ng","cases","assume","use"]} -{"prover":"Minlog","tactic_proof":"use simp assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_916","id":89415,"instance":916,"goal":"all v q = q","context":["use","simp","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ind rw use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_917","id":89416,"instance":917,"goal":"all f + 0 = f","context":["ind","rw","use","cases"]} -{"prover":"Minlog","tactic_proof":"intro rw elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_918","id":89417,"instance":918,"goal":"all y x, y + x = x + y","context":["intro","rw","elim","rw"]} -{"prover":"Minlog","tactic_proof":"rw elim ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_919","id":89418,"instance":919,"goal":"all g n p, (g + n) + p = g + (n + p)","context":["rw","elim","ng","defn"]} -{"prover":"Minlog","tactic_proof":"cases rw assume ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_920","id":89419,"instance":920,"goal":"all p, p * 0 = 0","context":["cases","rw","assume","ind"]} -{"prover":"Minlog","tactic_proof":"cases rw intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_921","id":89420,"instance":921,"goal":"all g, g * 1 = g","context":["cases","rw","intro","simp"]} -{"prover":"Minlog","tactic_proof":"ind assume defn assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_922","id":89421,"instance":922,"goal":"all f y e, (f * y) * e = f * (y * e)","context":["ind","assume","defn","assume"]} -{"prover":"Minlog","tactic_proof":"elim ind simplify ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_923","id":89422,"instance":923,"goal":"all p u, p * u = u * p","context":["elim","ind","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"assume elim defn simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_924","id":89423,"instance":924,"goal":"all u a d, u * (a + d) = u * a + u * d","context":["assume","elim","defn","simp"]} -{"prover":"Minlog","tactic_proof":"use ind ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_925","id":89424,"instance":925,"goal":"all y, y <= y","context":["use","ind","ind","ind"]} -{"prover":"Minlog","tactic_proof":"use elim assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_926","id":89425,"instance":926,"goal":"all y u c, y <= u -> u <= c -> y <= c","context":["use","elim","assume","elim"]} -{"prover":"Minlog","tactic_proof":"simplify defn ind use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_927","id":89426,"instance":927,"goal":"all p, not p < p","context":["simplify","defn","ind","use"]} -{"prover":"Minlog","tactic_proof":"rw use simp use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_928","id":89427,"instance":928,"goal":"all y h, Even y -> Even h -> Even (y + h)","context":["rw","use","simp","use"]} -{"prover":"Minlog","tactic_proof":"elim ng intro elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_929","id":89428,"instance":929,"goal":"all h, h | h","context":["elim","ng","intro","elim"]} -{"prover":"Minlog","tactic_proof":"cases rw intro intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_930","id":89429,"instance":930,"goal":"all d, Fact d > 0","context":["cases","rw","intro","intro"]} -{"prover":"Minlog","tactic_proof":"defn ng simplify elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_931","id":89430,"instance":931,"goal":"all u n = n","context":["defn","ng","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"ng ng ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_932","id":89431,"instance":932,"goal":"all h + 0 = h","context":["ng","ng","ng","assume"]} -{"prover":"Minlog","tactic_proof":"intro cases elim use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_933","id":89432,"instance":933,"goal":"all y q, y + q = q + y","context":["intro","cases","elim","use"]} -{"prover":"Minlog","tactic_proof":"ind intro elim cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_934","id":89433,"instance":934,"goal":"all n b w, (n + b) + w = n + (b + w)","context":["ind","intro","elim","cases"]} -{"prover":"Minlog","tactic_proof":"rw use simplify use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_935","id":89434,"instance":935,"goal":"all y, y * 0 = 0","context":["rw","use","simplify","use"]} -{"prover":"Minlog","tactic_proof":"defn ng simp use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_936","id":89435,"instance":936,"goal":"all c, c * 1 = c","context":["defn","ng","simp","use"]} -{"prover":"Minlog","tactic_proof":"elim rw ind intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_937","id":89436,"instance":937,"goal":"all g x v, (g * x) * v = g * (x * v)","context":["elim","rw","ind","intro"]} -{"prover":"Minlog","tactic_proof":"ind ng defn ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_938","id":89437,"instance":938,"goal":"all y a, y * a = a * y","context":["ind","ng","defn","ind"]} -{"prover":"Minlog","tactic_proof":"elim rw ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_939","id":89438,"instance":939,"goal":"all c x p, c * (x + p) = c * x + c * p","context":["elim","rw","ng","intro"]} -{"prover":"Minlog","tactic_proof":"ind ind cases ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_940","id":89439,"instance":940,"goal":"all x, x <= x","context":["ind","ind","cases","ind"]} -{"prover":"Minlog","tactic_proof":"elim simplify defn simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_941","id":89440,"instance":941,"goal":"all h b c, h <= b -> b <= c -> h <= c","context":["elim","simplify","defn","simp"]} -{"prover":"Minlog","tactic_proof":"cases defn intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_942","id":89441,"instance":942,"goal":"all n, not n < n","context":["cases","defn","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ind elim cases simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_943","id":89442,"instance":943,"goal":"all z f, Even z -> Even f -> Even (z + f)","context":["ind","elim","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"assume ng ng ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_944","id":89443,"instance":944,"goal":"all e, e | e","context":["assume","ng","ng","ind"]} -{"prover":"Minlog","tactic_proof":"assume defn simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_945","id":89444,"instance":945,"goal":"all g, Fact g > 0","context":["assume","defn","simp","rw"]} -{"prover":"Minlog","tactic_proof":"intro rw ind ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_946","id":89445,"instance":946,"goal":"all p n = n","context":["intro","rw","ind","ng"]} -{"prover":"Minlog","tactic_proof":"elim simplify rw ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_947","id":89446,"instance":947,"goal":"all u + 0 = u","context":["elim","simplify","rw","ng"]} -{"prover":"Minlog","tactic_proof":"ind cases intro simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_948","id":89447,"instance":948,"goal":"all x e, x + e = e + x","context":["ind","cases","intro","simp"]} -{"prover":"Minlog","tactic_proof":"assume rw simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_949","id":89448,"instance":949,"goal":"all c d h, (c + d) + h = c + (d + h)","context":["assume","rw","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"ng defn ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_950","id":89449,"instance":950,"goal":"all d, d * 0 = 0","context":["ng","defn","ng","ind"]} -{"prover":"Minlog","tactic_proof":"use assume ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_951","id":89450,"instance":951,"goal":"all a, a * 1 = a","context":["use","assume","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"intro defn simp cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_952","id":89451,"instance":952,"goal":"all h x c, (h * x) * c = h * (x * c)","context":["intro","defn","simp","cases"]} -{"prover":"Minlog","tactic_proof":"ng use cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_953","id":89452,"instance":953,"goal":"all v u, v * u = u * v","context":["ng","use","cases","use"]} -{"prover":"Minlog","tactic_proof":"simp ind simp cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_954","id":89453,"instance":954,"goal":"all a u z, a * (u + z) = a * u + a * z","context":["simp","ind","simp","cases"]} -{"prover":"Minlog","tactic_proof":"simp rw rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_955","id":89454,"instance":955,"goal":"all e, e <= e","context":["simp","rw","rw","simp"]} -{"prover":"Minlog","tactic_proof":"intro assume assume ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_956","id":89455,"instance":956,"goal":"all y p w, y <= p -> p <= w -> y <= w","context":["intro","assume","assume","ng"]} -{"prover":"Minlog","tactic_proof":"rw simplify cases simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_957","id":89456,"instance":957,"goal":"all p, not p < p","context":["rw","simplify","cases","simp"]} -{"prover":"Minlog","tactic_proof":"intro cases use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_958","id":89457,"instance":958,"goal":"all a h, Even a -> Even h -> Even (a + h)","context":["intro","cases","use","intro"]} -{"prover":"Minlog","tactic_proof":"cases simplify defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_959","id":89458,"instance":959,"goal":"all e, e | e","context":["cases","simplify","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"ind simplify elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_960","id":89459,"instance":960,"goal":"all x, Fact x > 0","context":["ind","simplify","elim","use"]} -{"prover":"Minlog","tactic_proof":"ng cases use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_961","id":89460,"instance":961,"goal":"all x a = a","context":["ng","cases","use","defn"]} -{"prover":"Minlog","tactic_proof":"rw ng simplify simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_962","id":89461,"instance":962,"goal":"all c + 0 = c","context":["rw","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"assume intro cases cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_963","id":89462,"instance":963,"goal":"all q y, q + y = y + q","context":["assume","intro","cases","cases"]} -{"prover":"Minlog","tactic_proof":"elim defn assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_964","id":89463,"instance":964,"goal":"all a m p, (a + m) + p = a + (m + p)","context":["elim","defn","assume","intro"]} -{"prover":"Minlog","tactic_proof":"cases defn simp simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_965","id":89464,"instance":965,"goal":"all z, z * 0 = 0","context":["cases","defn","simp","simp"]} -{"prover":"Minlog","tactic_proof":"elim elim rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_966","id":89465,"instance":966,"goal":"all b, b * 1 = b","context":["elim","elim","rw","rw"]} -{"prover":"Minlog","tactic_proof":"ind cases intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_967","id":89466,"instance":967,"goal":"all p e z, (p * e) * z = p * (e * z)","context":["ind","cases","intro","cases"]} -{"prover":"Minlog","tactic_proof":"defn assume simplify assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_968","id":89467,"instance":968,"goal":"all h p, h * p = p * h","context":["defn","assume","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify cases rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_969","id":89468,"instance":969,"goal":"all v b q, v * (b + q) = v * b + v * q","context":["assume","simplify","cases","rw"]} -{"prover":"Minlog","tactic_proof":"defn simp simplify defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_970","id":89469,"instance":970,"goal":"all y, y <= y","context":["defn","simp","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"use use ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_971","id":89470,"instance":971,"goal":"all e u v, e <= u -> u <= v -> e <= v","context":["use","use","ng","ind"]} -{"prover":"Minlog","tactic_proof":"ind ng elim simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_972","id":89471,"instance":972,"goal":"all h, not h < h","context":["ind","ng","elim","simp"]} -{"prover":"Minlog","tactic_proof":"intro defn assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_973","id":89472,"instance":973,"goal":"all b f, Even b -> Even f -> Even (b + f)","context":["intro","defn","assume","use"]} -{"prover":"Minlog","tactic_proof":"intro assume cases elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_974","id":89473,"instance":974,"goal":"all v, v | v","context":["intro","assume","cases","elim"]} -{"prover":"Minlog","tactic_proof":"ind rw use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_975","id":89474,"instance":975,"goal":"all n, Fact n > 0","context":["ind","rw","use","ng"]} -{"prover":"Minlog","tactic_proof":"cases simp cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_976","id":89475,"instance":976,"goal":"all e m = m","context":["cases","simp","cases","elim"]} -{"prover":"Minlog","tactic_proof":"ind use simp simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_977","id":89476,"instance":977,"goal":"all v + 0 = v","context":["ind","use","simp","simp"]} -{"prover":"Minlog","tactic_proof":"cases intro rw ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_978","id":89477,"instance":978,"goal":"all h w, h + w = w + h","context":["cases","intro","rw","ind"]} -{"prover":"Minlog","tactic_proof":"use elim elim simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_979","id":89478,"instance":979,"goal":"all v f n, (v + f) + n = v + (f + n)","context":["use","elim","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"ind rw use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_980","id":89479,"instance":980,"goal":"all m, m * 0 = 0","context":["ind","rw","use","ng"]} -{"prover":"Minlog","tactic_proof":"elim intro intro elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_981","id":89480,"instance":981,"goal":"all y, y * 1 = y","context":["elim","intro","intro","elim"]} -{"prover":"Minlog","tactic_proof":"simplify rw ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_982","id":89481,"instance":982,"goal":"all b e q, (b * e) * q = b * (e * q)","context":["simplify","rw","ind","defn"]} -{"prover":"Minlog","tactic_proof":"simp elim assume defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_983","id":89482,"instance":983,"goal":"all h y, h * y = y * h","context":["simp","elim","assume","defn"]} -{"prover":"Minlog","tactic_proof":"ng assume assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_984","id":89483,"instance":984,"goal":"all a p y, a * (p + y) = a * p + a * y","context":["ng","assume","assume","use"]} -{"prover":"Minlog","tactic_proof":"simp assume ng simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_985","id":89484,"instance":985,"goal":"all f, f <= f","context":["simp","assume","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"cases simplify use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_986","id":89485,"instance":986,"goal":"all w b h, w <= b -> b <= h -> w <= h","context":["cases","simplify","use","rw"]} -{"prover":"Minlog","tactic_proof":"defn simp use simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_987","id":89486,"instance":987,"goal":"all d, not d < d","context":["defn","simp","use","simp"]} -{"prover":"Minlog","tactic_proof":"use elim cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_988","id":89487,"instance":988,"goal":"all m q, Even m -> Even q -> Even (m + q)","context":["use","elim","cases","use"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_989","id":89488,"instance":989,"goal":"all e, e | e","context":["defn","simplify","rw","defn"]} -{"prover":"Minlog","tactic_proof":"cases defn defn intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_990","id":89489,"instance":990,"goal":"all y, Fact y > 0","context":["cases","defn","defn","intro"]} -{"prover":"Minlog","tactic_proof":"use defn elim simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_991","id":89490,"instance":991,"goal":"all w x = x","context":["use","defn","elim","simp"]} -{"prover":"Minlog","tactic_proof":"defn rw intro defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_992","id":89491,"instance":992,"goal":"all v + 0 = v","context":["defn","rw","intro","defn"]} -{"prover":"Minlog","tactic_proof":"elim elim elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_993","id":89492,"instance":993,"goal":"all w b, w + b = b + w","context":["elim","elim","elim","intro"]} -{"prover":"Minlog","tactic_proof":"ng use cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_994","id":89493,"instance":994,"goal":"all y c e, (y + c) + e = y + (c + e)","context":["ng","use","cases","simp"]} -{"prover":"Minlog","tactic_proof":"use elim intro ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_995","id":89494,"instance":995,"goal":"all b, b * 0 = 0","context":["use","elim","intro","ng"]} -{"prover":"Minlog","tactic_proof":"simplify simp assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_996","id":89495,"instance":996,"goal":"all b, b * 1 = b","context":["simplify","simp","assume","assume"]} -{"prover":"Minlog","tactic_proof":"cases simplify elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_997","id":89496,"instance":997,"goal":"all n z h, (n * z) * h = n * (z * h)","context":["cases","simplify","elim","intro"]} -{"prover":"Minlog","tactic_proof":"assume use simp ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_998","id":89497,"instance":998,"goal":"all f g, f * g = g * f","context":["assume","use","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng ind ind ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_999","id":89498,"instance":999,"goal":"all h v q, h * (v + q) = h * v + h * q","context":["ng","ind","ind","ng"]} -{"prover":"Minlog","tactic_proof":"rw defn use simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1000","id":89499,"instance":1000,"goal":"all y, y <= y","context":["rw","defn","use","simp"]} -{"prover":"Minlog","tactic_proof":"defn rw defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1001","id":89500,"instance":1001,"goal":"all y c d, y <= c -> c <= d -> y <= d","context":["defn","rw","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"cases intro ng use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1002","id":89501,"instance":1002,"goal":"all u, not u < u","context":["cases","intro","ng","use"]} -{"prover":"Minlog","tactic_proof":"assume use ng elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1003","id":89502,"instance":1003,"goal":"all x w, Even x -> Even w -> Even (x + w)","context":["assume","use","ng","elim"]} -{"prover":"Minlog","tactic_proof":"cases ng simp assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1004","id":89503,"instance":1004,"goal":"all u, u | u","context":["cases","ng","simp","assume"]} -{"prover":"Minlog","tactic_proof":"intro rw elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1005","id":89504,"instance":1005,"goal":"all h, Fact h > 0","context":["intro","rw","elim","ind"]} -{"prover":"Minlog","tactic_proof":"simp intro ng rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1006","id":89505,"instance":1006,"goal":"all p n = n","context":["simp","intro","ng","rw"]} -{"prover":"Minlog","tactic_proof":"use ind simp simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1007","id":89506,"instance":1007,"goal":"all w + 0 = w","context":["use","ind","simp","simp"]} -{"prover":"Minlog","tactic_proof":"simp use simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1008","id":89507,"instance":1008,"goal":"all v c, v + c = c + v","context":["simp","use","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"rw ind rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1009","id":89508,"instance":1009,"goal":"all d n v, (d + n) + v = d + (n + v)","context":["rw","ind","rw","assume"]} -{"prover":"Minlog","tactic_proof":"assume elim simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1010","id":89509,"instance":1010,"goal":"all m, m * 0 = 0","context":["assume","elim","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"assume assume assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1011","id":89510,"instance":1011,"goal":"all g, g * 1 = g","context":["assume","assume","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ind rw ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1012","id":89511,"instance":1012,"goal":"all a h p, (a * h) * p = a * (h * p)","context":["ind","rw","ng","assume"]} -{"prover":"Minlog","tactic_proof":"cases simplify assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1013","id":89512,"instance":1013,"goal":"all u a, u * a = a * u","context":["cases","simplify","assume","ng"]} -{"prover":"Minlog","tactic_proof":"simp defn simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1014","id":89513,"instance":1014,"goal":"all d e g, d * (e + g) = d * e + d * g","context":["simp","defn","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"simp use intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1015","id":89514,"instance":1015,"goal":"all m, m <= m","context":["simp","use","intro","cases"]} -{"prover":"Minlog","tactic_proof":"assume ng simp defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1016","id":89515,"instance":1016,"goal":"all f y g, f <= y -> y <= g -> f <= g","context":["assume","ng","simp","defn"]} -{"prover":"Minlog","tactic_proof":"use rw defn intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1017","id":89516,"instance":1017,"goal":"all x, not x < x","context":["use","rw","defn","intro"]} -{"prover":"Minlog","tactic_proof":"ind ind use ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1018","id":89517,"instance":1018,"goal":"all n x, Even n -> Even x -> Even (n + x)","context":["ind","ind","use","ind"]} -{"prover":"Minlog","tactic_proof":"simplify use ng rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1019","id":89518,"instance":1019,"goal":"all m, m | m","context":["simplify","use","ng","rw"]} -{"prover":"Minlog","tactic_proof":"assume defn intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1020","id":89519,"instance":1020,"goal":"all h, Fact h > 0","context":["assume","defn","intro","defn"]} -{"prover":"Minlog","tactic_proof":"cases simp use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1021","id":89520,"instance":1021,"goal":"all f q = q","context":["cases","simp","use","intro"]} -{"prover":"Minlog","tactic_proof":"rw ng simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1022","id":89521,"instance":1022,"goal":"all g + 0 = g","context":["rw","ng","simp","elim"]} -{"prover":"Minlog","tactic_proof":"cases assume ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1023","id":89522,"instance":1023,"goal":"all z c, z + c = c + z","context":["cases","assume","ng","assume"]} -{"prover":"Minlog","tactic_proof":"assume intro ind simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1024","id":89523,"instance":1024,"goal":"all q h m, (q + h) + m = q + (h + m)","context":["assume","intro","ind","simp"]} -{"prover":"Minlog","tactic_proof":"simplify defn ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1025","id":89524,"instance":1025,"goal":"all n, n * 0 = 0","context":["simplify","defn","ng","assume"]} -{"prover":"Minlog","tactic_proof":"rw defn defn cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1026","id":89525,"instance":1026,"goal":"all e, e * 1 = e","context":["rw","defn","defn","cases"]} -{"prover":"Minlog","tactic_proof":"rw use intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1027","id":89526,"instance":1027,"goal":"all q f g, (q * f) * g = q * (f * g)","context":["rw","use","intro","assume"]} -{"prover":"Minlog","tactic_proof":"elim simp cases elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1028","id":89527,"instance":1028,"goal":"all u a, u * a = a * u","context":["elim","simp","cases","elim"]} -{"prover":"Minlog","tactic_proof":"elim simp use assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1029","id":89528,"instance":1029,"goal":"all c m u, c * (m + u) = c * m + c * u","context":["elim","simp","use","assume"]} -{"prover":"Minlog","tactic_proof":"defn rw assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1030","id":89529,"instance":1030,"goal":"all v, v <= v","context":["defn","rw","assume","ng"]} -{"prover":"Minlog","tactic_proof":"use rw defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1031","id":89530,"instance":1031,"goal":"all m u v, m <= u -> u <= v -> m <= v","context":["use","rw","defn","simp"]} -{"prover":"Minlog","tactic_proof":"use use intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1032","id":89531,"instance":1032,"goal":"all d, not d < d","context":["use","use","intro","use"]} -{"prover":"Minlog","tactic_proof":"defn ng intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1033","id":89532,"instance":1033,"goal":"all q v, Even q -> Even v -> Even (q + v)","context":["defn","ng","intro","ind"]} -{"prover":"Minlog","tactic_proof":"simplify ind ind use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1034","id":89533,"instance":1034,"goal":"all m, m | m","context":["simplify","ind","ind","use"]} -{"prover":"Minlog","tactic_proof":"rw ng intro defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1035","id":89534,"instance":1035,"goal":"all m, Fact m > 0","context":["rw","ng","intro","defn"]} -{"prover":"Minlog","tactic_proof":"use intro simp intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1036","id":89535,"instance":1036,"goal":"all d v = v","context":["use","intro","simp","intro"]} -{"prover":"Minlog","tactic_proof":"use use defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1037","id":89536,"instance":1037,"goal":"all d + 0 = d","context":["use","use","defn","elim"]} -{"prover":"Minlog","tactic_proof":"defn ng intro cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1038","id":89537,"instance":1038,"goal":"all g v, g + v = v + g","context":["defn","ng","intro","cases"]} -{"prover":"Minlog","tactic_proof":"assume rw use ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1039","id":89538,"instance":1039,"goal":"all y c x, (y + c) + x = y + (c + x)","context":["assume","rw","use","ind"]} -{"prover":"Minlog","tactic_proof":"simplify ng use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1040","id":89539,"instance":1040,"goal":"all n, n * 0 = 0","context":["simplify","ng","use","elim"]} -{"prover":"Minlog","tactic_proof":"cases intro defn ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1041","id":89540,"instance":1041,"goal":"all d, d * 1 = d","context":["cases","intro","defn","ind"]} -{"prover":"Minlog","tactic_proof":"use cases ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1042","id":89541,"instance":1042,"goal":"all x p v, (x * p) * v = x * (p * v)","context":["use","cases","ng","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1043","id":89542,"instance":1043,"goal":"all d m, d * m = m * d","context":["ind","assume","use","ng"]} -{"prover":"Minlog","tactic_proof":"cases intro cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1044","id":89543,"instance":1044,"goal":"all e y z, e * (y + z) = e * y + e * z","context":["cases","intro","cases","elim"]} -{"prover":"Minlog","tactic_proof":"rw elim elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1045","id":89544,"instance":1045,"goal":"all x, x <= x","context":["rw","elim","elim","intro"]} -{"prover":"Minlog","tactic_proof":"use intro intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1046","id":89545,"instance":1046,"goal":"all m w f, m <= w -> w <= f -> m <= f","context":["use","intro","intro","ng"]} -{"prover":"Minlog","tactic_proof":"elim assume ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1047","id":89546,"instance":1047,"goal":"all b, not b < b","context":["elim","assume","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simplify intro rw defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1048","id":89547,"instance":1048,"goal":"all a g, Even a -> Even g -> Even (a + g)","context":["simplify","intro","rw","defn"]} -{"prover":"Minlog","tactic_proof":"defn defn elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1049","id":89548,"instance":1049,"goal":"all g, g | g","context":["defn","defn","elim","use"]} -{"prover":"Minlog","tactic_proof":"rw intro assume intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1050","id":89549,"instance":1050,"goal":"all x, Fact x > 0","context":["rw","intro","assume","intro"]} -{"prover":"Minlog","tactic_proof":"cases use ng cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1051","id":89550,"instance":1051,"goal":"all d q = q","context":["cases","use","ng","cases"]} -{"prover":"Minlog","tactic_proof":"ind ind elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1052","id":89551,"instance":1052,"goal":"all b + 0 = b","context":["ind","ind","elim","elim"]} -{"prover":"Minlog","tactic_proof":"use cases intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1053","id":89552,"instance":1053,"goal":"all q b, q + b = b + q","context":["use","cases","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ng cases use cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1054","id":89553,"instance":1054,"goal":"all v h g, (v + h) + g = v + (h + g)","context":["ng","cases","use","cases"]} -{"prover":"Minlog","tactic_proof":"simp elim assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1055","id":89554,"instance":1055,"goal":"all h, h * 0 = 0","context":["simp","elim","assume","defn"]} -{"prover":"Minlog","tactic_proof":"simplify elim defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1056","id":89555,"instance":1056,"goal":"all z, z * 1 = z","context":["simplify","elim","defn","cases"]} -{"prover":"Minlog","tactic_proof":"use assume defn defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1057","id":89556,"instance":1057,"goal":"all m c y, (m * c) * y = m * (c * y)","context":["use","assume","defn","defn"]} -{"prover":"Minlog","tactic_proof":"ng rw ng elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1058","id":89557,"instance":1058,"goal":"all c b, c * b = b * c","context":["ng","rw","ng","elim"]} -{"prover":"Minlog","tactic_proof":"elim ind assume cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1059","id":89558,"instance":1059,"goal":"all h d w, h * (d + w) = h * d + h * w","context":["elim","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"defn simp rw defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1060","id":89559,"instance":1060,"goal":"all f, f <= f","context":["defn","simp","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ind defn intro ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1061","id":89560,"instance":1061,"goal":"all p h f, p <= h -> h <= f -> p <= f","context":["ind","defn","intro","ng"]} -{"prover":"Minlog","tactic_proof":"elim simp elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1062","id":89561,"instance":1062,"goal":"all a, not a < a","context":["elim","simp","elim","ind"]} -{"prover":"Minlog","tactic_proof":"intro cases intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1063","id":89562,"instance":1063,"goal":"all d y, Even d -> Even y -> Even (d + y)","context":["intro","cases","intro","defn"]} -{"prover":"Minlog","tactic_proof":"rw cases ind cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1064","id":89563,"instance":1064,"goal":"all b, b | b","context":["rw","cases","ind","cases"]} -{"prover":"Minlog","tactic_proof":"cases intro rw rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1065","id":89564,"instance":1065,"goal":"all w, Fact w > 0","context":["cases","intro","rw","rw"]} -{"prover":"Minlog","tactic_proof":"simplify rw elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1066","id":89565,"instance":1066,"goal":"all h c = c","context":["simplify","rw","elim","rw"]} -{"prover":"Minlog","tactic_proof":"use intro cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1067","id":89566,"instance":1067,"goal":"all f + 0 = f","context":["use","intro","cases","intro"]} -{"prover":"Minlog","tactic_proof":"intro defn elim rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1068","id":89567,"instance":1068,"goal":"all g p, g + p = p + g","context":["intro","defn","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ng intro ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1069","id":89568,"instance":1069,"goal":"all y d m, (y + d) + m = y + (d + m)","context":["ng","intro","ind","rw"]} -{"prover":"Minlog","tactic_proof":"ind defn ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1070","id":89569,"instance":1070,"goal":"all z, z * 0 = 0","context":["ind","defn","ng","defn"]} -{"prover":"Minlog","tactic_proof":"assume ind ind simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1071","id":89570,"instance":1071,"goal":"all u, u * 1 = u","context":["assume","ind","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"rw assume simplify defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1072","id":89571,"instance":1072,"goal":"all q p e, (q * p) * e = q * (p * e)","context":["rw","assume","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"intro cases simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1073","id":89572,"instance":1073,"goal":"all g b, g * b = b * g","context":["intro","cases","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"ng cases cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1074","id":89573,"instance":1074,"goal":"all y m f, y * (m + f) = y * m + y * f","context":["ng","cases","cases","ng"]} -{"prover":"Minlog","tactic_proof":"intro ng ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1075","id":89574,"instance":1075,"goal":"all y, y <= y","context":["intro","ng","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro rw intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1076","id":89575,"instance":1076,"goal":"all c q p, c <= q -> q <= p -> c <= p","context":["intro","rw","intro","ng"]} -{"prover":"Minlog","tactic_proof":"cases cases simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1077","id":89576,"instance":1077,"goal":"all g, not g < g","context":["cases","cases","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"simp simp elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1078","id":89577,"instance":1078,"goal":"all n q, Even n -> Even q -> Even (n + q)","context":["simp","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"intro rw ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1079","id":89578,"instance":1079,"goal":"all g, g | g","context":["intro","rw","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume elim cases ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1080","id":89579,"instance":1080,"goal":"all e, Fact e > 0","context":["assume","elim","cases","ind"]} -{"prover":"Minlog","tactic_proof":"simp simp ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1081","id":89580,"instance":1081,"goal":"all f v = v","context":["simp","simp","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simp use rw cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1082","id":89581,"instance":1082,"goal":"all q + 0 = q","context":["simp","use","rw","cases"]} -{"prover":"Minlog","tactic_proof":"simplify assume ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1083","id":89582,"instance":1083,"goal":"all w x, w + x = x + w","context":["simplify","assume","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"intro ng intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1084","id":89583,"instance":1084,"goal":"all c y e, (c + y) + e = c + (y + e)","context":["intro","ng","intro","assume"]} -{"prover":"Minlog","tactic_proof":"defn elim rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1085","id":89584,"instance":1085,"goal":"all d, d * 0 = 0","context":["defn","elim","rw","ng"]} -{"prover":"Minlog","tactic_proof":"defn use simplify use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1086","id":89585,"instance":1086,"goal":"all g, g * 1 = g","context":["defn","use","simplify","use"]} -{"prover":"Minlog","tactic_proof":"simplify cases ind ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1087","id":89586,"instance":1087,"goal":"all e m v, (e * m) * v = e * (m * v)","context":["simplify","cases","ind","ng"]} -{"prover":"Minlog","tactic_proof":"ind simplify use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1088","id":89587,"instance":1088,"goal":"all x m, x * m = m * x","context":["ind","simplify","use","simplify"]} -{"prover":"Minlog","tactic_proof":"rw intro defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1089","id":89588,"instance":1089,"goal":"all z f h, z * (f + h) = z * f + z * h","context":["rw","intro","defn","assume"]} -{"prover":"Minlog","tactic_proof":"intro assume ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1090","id":89589,"instance":1090,"goal":"all p, p <= p","context":["intro","assume","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simp defn rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1091","id":89590,"instance":1091,"goal":"all w p x, w <= p -> p <= x -> w <= x","context":["assume","simp","defn","rw"]} -{"prover":"Minlog","tactic_proof":"simplify use assume ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1092","id":89591,"instance":1092,"goal":"all y, not y < y","context":["simplify","use","assume","ng"]} -{"prover":"Minlog","tactic_proof":"cases rw ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1093","id":89592,"instance":1093,"goal":"all y z, Even y -> Even z -> Even (y + z)","context":["cases","rw","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simplify intro defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1094","id":89593,"instance":1094,"goal":"all x, x | x","context":["simplify","intro","defn","rw"]} -{"prover":"Minlog","tactic_proof":"ind rw defn elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1095","id":89594,"instance":1095,"goal":"all x, Fact x > 0","context":["ind","rw","defn","elim"]} -{"prover":"Minlog","tactic_proof":"cases cases ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1096","id":89595,"instance":1096,"goal":"all n a = a","context":["cases","cases","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simp simp ng simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1097","id":89596,"instance":1097,"goal":"all e + 0 = e","context":["simp","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume defn defn elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1098","id":89597,"instance":1098,"goal":"all u v, u + v = v + u","context":["assume","defn","defn","elim"]} -{"prover":"Minlog","tactic_proof":"simplify use ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1099","id":89598,"instance":1099,"goal":"all m b y, (m + b) + y = m + (b + y)","context":["simplify","use","ng","ind"]} -{"prover":"Minlog","tactic_proof":"intro ind cases cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1100","id":89599,"instance":1100,"goal":"all n, n * 0 = 0","context":["intro","ind","cases","cases"]} -{"prover":"Minlog","tactic_proof":"defn simp simplify assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1101","id":89600,"instance":1101,"goal":"all g, g * 1 = g","context":["defn","simp","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1102","id":89601,"instance":1102,"goal":"all f h n, (f * h) * n = f * (h * n)","context":["elim","ng","use","simp"]} -{"prover":"Minlog","tactic_proof":"defn ng intro assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1103","id":89602,"instance":1103,"goal":"all m x, m * x = x * m","context":["defn","ng","intro","assume"]} -{"prover":"Minlog","tactic_proof":"simplify ind cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1104","id":89603,"instance":1104,"goal":"all d c z, d * (c + z) = d * c + d * z","context":["simplify","ind","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use elim cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1105","id":89604,"instance":1105,"goal":"all d, d <= d","context":["use","elim","cases","assume"]} -{"prover":"Minlog","tactic_proof":"assume defn cases ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1106","id":89605,"instance":1106,"goal":"all x m p, x <= m -> m <= p -> x <= p","context":["assume","defn","cases","ng"]} -{"prover":"Minlog","tactic_proof":"simp simplify intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1107","id":89606,"instance":1107,"goal":"all y, not y < y","context":["simp","simplify","intro","assume"]} -{"prover":"Minlog","tactic_proof":"use cases intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1108","id":89607,"instance":1108,"goal":"all g u, Even g -> Even u -> Even (g + u)","context":["use","cases","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ng elim simp intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1109","id":89608,"instance":1109,"goal":"all x, x | x","context":["ng","elim","simp","intro"]} -{"prover":"Minlog","tactic_proof":"assume simp cases cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1110","id":89609,"instance":1110,"goal":"all m, Fact m > 0","context":["assume","simp","cases","cases"]} -{"prover":"Minlog","tactic_proof":"simplify rw ind ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1111","id":89610,"instance":1111,"goal":"all a p = p","context":["simplify","rw","ind","ng"]} -{"prover":"Minlog","tactic_proof":"cases ind elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1112","id":89611,"instance":1112,"goal":"all n + 0 = n","context":["cases","ind","elim","assume"]} -{"prover":"Minlog","tactic_proof":"simp defn ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1113","id":89612,"instance":1113,"goal":"all a u, a + u = u + a","context":["simp","defn","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"intro use intro intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1114","id":89613,"instance":1114,"goal":"all n u z, (n + u) + z = n + (u + z)","context":["intro","use","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ind rw simp rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1115","id":89614,"instance":1115,"goal":"all m, m * 0 = 0","context":["ind","rw","simp","rw"]} -{"prover":"Minlog","tactic_proof":"use ng intro ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1116","id":89615,"instance":1116,"goal":"all b, b * 1 = b","context":["use","ng","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ind assume simplify use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1117","id":89616,"instance":1117,"goal":"all u w c, (u * w) * c = u * (w * c)","context":["ind","assume","simplify","use"]} -{"prover":"Minlog","tactic_proof":"intro defn cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1118","id":89617,"instance":1118,"goal":"all x a, x * a = a * x","context":["intro","defn","cases","cases"]} -{"prover":"Minlog","tactic_proof":"ng intro elim use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1119","id":89618,"instance":1119,"goal":"all a z u, a * (z + u) = a * z + a * u","context":["ng","intro","elim","use"]} -{"prover":"Minlog","tactic_proof":"cases simp rw ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1120","id":89619,"instance":1120,"goal":"all a, a <= a","context":["cases","simp","rw","ng"]} -{"prover":"Minlog","tactic_proof":"intro assume intro rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1121","id":89620,"instance":1121,"goal":"all a z p, a <= z -> z <= p -> a <= p","context":["intro","assume","intro","rw"]} -{"prover":"Minlog","tactic_proof":"intro defn use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1122","id":89621,"instance":1122,"goal":"all g, not g < g","context":["intro","defn","use","cases"]} -{"prover":"Minlog","tactic_proof":"simplify rw use simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1123","id":89622,"instance":1123,"goal":"all p x, Even p -> Even x -> Even (p + x)","context":["simplify","rw","use","simplify"]} -{"prover":"Minlog","tactic_proof":"rw ind rw ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1124","id":89623,"instance":1124,"goal":"all n, n | n","context":["rw","ind","rw","ind"]} -{"prover":"Minlog","tactic_proof":"simp ng ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1125","id":89624,"instance":1125,"goal":"all h, Fact h > 0","context":["simp","ng","ind","rw"]} -{"prover":"Minlog","tactic_proof":"assume elim rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1126","id":89625,"instance":1126,"goal":"all x w = w","context":["assume","elim","rw","simp"]} -{"prover":"Minlog","tactic_proof":"simp elim cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1127","id":89626,"instance":1127,"goal":"all w + 0 = w","context":["simp","elim","cases","cases"]} -{"prover":"Minlog","tactic_proof":"assume ind simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1128","id":89627,"instance":1128,"goal":"all b m, b + m = m + b","context":["assume","ind","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ind use use intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1129","id":89628,"instance":1129,"goal":"all u h g, (u + h) + g = u + (h + g)","context":["ind","use","use","intro"]} -{"prover":"Minlog","tactic_proof":"rw ng simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1130","id":89629,"instance":1130,"goal":"all q, q * 0 = 0","context":["rw","ng","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng simplify use rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1131","id":89630,"instance":1131,"goal":"all d, d * 1 = d","context":["ng","simplify","use","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn rw use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1132","id":89631,"instance":1132,"goal":"all y u n, (y * u) * n = y * (u * n)","context":["ng","defn","rw","use"]} -{"prover":"Minlog","tactic_proof":"cases rw rw ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1133","id":89632,"instance":1133,"goal":"all g x, g * x = x * g","context":["cases","rw","rw","ind"]} -{"prover":"Minlog","tactic_proof":"simp cases assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1134","id":89633,"instance":1134,"goal":"all w g f, w * (g + f) = w * g + w * f","context":["simp","cases","assume","defn"]} -{"prover":"Minlog","tactic_proof":"elim assume assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1135","id":89634,"instance":1135,"goal":"all a, a <= a","context":["elim","assume","assume","intro"]} -{"prover":"Minlog","tactic_proof":"cases assume ng simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1136","id":89635,"instance":1136,"goal":"all g f n, g <= f -> f <= n -> g <= n","context":["cases","assume","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify elim simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1137","id":89636,"instance":1137,"goal":"all q, not q < q","context":["simplify","elim","simp","use"]} -{"prover":"Minlog","tactic_proof":"assume simp rw simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1138","id":89637,"instance":1138,"goal":"all f e, Even f -> Even e -> Even (f + e)","context":["assume","simp","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"cases use intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1139","id":89638,"instance":1139,"goal":"all a, a | a","context":["cases","use","intro","cases"]} -{"prover":"Minlog","tactic_proof":"intro cases simplify cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1140","id":89639,"instance":1140,"goal":"all q, Fact q > 0","context":["intro","cases","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"use assume defn ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1141","id":89640,"instance":1141,"goal":"all p q = q","context":["use","assume","defn","ind"]} -{"prover":"Minlog","tactic_proof":"ind use simp intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1142","id":89641,"instance":1142,"goal":"all g + 0 = g","context":["ind","use","simp","intro"]} -{"prover":"Minlog","tactic_proof":"simp rw ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1143","id":89642,"instance":1143,"goal":"all b p, b + p = p + b","context":["simp","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simp assume use defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1144","id":89643,"instance":1144,"goal":"all e z w, (e + z) + w = e + (z + w)","context":["simp","assume","use","defn"]} -{"prover":"Minlog","tactic_proof":"assume defn simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1145","id":89644,"instance":1145,"goal":"all b, b * 0 = 0","context":["assume","defn","simp","use"]} -{"prover":"Minlog","tactic_proof":"simplify simplify cases assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1146","id":89645,"instance":1146,"goal":"all e, e * 1 = e","context":["simplify","simplify","cases","assume"]} -{"prover":"Minlog","tactic_proof":"ng simplify simp simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1147","id":89646,"instance":1147,"goal":"all p c b, (p * c) * b = p * (c * b)","context":["ng","simplify","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"rw rw rw ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1148","id":89647,"instance":1148,"goal":"all z c, z * c = c * z","context":["rw","rw","rw","ng"]} -{"prover":"Minlog","tactic_proof":"simplify use defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1149","id":89648,"instance":1149,"goal":"all e c a, e * (c + a) = e * c + e * a","context":["simplify","use","defn","elim"]} -{"prover":"Minlog","tactic_proof":"ng use ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1150","id":89649,"instance":1150,"goal":"all c, c <= c","context":["ng","use","ind","defn"]} -{"prover":"Minlog","tactic_proof":"ng simp defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1151","id":89650,"instance":1151,"goal":"all g h a, g <= h -> h <= a -> g <= a","context":["ng","simp","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"cases cases ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1152","id":89651,"instance":1152,"goal":"all c, not c < c","context":["cases","cases","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ng defn ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1153","id":89652,"instance":1153,"goal":"all h y, Even h -> Even y -> Even (h + y)","context":["ng","defn","ng","cases"]} -{"prover":"Minlog","tactic_proof":"assume cases elim defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1154","id":89653,"instance":1154,"goal":"all b, b | b","context":["assume","cases","elim","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1155","id":89654,"instance":1155,"goal":"all c, Fact c > 0","context":["ng","defn","elim","intro"]} -{"prover":"Minlog","tactic_proof":"assume ind assume cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1156","id":89655,"instance":1156,"goal":"all y u = u","context":["assume","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"ind simp elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1157","id":89656,"instance":1157,"goal":"all x + 0 = x","context":["ind","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"simp ind ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1158","id":89657,"instance":1158,"goal":"all f b, f + b = b + f","context":["simp","ind","ind","defn"]} -{"prover":"Minlog","tactic_proof":"defn simp assume rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1159","id":89658,"instance":1159,"goal":"all n p q, (n + p) + q = n + (p + q)","context":["defn","simp","assume","rw"]} -{"prover":"Minlog","tactic_proof":"cases rw rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1160","id":89659,"instance":1160,"goal":"all g, g * 0 = 0","context":["cases","rw","rw","simp"]} -{"prover":"Minlog","tactic_proof":"elim cases simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1161","id":89660,"instance":1161,"goal":"all b, b * 1 = b","context":["elim","cases","simplify","use"]} -{"prover":"Minlog","tactic_proof":"defn ng intro rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1162","id":89661,"instance":1162,"goal":"all u p a, (u * p) * a = u * (p * a)","context":["defn","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"assume assume use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1163","id":89662,"instance":1163,"goal":"all p v, p * v = v * p","context":["assume","assume","use","rw"]} -{"prover":"Minlog","tactic_proof":"intro rw intro cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1164","id":89663,"instance":1164,"goal":"all f g y, f * (g + y) = f * g + f * y","context":["intro","rw","intro","cases"]} -{"prover":"Minlog","tactic_proof":"use ng defn intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1165","id":89664,"instance":1165,"goal":"all q, q <= q","context":["use","ng","defn","intro"]} -{"prover":"Minlog","tactic_proof":"simplify intro simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1166","id":89665,"instance":1166,"goal":"all h x n, h <= x -> x <= n -> h <= n","context":["simplify","intro","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"assume rw ind ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1167","id":89666,"instance":1167,"goal":"all m, not m < m","context":["assume","rw","ind","ind"]} -{"prover":"Minlog","tactic_proof":"elim rw assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1168","id":89667,"instance":1168,"goal":"all d e, Even d -> Even e -> Even (d + e)","context":["elim","rw","assume","assume"]} -{"prover":"Minlog","tactic_proof":"cases cases defn elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1169","id":89668,"instance":1169,"goal":"all b, b | b","context":["cases","cases","defn","elim"]} -{"prover":"Minlog","tactic_proof":"intro ind rw defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1170","id":89669,"instance":1170,"goal":"all g, Fact g > 0","context":["intro","ind","rw","defn"]} -{"prover":"Minlog","tactic_proof":"simp ind ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1171","id":89670,"instance":1171,"goal":"all c n = n","context":["simp","ind","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"cases cases use cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1172","id":89671,"instance":1172,"goal":"all c + 0 = c","context":["cases","cases","use","cases"]} -{"prover":"Minlog","tactic_proof":"simp elim cases defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1173","id":89672,"instance":1173,"goal":"all c a, c + a = a + c","context":["simp","elim","cases","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn simplify ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1174","id":89673,"instance":1174,"goal":"all h z x, (h + z) + x = h + (z + x)","context":["ng","defn","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"assume use ind ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1175","id":89674,"instance":1175,"goal":"all u, u * 0 = 0","context":["assume","use","ind","ng"]} -{"prover":"Minlog","tactic_proof":"intro rw simplify ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1176","id":89675,"instance":1176,"goal":"all a, a * 1 = a","context":["intro","rw","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ind simp cases elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1177","id":89676,"instance":1177,"goal":"all a n c, (a * n) * c = a * (n * c)","context":["ind","simp","cases","elim"]} -{"prover":"Minlog","tactic_proof":"rw ind use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1178","id":89677,"instance":1178,"goal":"all m f, m * f = f * m","context":["rw","ind","use","defn"]} -{"prover":"Minlog","tactic_proof":"ind cases simp elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1179","id":89678,"instance":1179,"goal":"all z b v, z * (b + v) = z * b + z * v","context":["ind","cases","simp","elim"]} -{"prover":"Minlog","tactic_proof":"rw cases cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1180","id":89679,"instance":1180,"goal":"all u, u <= u","context":["rw","cases","cases","intro"]} -{"prover":"Minlog","tactic_proof":"simplify simplify elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1181","id":89680,"instance":1181,"goal":"all v p u, v <= p -> p <= u -> v <= u","context":["simplify","simplify","elim","ind"]} -{"prover":"Minlog","tactic_proof":"elim intro cases intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1182","id":89681,"instance":1182,"goal":"all v, not v < v","context":["elim","intro","cases","intro"]} -{"prover":"Minlog","tactic_proof":"use defn defn rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1183","id":89682,"instance":1183,"goal":"all w y, Even w -> Even y -> Even (w + y)","context":["use","defn","defn","rw"]} -{"prover":"Minlog","tactic_proof":"rw intro intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1184","id":89683,"instance":1184,"goal":"all c, c | c","context":["rw","intro","intro","cases"]} -{"prover":"Minlog","tactic_proof":"simp use elim rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1185","id":89684,"instance":1185,"goal":"all n, Fact n > 0","context":["simp","use","elim","rw"]} -{"prover":"Minlog","tactic_proof":"cases cases intro ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1186","id":89685,"instance":1186,"goal":"all x q = q","context":["cases","cases","intro","ind"]} -{"prover":"Minlog","tactic_proof":"intro ind ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1187","id":89686,"instance":1187,"goal":"all d + 0 = d","context":["intro","ind","ng","simp"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1188","id":89687,"instance":1188,"goal":"all f p, f + p = p + f","context":["ind","ng","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"ind simplify simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1189","id":89688,"instance":1189,"goal":"all e y a, (e + y) + a = e + (y + a)","context":["ind","simplify","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"intro defn use simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1190","id":89689,"instance":1190,"goal":"all n, n * 0 = 0","context":["intro","defn","use","simplify"]} -{"prover":"Minlog","tactic_proof":"rw intro cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1191","id":89690,"instance":1191,"goal":"all e, e * 1 = e","context":["rw","intro","cases","defn"]} -{"prover":"Minlog","tactic_proof":"simplify ng simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1192","id":89691,"instance":1192,"goal":"all y p h, (y * p) * h = y * (p * h)","context":["simplify","ng","simp","ind"]} -{"prover":"Minlog","tactic_proof":"assume use ng use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1193","id":89692,"instance":1193,"goal":"all b p, b * p = p * b","context":["assume","use","ng","use"]} -{"prover":"Minlog","tactic_proof":"use use use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1194","id":89693,"instance":1194,"goal":"all e m c, e * (m + c) = e * m + e * c","context":["use","use","use","use"]} -{"prover":"Minlog","tactic_proof":"defn ng assume elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1195","id":89694,"instance":1195,"goal":"all n, n <= n","context":["defn","ng","assume","elim"]} -{"prover":"Minlog","tactic_proof":"simp simp simp simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1196","id":89695,"instance":1196,"goal":"all x w m, x <= w -> w <= m -> x <= m","context":["simp","simp","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ind elim cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1197","id":89696,"instance":1197,"goal":"all b, not b < b","context":["ind","elim","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"elim assume elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1198","id":89697,"instance":1198,"goal":"all m c, Even m -> Even c -> Even (m + c)","context":["elim","assume","elim","intro"]} -{"prover":"Minlog","tactic_proof":"use ind elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1199","id":89698,"instance":1199,"goal":"all x, x | x","context":["use","ind","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"elim cases cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1200","id":89699,"instance":1200,"goal":"all b, Fact b > 0","context":["elim","cases","cases","assume"]} -{"prover":"Minlog","tactic_proof":"rw rw simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1201","id":89700,"instance":1201,"goal":"all c a = a","context":["rw","rw","simp","ind"]} -{"prover":"Minlog","tactic_proof":"assume rw simp defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1202","id":89701,"instance":1202,"goal":"all a + 0 = a","context":["assume","rw","simp","defn"]} -{"prover":"Minlog","tactic_proof":"ind assume defn simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1203","id":89702,"instance":1203,"goal":"all w y, w + y = y + w","context":["ind","assume","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify simplify rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1204","id":89703,"instance":1204,"goal":"all b u z, (b + u) + z = b + (u + z)","context":["simplify","simplify","rw","assume"]} -{"prover":"Minlog","tactic_proof":"defn assume rw use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1205","id":89704,"instance":1205,"goal":"all y, y * 0 = 0","context":["defn","assume","rw","use"]} -{"prover":"Minlog","tactic_proof":"cases cases assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1206","id":89705,"instance":1206,"goal":"all y, y * 1 = y","context":["cases","cases","assume","defn"]} -{"prover":"Minlog","tactic_proof":"cases assume assume simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1207","id":89706,"instance":1207,"goal":"all z b d, (z * b) * d = z * (b * d)","context":["cases","assume","assume","simp"]} -{"prover":"Minlog","tactic_proof":"cases simplify defn use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1208","id":89707,"instance":1208,"goal":"all y b, y * b = b * y","context":["cases","simplify","defn","use"]} -{"prover":"Minlog","tactic_proof":"simp use defn defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1209","id":89708,"instance":1209,"goal":"all h n q, h * (n + q) = h * n + h * q","context":["simp","use","defn","defn"]} -{"prover":"Minlog","tactic_proof":"cases cases cases ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1210","id":89709,"instance":1210,"goal":"all d, d <= d","context":["cases","cases","cases","ind"]} -{"prover":"Minlog","tactic_proof":"rw cases rw intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1211","id":89710,"instance":1211,"goal":"all c w f, c <= w -> w <= f -> c <= f","context":["rw","cases","rw","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases simp rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1212","id":89711,"instance":1212,"goal":"all b, not b < b","context":["intro","cases","simp","rw"]} -{"prover":"Minlog","tactic_proof":"elim assume use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1213","id":89712,"instance":1213,"goal":"all q b, Even q -> Even b -> Even (q + b)","context":["elim","assume","use","simp"]} -{"prover":"Minlog","tactic_proof":"rw cases ng rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1214","id":89713,"instance":1214,"goal":"all f, f | f","context":["rw","cases","ng","rw"]} -{"prover":"Minlog","tactic_proof":"intro simp use ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1215","id":89714,"instance":1215,"goal":"all h, Fact h > 0","context":["intro","simp","use","ind"]} -{"prover":"Minlog","tactic_proof":"rw cases ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1216","id":89715,"instance":1216,"goal":"all d b = b","context":["rw","cases","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"elim use simplify ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1217","id":89716,"instance":1217,"goal":"all y + 0 = y","context":["elim","use","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ng use defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1218","id":89717,"instance":1218,"goal":"all y f, y + f = f + y","context":["ng","use","defn","rw"]} -{"prover":"Minlog","tactic_proof":"ind ind intro ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1219","id":89718,"instance":1219,"goal":"all n g v, (n + g) + v = n + (g + v)","context":["ind","ind","intro","ng"]} -{"prover":"Minlog","tactic_proof":"cases ng rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1220","id":89719,"instance":1220,"goal":"all g, g * 0 = 0","context":["cases","ng","rw","assume"]} -{"prover":"Minlog","tactic_proof":"use simplify simp rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1221","id":89720,"instance":1221,"goal":"all n, n * 1 = n","context":["use","simplify","simp","rw"]} -{"prover":"Minlog","tactic_proof":"ng elim defn assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1222","id":89721,"instance":1222,"goal":"all n b u, (n * b) * u = n * (b * u)","context":["ng","elim","defn","assume"]} -{"prover":"Minlog","tactic_proof":"assume elim ind elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1223","id":89722,"instance":1223,"goal":"all b n, b * n = n * b","context":["assume","elim","ind","elim"]} -{"prover":"Minlog","tactic_proof":"use elim rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1224","id":89723,"instance":1224,"goal":"all g n z, g * (n + z) = g * n + g * z","context":["use","elim","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp use ind defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1225","id":89724,"instance":1225,"goal":"all f, f <= f","context":["simp","use","ind","defn"]} -{"prover":"Minlog","tactic_proof":"use assume ind assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1226","id":89725,"instance":1226,"goal":"all d x a, d <= x -> x <= a -> d <= a","context":["use","assume","ind","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1227","id":89726,"instance":1227,"goal":"all d, not d < d","context":["intro","intro","elim","assume"]} -{"prover":"Minlog","tactic_proof":"ng assume intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1228","id":89727,"instance":1228,"goal":"all f e, Even f -> Even e -> Even (f + e)","context":["ng","assume","intro","use"]} -{"prover":"Minlog","tactic_proof":"rw elim rw cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1229","id":89728,"instance":1229,"goal":"all z, z | z","context":["rw","elim","rw","cases"]} -{"prover":"Minlog","tactic_proof":"simplify use ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1230","id":89729,"instance":1230,"goal":"all g, Fact g > 0","context":["simplify","use","ng","ind"]} -{"prover":"Minlog","tactic_proof":"ind cases simplify ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1231","id":89730,"instance":1231,"goal":"all a m = m","context":["ind","cases","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"ind use ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1232","id":89731,"instance":1232,"goal":"all p + 0 = p","context":["ind","use","ng","defn"]} -{"prover":"Minlog","tactic_proof":"elim cases intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1233","id":89732,"instance":1233,"goal":"all z w, z + w = w + z","context":["elim","cases","intro","elim"]} -{"prover":"Minlog","tactic_proof":"rw ind simp elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1234","id":89733,"instance":1234,"goal":"all p e u, (p + e) + u = p + (e + u)","context":["rw","ind","simp","elim"]} -{"prover":"Minlog","tactic_proof":"cases defn use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1235","id":89734,"instance":1235,"goal":"all q, q * 0 = 0","context":["cases","defn","use","elim"]} -{"prover":"Minlog","tactic_proof":"defn cases use rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1236","id":89735,"instance":1236,"goal":"all h, h * 1 = h","context":["defn","cases","use","rw"]} -{"prover":"Minlog","tactic_proof":"intro assume rw cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1237","id":89736,"instance":1237,"goal":"all x w n, (x * w) * n = x * (w * n)","context":["intro","assume","rw","cases"]} -{"prover":"Minlog","tactic_proof":"cases assume elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1238","id":89737,"instance":1238,"goal":"all w y, w * y = y * w","context":["cases","assume","elim","cases"]} -{"prover":"Minlog","tactic_proof":"elim defn simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1239","id":89738,"instance":1239,"goal":"all m b g, m * (b + g) = m * b + m * g","context":["elim","defn","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"elim defn elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1240","id":89739,"instance":1240,"goal":"all z, z <= z","context":["elim","defn","elim","assume"]} -{"prover":"Minlog","tactic_proof":"simp simplify cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1241","id":89740,"instance":1241,"goal":"all g m f, g <= m -> m <= f -> g <= f","context":["simp","simplify","cases","intro"]} -{"prover":"Minlog","tactic_proof":"assume intro intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1242","id":89741,"instance":1242,"goal":"all h, not h < h","context":["assume","intro","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"ind defn simp intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1243","id":89742,"instance":1243,"goal":"all a v, Even a -> Even v -> Even (a + v)","context":["ind","defn","simp","intro"]} -{"prover":"Minlog","tactic_proof":"use simplify elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1244","id":89743,"instance":1244,"goal":"all g, g | g","context":["use","simplify","elim","elim"]} -{"prover":"Minlog","tactic_proof":"cases intro use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1245","id":89744,"instance":1245,"goal":"all a, Fact a > 0","context":["cases","intro","use","intro"]} -{"prover":"Minlog","tactic_proof":"elim rw assume defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1246","id":89745,"instance":1246,"goal":"all d u = u","context":["elim","rw","assume","defn"]} -{"prover":"Minlog","tactic_proof":"use simplify simp cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1247","id":89746,"instance":1247,"goal":"all e + 0 = e","context":["use","simplify","simp","cases"]} -{"prover":"Minlog","tactic_proof":"use simplify assume use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1248","id":89747,"instance":1248,"goal":"all z q, z + q = q + z","context":["use","simplify","assume","use"]} -{"prover":"Minlog","tactic_proof":"use use assume ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1249","id":89748,"instance":1249,"goal":"all e q a, (e + q) + a = e + (q + a)","context":["use","use","assume","ng"]} -{"prover":"Minlog","tactic_proof":"ng rw simp defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1250","id":89749,"instance":1250,"goal":"all q, q * 0 = 0","context":["ng","rw","simp","defn"]} -{"prover":"Minlog","tactic_proof":"simplify simp elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1251","id":89750,"instance":1251,"goal":"all v, v * 1 = v","context":["simplify","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"defn cases ng ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1252","id":89751,"instance":1252,"goal":"all d q p, (d * q) * p = d * (q * p)","context":["defn","cases","ng","ng"]} -{"prover":"Minlog","tactic_proof":"simplify simp elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1253","id":89752,"instance":1253,"goal":"all z n, z * n = n * z","context":["simplify","simp","elim","rw"]} -{"prover":"Minlog","tactic_proof":"intro simplify ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1254","id":89753,"instance":1254,"goal":"all x q v, x * (q + v) = x * q + x * v","context":["intro","simplify","ng","elim"]} -{"prover":"Minlog","tactic_proof":"simplify rw cases simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1255","id":89754,"instance":1255,"goal":"all g, g <= g","context":["simplify","rw","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"assume elim defn intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1256","id":89755,"instance":1256,"goal":"all q w n, q <= w -> w <= n -> q <= n","context":["assume","elim","defn","intro"]} -{"prover":"Minlog","tactic_proof":"assume ng elim defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1257","id":89756,"instance":1257,"goal":"all g, not g < g","context":["assume","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"elim defn use elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1258","id":89757,"instance":1258,"goal":"all q b, Even q -> Even b -> Even (q + b)","context":["elim","defn","use","elim"]} -{"prover":"Minlog","tactic_proof":"rw intro defn intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1259","id":89758,"instance":1259,"goal":"all g, g | g","context":["rw","intro","defn","intro"]} -{"prover":"Minlog","tactic_proof":"cases ind simp simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1260","id":89759,"instance":1260,"goal":"all u, Fact u > 0","context":["cases","ind","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ind simp defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1261","id":89760,"instance":1261,"goal":"all x z = z","context":["cases","ind","simp","defn"]} -{"prover":"Minlog","tactic_proof":"use ind use rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1262","id":89761,"instance":1262,"goal":"all w + 0 = w","context":["use","ind","use","rw"]} -{"prover":"Minlog","tactic_proof":"use use defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1263","id":89762,"instance":1263,"goal":"all d y, d + y = y + d","context":["use","use","defn","elim"]} -{"prover":"Minlog","tactic_proof":"ng simplify ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1264","id":89763,"instance":1264,"goal":"all c y x, (c + y) + x = c + (y + x)","context":["ng","simplify","ng","cases"]} -{"prover":"Minlog","tactic_proof":"elim rw rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1265","id":89764,"instance":1265,"goal":"all y, y * 0 = 0","context":["elim","rw","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simplify cases elim assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1266","id":89765,"instance":1266,"goal":"all h, h * 1 = h","context":["simplify","cases","elim","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simplify rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1267","id":89766,"instance":1267,"goal":"all m c u, (m * c) * u = m * (c * u)","context":["simplify","simplify","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"ind defn ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1268","id":89767,"instance":1268,"goal":"all x m, x * m = m * x","context":["ind","defn","ind","elim"]} -{"prover":"Minlog","tactic_proof":"ng ind defn use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1269","id":89768,"instance":1269,"goal":"all n e p, n * (e + p) = n * e + n * p","context":["ng","ind","defn","use"]} -{"prover":"Minlog","tactic_proof":"simp use rw simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1270","id":89769,"instance":1270,"goal":"all q, q <= q","context":["simp","use","rw","simp"]} -{"prover":"Minlog","tactic_proof":"simp simp rw ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1271","id":89770,"instance":1271,"goal":"all a d q, a <= d -> d <= q -> a <= q","context":["simp","simp","rw","ind"]} -{"prover":"Minlog","tactic_proof":"elim ng intro cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1272","id":89771,"instance":1272,"goal":"all e, not e < e","context":["elim","ng","intro","cases"]} -{"prover":"Minlog","tactic_proof":"cases ind ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1273","id":89772,"instance":1273,"goal":"all m e, Even m -> Even e -> Even (m + e)","context":["cases","ind","ind","rw"]} -{"prover":"Minlog","tactic_proof":"assume use assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1274","id":89773,"instance":1274,"goal":"all h, h | h","context":["assume","use","assume","intro"]} -{"prover":"Minlog","tactic_proof":"ind ng use cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1275","id":89774,"instance":1275,"goal":"all e, Fact e > 0","context":["ind","ng","use","cases"]} -{"prover":"Minlog","tactic_proof":"assume rw defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1276","id":89775,"instance":1276,"goal":"all d v = v","context":["assume","rw","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"assume rw defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1277","id":89776,"instance":1277,"goal":"all h + 0 = h","context":["assume","rw","defn","simp"]} -{"prover":"Minlog","tactic_proof":"ng simplify cases elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1278","id":89777,"instance":1278,"goal":"all a v, a + v = v + a","context":["ng","simplify","cases","elim"]} -{"prover":"Minlog","tactic_proof":"cases simp simp simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1279","id":89778,"instance":1279,"goal":"all b e v, (b + e) + v = b + (e + v)","context":["cases","simp","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ng rw simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1280","id":89779,"instance":1280,"goal":"all q, q * 0 = 0","context":["ng","rw","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"ind assume intro simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1281","id":89780,"instance":1281,"goal":"all g, g * 1 = g","context":["ind","assume","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp intro elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1282","id":89781,"instance":1282,"goal":"all c a w, (c * a) * w = c * (a * w)","context":["simp","intro","elim","elim"]} -{"prover":"Minlog","tactic_proof":"intro elim ng ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1283","id":89782,"instance":1283,"goal":"all e g, e * g = g * e","context":["intro","elim","ng","ind"]} -{"prover":"Minlog","tactic_proof":"defn rw ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1284","id":89783,"instance":1284,"goal":"all h y z, h * (y + z) = h * y + h * z","context":["defn","rw","ng","cases"]} -{"prover":"Minlog","tactic_proof":"use simplify intro elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1285","id":89784,"instance":1285,"goal":"all a, a <= a","context":["use","simplify","intro","elim"]} -{"prover":"Minlog","tactic_proof":"defn cases cases simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1286","id":89785,"instance":1286,"goal":"all f x z, f <= x -> x <= z -> f <= z","context":["defn","cases","cases","simp"]} -{"prover":"Minlog","tactic_proof":"elim rw ind cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1287","id":89786,"instance":1287,"goal":"all z, not z < z","context":["elim","rw","ind","cases"]} -{"prover":"Minlog","tactic_proof":"defn cases defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1288","id":89787,"instance":1288,"goal":"all m h, Even m -> Even h -> Even (m + h)","context":["defn","cases","defn","cases"]} -{"prover":"Minlog","tactic_proof":"elim ind ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1289","id":89788,"instance":1289,"goal":"all p, p | p","context":["elim","ind","ng","ind"]} -{"prover":"Minlog","tactic_proof":"simplify elim elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1290","id":89789,"instance":1290,"goal":"all w, Fact w > 0","context":["simplify","elim","elim","intro"]} -{"prover":"Minlog","tactic_proof":"simp intro simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1291","id":89790,"instance":1291,"goal":"all e p = p","context":["simp","intro","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"defn defn ind assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1292","id":89791,"instance":1292,"goal":"all n + 0 = n","context":["defn","defn","ind","assume"]} -{"prover":"Minlog","tactic_proof":"ng assume intro elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1293","id":89792,"instance":1293,"goal":"all w h, w + h = h + w","context":["ng","assume","intro","elim"]} -{"prover":"Minlog","tactic_proof":"cases simp simp defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1294","id":89793,"instance":1294,"goal":"all f v z, (f + v) + z = f + (v + z)","context":["cases","simp","simp","defn"]} -{"prover":"Minlog","tactic_proof":"elim rw ng intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1295","id":89794,"instance":1295,"goal":"all a, a * 0 = 0","context":["elim","rw","ng","intro"]} -{"prover":"Minlog","tactic_proof":"simplify simplify ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1296","id":89795,"instance":1296,"goal":"all e, e * 1 = e","context":["simplify","simplify","ind","rw"]} -{"prover":"Minlog","tactic_proof":"cases intro defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1297","id":89796,"instance":1297,"goal":"all g h f, (g * h) * f = g * (h * f)","context":["cases","intro","defn","cases"]} -{"prover":"Minlog","tactic_proof":"rw defn rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1298","id":89797,"instance":1298,"goal":"all g f, g * f = f * g","context":["rw","defn","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases rw cases ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1299","id":89798,"instance":1299,"goal":"all m w z, m * (w + z) = m * w + m * z","context":["cases","rw","cases","ng"]} -{"prover":"Minlog","tactic_proof":"ind simp ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1300","id":89799,"instance":1300,"goal":"all f, f <= f","context":["ind","simp","ind","ind"]} -{"prover":"Minlog","tactic_proof":"intro elim simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1301","id":89800,"instance":1301,"goal":"all b y p, b <= y -> y <= p -> b <= p","context":["intro","elim","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"rw use use cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1302","id":89801,"instance":1302,"goal":"all m, not m < m","context":["rw","use","use","cases"]} -{"prover":"Minlog","tactic_proof":"assume rw ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1303","id":89802,"instance":1303,"goal":"all a b, Even a -> Even b -> Even (a + b)","context":["assume","rw","ng","ng"]} -{"prover":"Minlog","tactic_proof":"intro simplify elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1304","id":89803,"instance":1304,"goal":"all h, h | h","context":["intro","simplify","elim","use"]} -{"prover":"Minlog","tactic_proof":"elim rw ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1305","id":89804,"instance":1305,"goal":"all g, Fact g > 0","context":["elim","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"cases assume simplify assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1306","id":89805,"instance":1306,"goal":"all g y = y","context":["cases","assume","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"simplify intro rw ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1307","id":89806,"instance":1307,"goal":"all c + 0 = c","context":["simplify","intro","rw","ind"]} -{"prover":"Minlog","tactic_proof":"elim use elim intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1308","id":89807,"instance":1308,"goal":"all m a, m + a = a + m","context":["elim","use","elim","intro"]} -{"prover":"Minlog","tactic_proof":"cases ind assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1309","id":89808,"instance":1309,"goal":"all b h w, (b + h) + w = b + (h + w)","context":["cases","ind","assume","simp"]} -{"prover":"Minlog","tactic_proof":"assume ind simplify ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1310","id":89809,"instance":1310,"goal":"all n, n * 0 = 0","context":["assume","ind","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"ng intro assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1311","id":89810,"instance":1311,"goal":"all v, v * 1 = v","context":["ng","intro","assume","elim"]} -{"prover":"Minlog","tactic_proof":"defn elim use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1312","id":89811,"instance":1312,"goal":"all h b x, (h * b) * x = h * (b * x)","context":["defn","elim","use","simp"]} -{"prover":"Minlog","tactic_proof":"simp intro ng ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1313","id":89812,"instance":1313,"goal":"all v z, v * z = z * v","context":["simp","intro","ng","ind"]} -{"prover":"Minlog","tactic_proof":"assume elim ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1314","id":89813,"instance":1314,"goal":"all q w m, q * (w + m) = q * w + q * m","context":["assume","elim","ng","defn"]} -{"prover":"Minlog","tactic_proof":"elim elim simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1315","id":89814,"instance":1315,"goal":"all f, f <= f","context":["elim","elim","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ng rw simplify use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1316","id":89815,"instance":1316,"goal":"all y x w, y <= x -> x <= w -> y <= w","context":["ng","rw","simplify","use"]} -{"prover":"Minlog","tactic_proof":"ind cases assume use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1317","id":89816,"instance":1317,"goal":"all z, not z < z","context":["ind","cases","assume","use"]} -{"prover":"Minlog","tactic_proof":"ng ind simplify use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1318","id":89817,"instance":1318,"goal":"all g n, Even g -> Even n -> Even (g + n)","context":["ng","ind","simplify","use"]} -{"prover":"Minlog","tactic_proof":"ng simplify intro cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1319","id":89818,"instance":1319,"goal":"all a, a | a","context":["ng","simplify","intro","cases"]} -{"prover":"Minlog","tactic_proof":"cases simplify rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1320","id":89819,"instance":1320,"goal":"all d, Fact d > 0","context":["cases","simplify","rw","assume"]} -{"prover":"Minlog","tactic_proof":"intro cases rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1321","id":89820,"instance":1321,"goal":"all e q = q","context":["intro","cases","rw","assume"]} -{"prover":"Minlog","tactic_proof":"rw intro rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1322","id":89821,"instance":1322,"goal":"all d + 0 = d","context":["rw","intro","rw","ng"]} -{"prover":"Minlog","tactic_proof":"intro cases defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1323","id":89822,"instance":1323,"goal":"all v f, v + f = f + v","context":["intro","cases","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify simplify cases simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1324","id":89823,"instance":1324,"goal":"all n u p, (n + u) + p = n + (u + p)","context":["simplify","simplify","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"simp defn intro intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1325","id":89824,"instance":1325,"goal":"all f, f * 0 = 0","context":["simp","defn","intro","intro"]} -{"prover":"Minlog","tactic_proof":"elim simp defn rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1326","id":89825,"instance":1326,"goal":"all u, u * 1 = u","context":["elim","simp","defn","rw"]} -{"prover":"Minlog","tactic_proof":"cases cases rw intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1327","id":89826,"instance":1327,"goal":"all g n u, (g * n) * u = g * (n * u)","context":["cases","cases","rw","intro"]} -{"prover":"Minlog","tactic_proof":"rw simplify use rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1328","id":89827,"instance":1328,"goal":"all n p, n * p = p * n","context":["rw","simplify","use","rw"]} -{"prover":"Minlog","tactic_proof":"ng ng elim simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1329","id":89828,"instance":1329,"goal":"all x p u, x * (p + u) = x * p + x * u","context":["ng","ng","elim","simp"]} -{"prover":"Minlog","tactic_proof":"simplify elim assume simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1330","id":89829,"instance":1330,"goal":"all a, a <= a","context":["simplify","elim","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"ind simp elim elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1331","id":89830,"instance":1331,"goal":"all a z b, a <= z -> z <= b -> a <= b","context":["ind","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"intro intro simplify cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1332","id":89831,"instance":1332,"goal":"all h, not h < h","context":["intro","intro","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"ng elim ind defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1333","id":89832,"instance":1333,"goal":"all e g, Even e -> Even g -> Even (e + g)","context":["ng","elim","ind","defn"]} -{"prover":"Minlog","tactic_proof":"defn intro assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1334","id":89833,"instance":1334,"goal":"all w, w | w","context":["defn","intro","assume","intro"]} -{"prover":"Minlog","tactic_proof":"assume ng cases rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1335","id":89834,"instance":1335,"goal":"all f, Fact f > 0","context":["assume","ng","cases","rw"]} -{"prover":"Minlog","tactic_proof":"elim rw simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1336","id":89835,"instance":1336,"goal":"all e d = d","context":["elim","rw","simplify","use"]} -{"prover":"Minlog","tactic_proof":"defn cases cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1337","id":89836,"instance":1337,"goal":"all w + 0 = w","context":["defn","cases","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simp use ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1338","id":89837,"instance":1338,"goal":"all x a, x + a = a + x","context":["simplify","simp","use","ng"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1339","id":89838,"instance":1339,"goal":"all w q p, (w + q) + p = w + (q + p)","context":["intro","simplify","intro","cases"]} -{"prover":"Minlog","tactic_proof":"use elim elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1340","id":89839,"instance":1340,"goal":"all f, f * 0 = 0","context":["use","elim","elim","elim"]} -{"prover":"Minlog","tactic_proof":"assume simp elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1341","id":89840,"instance":1341,"goal":"all v, v * 1 = v","context":["assume","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"defn use cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1342","id":89841,"instance":1342,"goal":"all w v x, (w * v) * x = w * (v * x)","context":["defn","use","cases","intro"]} -{"prover":"Minlog","tactic_proof":"elim simplify ind intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1343","id":89842,"instance":1343,"goal":"all m y, m * y = y * m","context":["elim","simplify","ind","intro"]} -{"prover":"Minlog","tactic_proof":"intro elim intro use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1344","id":89843,"instance":1344,"goal":"all w q p, w * (q + p) = w * q + w * p","context":["intro","elim","intro","use"]} -{"prover":"Minlog","tactic_proof":"defn ng simp simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1345","id":89844,"instance":1345,"goal":"all v, v <= v","context":["defn","ng","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ind ng use defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1346","id":89845,"instance":1346,"goal":"all x v a, x <= v -> v <= a -> x <= a","context":["ind","ng","use","defn"]} -{"prover":"Minlog","tactic_proof":"defn cases intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1347","id":89846,"instance":1347,"goal":"all u, not u < u","context":["defn","cases","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ng ng ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1348","id":89847,"instance":1348,"goal":"all b y, Even b -> Even y -> Even (b + y)","context":["ng","ng","ng","defn"]} -{"prover":"Minlog","tactic_proof":"assume ind defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1349","id":89848,"instance":1349,"goal":"all u, u | u","context":["assume","ind","defn","cases"]} -{"prover":"Minlog","tactic_proof":"ng use use ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1350","id":89849,"instance":1350,"goal":"all u, Fact u > 0","context":["ng","use","use","ng"]} -{"prover":"Minlog","tactic_proof":"intro cases assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1351","id":89850,"instance":1351,"goal":"all n b = b","context":["intro","cases","assume","intro"]} -{"prover":"Minlog","tactic_proof":"use simp simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1352","id":89851,"instance":1352,"goal":"all g + 0 = g","context":["use","simp","simp","elim"]} -{"prover":"Minlog","tactic_proof":"use ind use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1353","id":89852,"instance":1353,"goal":"all y n, y + n = n + y","context":["use","ind","use","simp"]} -{"prover":"Minlog","tactic_proof":"simplify simp rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1354","id":89853,"instance":1354,"goal":"all m u x, (m + u) + x = m + (u + x)","context":["simplify","simp","rw","ng"]} -{"prover":"Minlog","tactic_proof":"cases simp ng intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1355","id":89854,"instance":1355,"goal":"all z, z * 0 = 0","context":["cases","simp","ng","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases elim intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1356","id":89855,"instance":1356,"goal":"all m, m * 1 = m","context":["intro","cases","elim","intro"]} -{"prover":"Minlog","tactic_proof":"elim simplify intro intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1357","id":89856,"instance":1357,"goal":"all h g a, (h * g) * a = h * (g * a)","context":["elim","simplify","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ind rw assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1358","id":89857,"instance":1358,"goal":"all h w, h * w = w * h","context":["ind","rw","assume","intro"]} -{"prover":"Minlog","tactic_proof":"simp intro cases ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1359","id":89858,"instance":1359,"goal":"all u h x, u * (h + x) = u * h + u * x","context":["simp","intro","cases","ng"]} -{"prover":"Minlog","tactic_proof":"rw assume cases use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1360","id":89859,"instance":1360,"goal":"all z, z <= z","context":["rw","assume","cases","use"]} -{"prover":"Minlog","tactic_proof":"simplify defn ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1361","id":89860,"instance":1361,"goal":"all q u f, q <= u -> u <= f -> q <= f","context":["simplify","defn","ng","ng"]} -{"prover":"Minlog","tactic_proof":"ng intro defn use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1362","id":89861,"instance":1362,"goal":"all w, not w < w","context":["ng","intro","defn","use"]} -{"prover":"Minlog","tactic_proof":"cases intro defn simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1363","id":89862,"instance":1363,"goal":"all x y, Even x -> Even y -> Even (x + y)","context":["cases","intro","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"elim assume rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1364","id":89863,"instance":1364,"goal":"all v, v | v","context":["elim","assume","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp cases intro elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1365","id":89864,"instance":1365,"goal":"all c, Fact c > 0","context":["simp","cases","intro","elim"]} -{"prover":"Minlog","tactic_proof":"cases ind simplify defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1366","id":89865,"instance":1366,"goal":"all b a = a","context":["cases","ind","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"rw elim ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1367","id":89866,"instance":1367,"goal":"all w + 0 = w","context":["rw","elim","ind","ind"]} -{"prover":"Minlog","tactic_proof":"elim elim use elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1368","id":89867,"instance":1368,"goal":"all f h, f + h = h + f","context":["elim","elim","use","elim"]} -{"prover":"Minlog","tactic_proof":"cases cases simplify use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1369","id":89868,"instance":1369,"goal":"all z a e, (z + a) + e = z + (a + e)","context":["cases","cases","simplify","use"]} -{"prover":"Minlog","tactic_proof":"cases elim cases simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1370","id":89869,"instance":1370,"goal":"all p, p * 0 = 0","context":["cases","elim","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"rw ng rw assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1371","id":89870,"instance":1371,"goal":"all m, m * 1 = m","context":["rw","ng","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases assume intro intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1372","id":89871,"instance":1372,"goal":"all h w y, (h * w) * y = h * (w * y)","context":["cases","assume","intro","intro"]} -{"prover":"Minlog","tactic_proof":"intro ng simp simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1373","id":89872,"instance":1373,"goal":"all q z, q * z = z * q","context":["intro","ng","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"use assume elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1374","id":89873,"instance":1374,"goal":"all m u g, m * (u + g) = m * u + m * g","context":["use","assume","elim","ind"]} -{"prover":"Minlog","tactic_proof":"defn cases use intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1375","id":89874,"instance":1375,"goal":"all e, e <= e","context":["defn","cases","use","intro"]} -{"prover":"Minlog","tactic_proof":"elim assume ng assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1376","id":89875,"instance":1376,"goal":"all x c h, x <= c -> c <= h -> x <= h","context":["elim","assume","ng","assume"]} -{"prover":"Minlog","tactic_proof":"intro simplify ind use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1377","id":89876,"instance":1377,"goal":"all m, not m < m","context":["intro","simplify","ind","use"]} -{"prover":"Minlog","tactic_proof":"simplify ind simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1378","id":89877,"instance":1378,"goal":"all a v, Even a -> Even v -> Even (a + v)","context":["simplify","ind","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify defn simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1379","id":89878,"instance":1379,"goal":"all p, p | p","context":["assume","simplify","defn","simp"]} -{"prover":"Minlog","tactic_proof":"cases elim intro ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1380","id":89879,"instance":1380,"goal":"all e, Fact e > 0","context":["cases","elim","intro","ind"]} -{"prover":"Minlog","tactic_proof":"rw ind defn elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1381","id":89880,"instance":1381,"goal":"all b h = h","context":["rw","ind","defn","elim"]} -{"prover":"Minlog","tactic_proof":"use cases simp cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1382","id":89881,"instance":1382,"goal":"all b + 0 = b","context":["use","cases","simp","cases"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simp assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1383","id":89882,"instance":1383,"goal":"all p q, p + q = q + p","context":["simplify","simplify","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1384","id":89883,"instance":1384,"goal":"all z a h, (z + a) + h = z + (a + h)","context":["assume","simplify","assume","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1385","id":89884,"instance":1385,"goal":"all v, v * 0 = 0","context":["intro","intro","use","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify use use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1386","id":89885,"instance":1386,"goal":"all p, p * 1 = p","context":["simplify","use","use","simplify"]} -{"prover":"Minlog","tactic_proof":"ind defn cases intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1387","id":89886,"instance":1387,"goal":"all n f e, (n * f) * e = n * (f * e)","context":["ind","defn","cases","intro"]} -{"prover":"Minlog","tactic_proof":"elim elim use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1388","id":89887,"instance":1388,"goal":"all h q, h * q = q * h","context":["elim","elim","use","ng"]} -{"prover":"Minlog","tactic_proof":"rw use assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1389","id":89888,"instance":1389,"goal":"all n m c, n * (m + c) = n * m + n * c","context":["rw","use","assume","assume"]} -{"prover":"Minlog","tactic_proof":"assume defn cases ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1390","id":89889,"instance":1390,"goal":"all b, b <= b","context":["assume","defn","cases","ind"]} -{"prover":"Minlog","tactic_proof":"elim ng intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1391","id":89890,"instance":1391,"goal":"all n p y, n <= p -> p <= y -> n <= y","context":["elim","ng","intro","cases"]} -{"prover":"Minlog","tactic_proof":"use rw rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1392","id":89891,"instance":1392,"goal":"all x, not x < x","context":["use","rw","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"assume defn rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1393","id":89892,"instance":1393,"goal":"all q v, Even q -> Even v -> Even (q + v)","context":["assume","defn","rw","rw"]} -{"prover":"Minlog","tactic_proof":"simp intro assume cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1394","id":89893,"instance":1394,"goal":"all c, c | c","context":["simp","intro","assume","cases"]} -{"prover":"Minlog","tactic_proof":"simp assume simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1395","id":89894,"instance":1395,"goal":"all f, Fact f > 0","context":["simp","assume","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"rw assume ind assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1396","id":89895,"instance":1396,"goal":"all f a = a","context":["rw","assume","ind","assume"]} -{"prover":"Minlog","tactic_proof":"elim defn simp simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1397","id":89896,"instance":1397,"goal":"all n + 0 = n","context":["elim","defn","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"rw defn cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1398","id":89897,"instance":1398,"goal":"all h m, h + m = m + h","context":["rw","defn","cases","ng"]} -{"prover":"Minlog","tactic_proof":"rw simplify defn simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1399","id":89898,"instance":1399,"goal":"all q d m, (q + d) + m = q + (d + m)","context":["rw","simplify","defn","simp"]} -{"prover":"Minlog","tactic_proof":"use simp use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1400","id":89899,"instance":1400,"goal":"all n, n * 0 = 0","context":["use","simp","use","assume"]} -{"prover":"Minlog","tactic_proof":"use ng use use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1401","id":89900,"instance":1401,"goal":"all m, m * 1 = m","context":["use","ng","use","use"]} -{"prover":"Minlog","tactic_proof":"rw ind ind simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1402","id":89901,"instance":1402,"goal":"all p v g, (p * v) * g = p * (v * g)","context":["rw","ind","ind","simp"]} -{"prover":"Minlog","tactic_proof":"use simp simplify assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1403","id":89902,"instance":1403,"goal":"all d m, d * m = m * d","context":["use","simp","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"rw intro simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1404","id":89903,"instance":1404,"goal":"all b v z, b * (v + z) = b * v + b * z","context":["rw","intro","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"defn defn simplify assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1405","id":89904,"instance":1405,"goal":"all h, h <= h","context":["defn","defn","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1406","id":89905,"instance":1406,"goal":"all z e a, z <= e -> e <= a -> z <= a","context":["defn","simplify","rw","simp"]} -{"prover":"Minlog","tactic_proof":"simplify intro ind simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1407","id":89906,"instance":1407,"goal":"all b, not b < b","context":["simplify","intro","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simp defn simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1408","id":89907,"instance":1408,"goal":"all u x, Even u -> Even x -> Even (u + x)","context":["assume","simp","defn","simp"]} -{"prover":"Minlog","tactic_proof":"simp defn elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1409","id":89908,"instance":1409,"goal":"all c, c | c","context":["simp","defn","elim","use"]} -{"prover":"Minlog","tactic_proof":"ng defn elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1410","id":89909,"instance":1410,"goal":"all b, Fact b > 0","context":["ng","defn","elim","cases"]} -{"prover":"Minlog","tactic_proof":"ng ng ind defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1411","id":89910,"instance":1411,"goal":"all u h = h","context":["ng","ng","ind","defn"]} -{"prover":"Minlog","tactic_proof":"cases assume elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1412","id":89911,"instance":1412,"goal":"all q + 0 = q","context":["cases","assume","elim","use"]} -{"prover":"Minlog","tactic_proof":"ng rw ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1413","id":89912,"instance":1413,"goal":"all x a, x + a = a + x","context":["ng","rw","ind","rw"]} -{"prover":"Minlog","tactic_proof":"simp elim elim ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1414","id":89913,"instance":1414,"goal":"all f b y, (f + b) + y = f + (b + y)","context":["simp","elim","elim","ind"]} -{"prover":"Minlog","tactic_proof":"ng cases cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1415","id":89914,"instance":1415,"goal":"all g, g * 0 = 0","context":["ng","cases","cases","use"]} -{"prover":"Minlog","tactic_proof":"rw ng simp elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1416","id":89915,"instance":1416,"goal":"all e, e * 1 = e","context":["rw","ng","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ind simp intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1417","id":89916,"instance":1417,"goal":"all y h q, (y * h) * q = y * (h * q)","context":["ind","simp","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ind assume intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1418","id":89917,"instance":1418,"goal":"all g f, g * f = f * g","context":["ind","assume","intro","ind"]} -{"prover":"Minlog","tactic_proof":"simp ind assume ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1419","id":89918,"instance":1419,"goal":"all g n h, g * (n + h) = g * n + g * h","context":["simp","ind","assume","ng"]} -{"prover":"Minlog","tactic_proof":"simplify ind simp simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1420","id":89919,"instance":1420,"goal":"all p, p <= p","context":["simplify","ind","simp","simp"]} -{"prover":"Minlog","tactic_proof":"elim assume rw intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1421","id":89920,"instance":1421,"goal":"all d q u, d <= q -> q <= u -> d <= u","context":["elim","assume","rw","intro"]} -{"prover":"Minlog","tactic_proof":"simplify ng simplify simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1422","id":89921,"instance":1422,"goal":"all a, not a < a","context":["simplify","ng","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"rw defn defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1423","id":89922,"instance":1423,"goal":"all g n, Even g -> Even n -> Even (g + n)","context":["rw","defn","defn","use"]} -{"prover":"Minlog","tactic_proof":"rw defn use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1424","id":89923,"instance":1424,"goal":"all z, z | z","context":["rw","defn","use","use"]} -{"prover":"Minlog","tactic_proof":"elim assume use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1425","id":89924,"instance":1425,"goal":"all f, Fact f > 0","context":["elim","assume","use","rw"]} -{"prover":"Minlog","tactic_proof":"elim ind defn rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1426","id":89925,"instance":1426,"goal":"all x a = a","context":["elim","ind","defn","rw"]} -{"prover":"Minlog","tactic_proof":"ind assume rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1427","id":89926,"instance":1427,"goal":"all v + 0 = v","context":["ind","assume","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ng cases simp defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1428","id":89927,"instance":1428,"goal":"all x p, x + p = p + x","context":["ng","cases","simp","defn"]} -{"prover":"Minlog","tactic_proof":"intro ind ind rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1429","id":89928,"instance":1429,"goal":"all x v h, (x + v) + h = x + (v + h)","context":["intro","ind","ind","rw"]} -{"prover":"Minlog","tactic_proof":"intro simp simp intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1430","id":89929,"instance":1430,"goal":"all x, x * 0 = 0","context":["intro","simp","simp","intro"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1431","id":89930,"instance":1431,"goal":"all a, a * 1 = a","context":["ind","ng","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"cases rw assume cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1432","id":89931,"instance":1432,"goal":"all q y c, (q * y) * c = q * (y * c)","context":["cases","rw","assume","cases"]} -{"prover":"Minlog","tactic_proof":"ng cases elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1433","id":89932,"instance":1433,"goal":"all e y, e * y = y * e","context":["ng","cases","elim","rw"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1434","id":89933,"instance":1434,"goal":"all y m g, y * (m + g) = y * m + y * g","context":["defn","simplify","rw","rw"]} -{"prover":"Minlog","tactic_proof":"intro ng ind ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1435","id":89934,"instance":1435,"goal":"all a, a <= a","context":["intro","ng","ind","ng"]} -{"prover":"Minlog","tactic_proof":"cases defn simplify elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1436","id":89935,"instance":1436,"goal":"all p d b, p <= d -> d <= b -> p <= b","context":["cases","defn","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"defn rw use assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1437","id":89936,"instance":1437,"goal":"all x, not x < x","context":["defn","rw","use","assume"]} -{"prover":"Minlog","tactic_proof":"ng ind defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1438","id":89937,"instance":1438,"goal":"all e a, Even e -> Even a -> Even (e + a)","context":["ng","ind","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"cases defn cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1439","id":89938,"instance":1439,"goal":"all p, p | p","context":["cases","defn","cases","ind"]} -{"prover":"Minlog","tactic_proof":"elim rw elim defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1440","id":89939,"instance":1440,"goal":"all z, Fact z > 0","context":["elim","rw","elim","defn"]} -{"prover":"Minlog","tactic_proof":"elim ind ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1441","id":89940,"instance":1441,"goal":"all m n = n","context":["elim","ind","ng","assume"]} -{"prover":"Minlog","tactic_proof":"elim simp ng ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1442","id":89941,"instance":1442,"goal":"all z + 0 = z","context":["elim","simp","ng","ind"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1443","id":89942,"instance":1443,"goal":"all f h, f + h = h + f","context":["simplify","simplify","simp","assume"]} -{"prover":"Minlog","tactic_proof":"rw defn assume rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1444","id":89943,"instance":1444,"goal":"all z w c, (z + w) + c = z + (w + c)","context":["rw","defn","assume","rw"]} -{"prover":"Minlog","tactic_proof":"simp ng cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1445","id":89944,"instance":1445,"goal":"all h, h * 0 = 0","context":["simp","ng","cases","assume"]} -{"prover":"Minlog","tactic_proof":"ng cases rw intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1446","id":89945,"instance":1446,"goal":"all z, z * 1 = z","context":["ng","cases","rw","intro"]} -{"prover":"Minlog","tactic_proof":"use elim intro ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1447","id":89946,"instance":1447,"goal":"all h y m, (h * y) * m = h * (y * m)","context":["use","elim","intro","ind"]} -{"prover":"Minlog","tactic_proof":"ind simp simplify ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1448","id":89947,"instance":1448,"goal":"all g d, g * d = d * g","context":["ind","simp","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"simplify use cases defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1449","id":89948,"instance":1449,"goal":"all v b y, v * (b + y) = v * b + v * y","context":["simplify","use","cases","defn"]} -{"prover":"Minlog","tactic_proof":"simplify ng simplify simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1450","id":89949,"instance":1450,"goal":"all b, b <= b","context":["simplify","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"intro simp defn defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1451","id":89950,"instance":1451,"goal":"all w x v, w <= x -> x <= v -> w <= v","context":["intro","simp","defn","defn"]} -{"prover":"Minlog","tactic_proof":"simp ind intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1452","id":89951,"instance":1452,"goal":"all h, not h < h","context":["simp","ind","intro","cases"]} -{"prover":"Minlog","tactic_proof":"cases intro simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1453","id":89952,"instance":1453,"goal":"all x c, Even x -> Even c -> Even (x + c)","context":["cases","intro","simp","use"]} -{"prover":"Minlog","tactic_proof":"ind rw intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1454","id":89953,"instance":1454,"goal":"all m, m | m","context":["ind","rw","intro","cases"]} -{"prover":"Minlog","tactic_proof":"simp use assume defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1455","id":89954,"instance":1455,"goal":"all z, Fact z > 0","context":["simp","use","assume","defn"]} -{"prover":"Minlog","tactic_proof":"simp cases elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1456","id":89955,"instance":1456,"goal":"all x u = u","context":["simp","cases","elim","cases"]} -{"prover":"Minlog","tactic_proof":"cases simplify ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1457","id":89956,"instance":1457,"goal":"all p + 0 = p","context":["cases","simplify","ng","assume"]} -{"prover":"Minlog","tactic_proof":"cases cases intro rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1458","id":89957,"instance":1458,"goal":"all w c, w + c = c + w","context":["cases","cases","intro","rw"]} -{"prover":"Minlog","tactic_proof":"simplify intro use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1459","id":89958,"instance":1459,"goal":"all w g x, (w + g) + x = w + (g + x)","context":["simplify","intro","use","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify assume rw assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1460","id":89959,"instance":1460,"goal":"all d, d * 0 = 0","context":["simplify","assume","rw","assume"]} -{"prover":"Minlog","tactic_proof":"rw elim assume elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1461","id":89960,"instance":1461,"goal":"all q, q * 1 = q","context":["rw","elim","assume","elim"]} -{"prover":"Minlog","tactic_proof":"ind intro use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1462","id":89961,"instance":1462,"goal":"all x c p, (x * c) * p = x * (c * p)","context":["ind","intro","use","defn"]} -{"prover":"Minlog","tactic_proof":"simp defn simplify ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1463","id":89962,"instance":1463,"goal":"all h a, h * a = a * h","context":["simp","defn","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"elim intro cases ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1464","id":89963,"instance":1464,"goal":"all m x q, m * (x + q) = m * x + m * q","context":["elim","intro","cases","ind"]} -{"prover":"Minlog","tactic_proof":"ind intro defn assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1465","id":89964,"instance":1465,"goal":"all c, c <= c","context":["ind","intro","defn","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro ind simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1466","id":89965,"instance":1466,"goal":"all h x f, h <= x -> x <= f -> h <= f","context":["intro","intro","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"ng ng rw simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1467","id":89966,"instance":1467,"goal":"all n, not n < n","context":["ng","ng","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"assume use elim cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1468","id":89967,"instance":1468,"goal":"all p n, Even p -> Even n -> Even (p + n)","context":["assume","use","elim","cases"]} -{"prover":"Minlog","tactic_proof":"simplify ind simplify ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1469","id":89968,"instance":1469,"goal":"all b, b | b","context":["simplify","ind","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"simplify elim intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1470","id":89969,"instance":1470,"goal":"all c, Fact c > 0","context":["simplify","elim","intro","ng"]} -{"prover":"Minlog","tactic_proof":"rw ind use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1471","id":89970,"instance":1471,"goal":"all x y = y","context":["rw","ind","use","simp"]} -{"prover":"Minlog","tactic_proof":"rw use simplify cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1472","id":89971,"instance":1472,"goal":"all d + 0 = d","context":["rw","use","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume simplify cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1473","id":89972,"instance":1473,"goal":"all z h, z + h = h + z","context":["ind","assume","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"simp use simp simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1474","id":89973,"instance":1474,"goal":"all e n f, (e + n) + f = e + (n + f)","context":["simp","use","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"use intro ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1475","id":89974,"instance":1475,"goal":"all n, n * 0 = 0","context":["use","intro","ind","intro"]} -{"prover":"Minlog","tactic_proof":"use intro assume cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1476","id":89975,"instance":1476,"goal":"all v, v * 1 = v","context":["use","intro","assume","cases"]} -{"prover":"Minlog","tactic_proof":"assume assume ind ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1477","id":89976,"instance":1477,"goal":"all g z n, (g * z) * n = g * (z * n)","context":["assume","assume","ind","ind"]} -{"prover":"Minlog","tactic_proof":"simplify ng assume simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1478","id":89977,"instance":1478,"goal":"all c m, c * m = m * c","context":["simplify","ng","assume","simp"]} -{"prover":"Minlog","tactic_proof":"simp simp ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1479","id":89978,"instance":1479,"goal":"all a h p, a * (h + p) = a * h + a * p","context":["simp","simp","ng","intro"]} -{"prover":"Minlog","tactic_proof":"elim use defn ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1480","id":89979,"instance":1480,"goal":"all y, y <= y","context":["elim","use","defn","ind"]} -{"prover":"Minlog","tactic_proof":"intro use rw intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1481","id":89980,"instance":1481,"goal":"all c n f, c <= n -> n <= f -> c <= f","context":["intro","use","rw","intro"]} -{"prover":"Minlog","tactic_proof":"ind assume ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1482","id":89981,"instance":1482,"goal":"all a, not a < a","context":["ind","assume","ng","cases"]} -{"prover":"Minlog","tactic_proof":"simp ind simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1483","id":89982,"instance":1483,"goal":"all x h, Even x -> Even h -> Even (x + h)","context":["simp","ind","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"rw elim elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1484","id":89983,"instance":1484,"goal":"all d, d | d","context":["rw","elim","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"cases simplify intro rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1485","id":89984,"instance":1485,"goal":"all m, Fact m > 0","context":["cases","simplify","intro","rw"]} -{"prover":"Minlog","tactic_proof":"simp cases elim ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1486","id":89985,"instance":1486,"goal":"all d p = p","context":["simp","cases","elim","ind"]} -{"prover":"Minlog","tactic_proof":"elim cases rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1487","id":89986,"instance":1487,"goal":"all c + 0 = c","context":["elim","cases","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp simp cases ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1488","id":89987,"instance":1488,"goal":"all g f, g + f = f + g","context":["simp","simp","cases","ind"]} -{"prover":"Minlog","tactic_proof":"ind intro use ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1489","id":89988,"instance":1489,"goal":"all m y h, (m + y) + h = m + (y + h)","context":["ind","intro","use","ind"]} -{"prover":"Minlog","tactic_proof":"cases assume rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1490","id":89989,"instance":1490,"goal":"all y, y * 0 = 0","context":["cases","assume","rw","ng"]} -{"prover":"Minlog","tactic_proof":"rw ng simp simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1491","id":89990,"instance":1491,"goal":"all v, v * 1 = v","context":["rw","ng","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ng intro ind use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1492","id":89991,"instance":1492,"goal":"all z u b, (z * u) * b = z * (u * b)","context":["ng","intro","ind","use"]} -{"prover":"Minlog","tactic_proof":"ind defn ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1493","id":89992,"instance":1493,"goal":"all w e, w * e = e * w","context":["ind","defn","ind","intro"]} -{"prover":"Minlog","tactic_proof":"cases elim cases defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1494","id":89993,"instance":1494,"goal":"all y h d, y * (h + d) = y * h + y * d","context":["cases","elim","cases","defn"]} -{"prover":"Minlog","tactic_proof":"simplify assume use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1495","id":89994,"instance":1495,"goal":"all z, z <= z","context":["simplify","assume","use","assume"]} -{"prover":"Minlog","tactic_proof":"elim ng simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1496","id":89995,"instance":1496,"goal":"all y z b, y <= z -> z <= b -> y <= b","context":["elim","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify elim ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1497","id":89996,"instance":1497,"goal":"all v, not v < v","context":["simplify","elim","ind","rw"]} -{"prover":"Minlog","tactic_proof":"simplify cases ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1498","id":89997,"instance":1498,"goal":"all u h, Even u -> Even h -> Even (u + h)","context":["simplify","cases","ng","cases"]} -{"prover":"Minlog","tactic_proof":"rw use cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1499","id":89998,"instance":1499,"goal":"all u, u | u","context":["rw","use","cases","cases"]} -{"prover":"Minlog","tactic_proof":"rw cases elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1500","id":89999,"instance":1500,"goal":"all h, Fact h > 0","context":["rw","cases","elim","rw"]} -{"prover":"Minlog","tactic_proof":"intro defn defn elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1501","id":90000,"instance":1501,"goal":"all x h = h","context":["intro","defn","defn","elim"]} -{"prover":"Minlog","tactic_proof":"use elim ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1502","id":90001,"instance":1502,"goal":"all w + 0 = w","context":["use","elim","ng","assume"]} -{"prover":"Minlog","tactic_proof":"cases simplify elim assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1503","id":90002,"instance":1503,"goal":"all q w, q + w = w + q","context":["cases","simplify","elim","assume"]} -{"prover":"Minlog","tactic_proof":"defn elim rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1504","id":90003,"instance":1504,"goal":"all q u v, (q + u) + v = q + (u + v)","context":["defn","elim","rw","ng"]} -{"prover":"Minlog","tactic_proof":"simplify cases defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1505","id":90004,"instance":1505,"goal":"all z, z * 0 = 0","context":["simplify","cases","defn","rw"]} -{"prover":"Minlog","tactic_proof":"cases defn elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1506","id":90005,"instance":1506,"goal":"all x, x * 1 = x","context":["cases","defn","elim","defn"]} -{"prover":"Minlog","tactic_proof":"assume ind assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1507","id":90006,"instance":1507,"goal":"all v g p, (v * g) * p = v * (g * p)","context":["assume","ind","assume","assume"]} -{"prover":"Minlog","tactic_proof":"elim assume cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1508","id":90007,"instance":1508,"goal":"all p c, p * c = c * p","context":["elim","assume","cases","cases"]} -{"prover":"Minlog","tactic_proof":"simplify rw ng defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1509","id":90008,"instance":1509,"goal":"all z x c, z * (x + c) = z * x + z * c","context":["simplify","rw","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro assume elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1510","id":90009,"instance":1510,"goal":"all e, e <= e","context":["intro","assume","elim","defn"]} -{"prover":"Minlog","tactic_proof":"simplify rw simplify ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1511","id":90010,"instance":1511,"goal":"all h g w, h <= g -> g <= w -> h <= w","context":["simplify","rw","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"defn defn cases intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1512","id":90011,"instance":1512,"goal":"all y, not y < y","context":["defn","defn","cases","intro"]} -{"prover":"Minlog","tactic_proof":"ng simplify ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1513","id":90012,"instance":1513,"goal":"all g p, Even g -> Even p -> Even (g + p)","context":["ng","simplify","ng","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simp ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1514","id":90013,"instance":1514,"goal":"all d, d | d","context":["simplify","simp","ng","ng"]} -{"prover":"Minlog","tactic_proof":"assume cases ind defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1515","id":90014,"instance":1515,"goal":"all n, Fact n > 0","context":["assume","cases","ind","defn"]} -{"prover":"Minlog","tactic_proof":"assume simp simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1516","id":90015,"instance":1516,"goal":"all x f = f","context":["assume","simp","simp","ind"]} -{"prover":"Minlog","tactic_proof":"simp simplify simp assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1517","id":90016,"instance":1517,"goal":"all f + 0 = f","context":["simp","simplify","simp","assume"]} -{"prover":"Minlog","tactic_proof":"rw use simp use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1518","id":90017,"instance":1518,"goal":"all p z, p + z = z + p","context":["rw","use","simp","use"]} -{"prover":"Minlog","tactic_proof":"ind elim use defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1519","id":90018,"instance":1519,"goal":"all p c w, (p + c) + w = p + (c + w)","context":["ind","elim","use","defn"]} -{"prover":"Minlog","tactic_proof":"ng simplify simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1520","id":90019,"instance":1520,"goal":"all g, g * 0 = 0","context":["ng","simplify","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ng simplify intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1521","id":90020,"instance":1521,"goal":"all b, b * 1 = b","context":["ng","simplify","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"ind intro cases rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1522","id":90021,"instance":1522,"goal":"all m z d, (m * z) * d = m * (z * d)","context":["ind","intro","cases","rw"]} -{"prover":"Minlog","tactic_proof":"ind cases rw intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1523","id":90022,"instance":1523,"goal":"all f u, f * u = u * f","context":["ind","cases","rw","intro"]} -{"prover":"Minlog","tactic_proof":"use ind elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1524","id":90023,"instance":1524,"goal":"all q b u, q * (b + u) = q * b + q * u","context":["use","ind","elim","use"]} -{"prover":"Minlog","tactic_proof":"rw elim elim use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1525","id":90024,"instance":1525,"goal":"all w, w <= w","context":["rw","elim","elim","use"]} -{"prover":"Minlog","tactic_proof":"elim intro defn elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1526","id":90025,"instance":1526,"goal":"all z c m, z <= c -> c <= m -> z <= m","context":["elim","intro","defn","elim"]} -{"prover":"Minlog","tactic_proof":"cases simplify elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1527","id":90026,"instance":1527,"goal":"all c, not c < c","context":["cases","simplify","elim","intro"]} -{"prover":"Minlog","tactic_proof":"intro ind use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1528","id":90027,"instance":1528,"goal":"all b u, Even b -> Even u -> Even (b + u)","context":["intro","ind","use","intro"]} -{"prover":"Minlog","tactic_proof":"defn cases assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1529","id":90028,"instance":1529,"goal":"all w, w | w","context":["defn","cases","assume","intro"]} -{"prover":"Minlog","tactic_proof":"cases use cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1530","id":90029,"instance":1530,"goal":"all q, Fact q > 0","context":["cases","use","cases","intro"]} -{"prover":"Minlog","tactic_proof":"simp ng defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1531","id":90030,"instance":1531,"goal":"all v w = w","context":["simp","ng","defn","assume"]} -{"prover":"Minlog","tactic_proof":"cases ind elim assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1532","id":90031,"instance":1532,"goal":"all q + 0 = q","context":["cases","ind","elim","assume"]} -{"prover":"Minlog","tactic_proof":"defn rw ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1533","id":90032,"instance":1533,"goal":"all m y, m + y = y + m","context":["defn","rw","ind","defn"]} -{"prover":"Minlog","tactic_proof":"use defn elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1534","id":90033,"instance":1534,"goal":"all d n q, (d + n) + q = d + (n + q)","context":["use","defn","elim","ng"]} -{"prover":"Minlog","tactic_proof":"simp defn rw intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1535","id":90034,"instance":1535,"goal":"all c, c * 0 = 0","context":["simp","defn","rw","intro"]} -{"prover":"Minlog","tactic_proof":"assume intro elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1536","id":90035,"instance":1536,"goal":"all h, h * 1 = h","context":["assume","intro","elim","ind"]} -{"prover":"Minlog","tactic_proof":"simp simp use simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1537","id":90036,"instance":1537,"goal":"all a u x, (a * u) * x = a * (u * x)","context":["simp","simp","use","simplify"]} -{"prover":"Minlog","tactic_proof":"intro simplify assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1538","id":90037,"instance":1538,"goal":"all e h, e * h = h * e","context":["intro","simplify","assume","assume"]} -{"prover":"Minlog","tactic_proof":"simp elim ng cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1539","id":90038,"instance":1539,"goal":"all y m z, y * (m + z) = y * m + y * z","context":["simp","elim","ng","cases"]} -{"prover":"Minlog","tactic_proof":"ng intro rw simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1540","id":90039,"instance":1540,"goal":"all h, h <= h","context":["ng","intro","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"elim ind ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1541","id":90040,"instance":1541,"goal":"all b v n, b <= v -> v <= n -> b <= n","context":["elim","ind","ind","cases"]} -{"prover":"Minlog","tactic_proof":"use simplify simplify simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1542","id":90041,"instance":1542,"goal":"all y, not y < y","context":["use","simplify","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"ng elim simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1543","id":90042,"instance":1543,"goal":"all a x, Even a -> Even x -> Even (a + x)","context":["ng","elim","simp","assume"]} -{"prover":"Minlog","tactic_proof":"rw intro ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1544","id":90043,"instance":1544,"goal":"all b, b | b","context":["rw","intro","ind","ind"]} -{"prover":"Minlog","tactic_proof":"rw ng elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1545","id":90044,"instance":1545,"goal":"all b, Fact b > 0","context":["rw","ng","elim","ind"]} -{"prover":"Minlog","tactic_proof":"intro defn intro ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1546","id":90045,"instance":1546,"goal":"all b p = p","context":["intro","defn","intro","ind"]} -{"prover":"Minlog","tactic_proof":"defn use rw cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1547","id":90046,"instance":1547,"goal":"all f + 0 = f","context":["defn","use","rw","cases"]} -{"prover":"Minlog","tactic_proof":"elim assume use cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1548","id":90047,"instance":1548,"goal":"all d q, d + q = q + d","context":["elim","assume","use","cases"]} -{"prover":"Minlog","tactic_proof":"cases ng elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1549","id":90048,"instance":1549,"goal":"all c x g, (c + x) + g = c + (x + g)","context":["cases","ng","elim","elim"]} -{"prover":"Minlog","tactic_proof":"rw rw ind assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1550","id":90049,"instance":1550,"goal":"all z, z * 0 = 0","context":["rw","rw","ind","assume"]} -{"prover":"Minlog","tactic_proof":"cases rw ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1551","id":90050,"instance":1551,"goal":"all q, q * 1 = q","context":["cases","rw","ind","cases"]} -{"prover":"Minlog","tactic_proof":"rw intro cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1552","id":90051,"instance":1552,"goal":"all v u h, (v * u) * h = v * (u * h)","context":["rw","intro","cases","elim"]} -{"prover":"Minlog","tactic_proof":"cases cases ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1553","id":90052,"instance":1553,"goal":"all v c, v * c = c * v","context":["cases","cases","ng","ind"]} -{"prover":"Minlog","tactic_proof":"cases use ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1554","id":90053,"instance":1554,"goal":"all h g f, h * (g + f) = h * g + h * f","context":["cases","use","ind","defn"]} -{"prover":"Minlog","tactic_proof":"cases ng simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1555","id":90054,"instance":1555,"goal":"all v, v <= v","context":["cases","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"ng cases ind assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1556","id":90055,"instance":1556,"goal":"all f u y, f <= u -> u <= y -> f <= y","context":["ng","cases","ind","assume"]} -{"prover":"Minlog","tactic_proof":"intro cases simp simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1557","id":90056,"instance":1557,"goal":"all p, not p < p","context":["intro","cases","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ng rw elim rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1558","id":90057,"instance":1558,"goal":"all e y, Even e -> Even y -> Even (e + y)","context":["ng","rw","elim","rw"]} -{"prover":"Minlog","tactic_proof":"cases rw intro elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1559","id":90058,"instance":1559,"goal":"all u, u | u","context":["cases","rw","intro","elim"]} -{"prover":"Minlog","tactic_proof":"cases use ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1560","id":90059,"instance":1560,"goal":"all v, Fact v > 0","context":["cases","use","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"simp simp rw cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1561","id":90060,"instance":1561,"goal":"all q f = f","context":["simp","simp","rw","cases"]} -{"prover":"Minlog","tactic_proof":"elim intro assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1562","id":90061,"instance":1562,"goal":"all e + 0 = e","context":["elim","intro","assume","rw"]} -{"prover":"Minlog","tactic_proof":"defn cases rw ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1563","id":90062,"instance":1563,"goal":"all q b, q + b = b + q","context":["defn","cases","rw","ng"]} -{"prover":"Minlog","tactic_proof":"ind simplify ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1564","id":90063,"instance":1564,"goal":"all g q c, (g + q) + c = g + (q + c)","context":["ind","simplify","ind","ind"]} -{"prover":"Minlog","tactic_proof":"rw simp defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1565","id":90064,"instance":1565,"goal":"all z, z * 0 = 0","context":["rw","simp","defn","simp"]} -{"prover":"Minlog","tactic_proof":"intro use ind cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1566","id":90065,"instance":1566,"goal":"all m, m * 1 = m","context":["intro","use","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simplify ind simp cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1567","id":90066,"instance":1567,"goal":"all b v f, (b * v) * f = b * (v * f)","context":["simplify","ind","simp","cases"]} -{"prover":"Minlog","tactic_proof":"simp assume assume defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1568","id":90067,"instance":1568,"goal":"all g y, g * y = y * g","context":["simp","assume","assume","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn defn intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1569","id":90068,"instance":1569,"goal":"all a m b, a * (m + b) = a * m + a * b","context":["ng","defn","defn","intro"]} -{"prover":"Minlog","tactic_proof":"use cases rw use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1570","id":90069,"instance":1570,"goal":"all c, c <= c","context":["use","cases","rw","use"]} -{"prover":"Minlog","tactic_proof":"assume use cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1571","id":90070,"instance":1571,"goal":"all p h z, p <= h -> h <= z -> p <= z","context":["assume","use","cases","ind"]} -{"prover":"Minlog","tactic_proof":"cases ind rw defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1572","id":90071,"instance":1572,"goal":"all h, not h < h","context":["cases","ind","rw","defn"]} -{"prover":"Minlog","tactic_proof":"cases simplify simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1573","id":90072,"instance":1573,"goal":"all n m, Even n -> Even m -> Even (n + m)","context":["cases","simplify","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"simp simplify use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1574","id":90073,"instance":1574,"goal":"all f, f | f","context":["simp","simplify","use","intro"]} -{"prover":"Minlog","tactic_proof":"cases ind use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1575","id":90074,"instance":1575,"goal":"all p, Fact p > 0","context":["cases","ind","use","intro"]} -{"prover":"Minlog","tactic_proof":"simplify elim defn ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1576","id":90075,"instance":1576,"goal":"all h q = q","context":["simplify","elim","defn","ng"]} -{"prover":"Minlog","tactic_proof":"cases intro elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1577","id":90076,"instance":1577,"goal":"all z + 0 = z","context":["cases","intro","elim","defn"]} -{"prover":"Minlog","tactic_proof":"elim intro simp assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1578","id":90077,"instance":1578,"goal":"all z n, z + n = n + z","context":["elim","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume simplify use defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1579","id":90078,"instance":1579,"goal":"all z p f, (z + p) + f = z + (p + f)","context":["assume","simplify","use","defn"]} -{"prover":"Minlog","tactic_proof":"defn simplify assume assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1580","id":90079,"instance":1580,"goal":"all a, a * 0 = 0","context":["defn","simplify","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ng elim ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1581","id":90080,"instance":1581,"goal":"all u, u * 1 = u","context":["ng","elim","ind","ind"]} -{"prover":"Minlog","tactic_proof":"cases use elim assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1582","id":90081,"instance":1582,"goal":"all g e d, (g * e) * d = g * (e * d)","context":["cases","use","elim","assume"]} -{"prover":"Minlog","tactic_proof":"assume ind elim defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1583","id":90082,"instance":1583,"goal":"all b d, b * d = d * b","context":["assume","ind","elim","defn"]} -{"prover":"Minlog","tactic_proof":"rw ng intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1584","id":90083,"instance":1584,"goal":"all m h a, m * (h + a) = m * h + m * a","context":["rw","ng","intro","assume"]} -{"prover":"Minlog","tactic_proof":"defn ng intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1585","id":90084,"instance":1585,"goal":"all q, q <= q","context":["defn","ng","intro","ng"]} -{"prover":"Minlog","tactic_proof":"simplify cases ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1586","id":90085,"instance":1586,"goal":"all h p b, h <= p -> p <= b -> h <= b","context":["simplify","cases","ind","elim"]} -{"prover":"Minlog","tactic_proof":"ng ng ind elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1587","id":90086,"instance":1587,"goal":"all y, not y < y","context":["ng","ng","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simp use ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1588","id":90087,"instance":1588,"goal":"all e a, Even e -> Even a -> Even (e + a)","context":["simp","use","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"cases simplify ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1589","id":90088,"instance":1589,"goal":"all z, z | z","context":["cases","simplify","ind","defn"]} -{"prover":"Minlog","tactic_proof":"cases cases ind assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1590","id":90089,"instance":1590,"goal":"all c, Fact c > 0","context":["cases","cases","ind","assume"]} -{"prover":"Minlog","tactic_proof":"assume use elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1591","id":90090,"instance":1591,"goal":"all q w = w","context":["assume","use","elim","intro"]} -{"prover":"Minlog","tactic_proof":"assume cases rw defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1592","id":90091,"instance":1592,"goal":"all y + 0 = y","context":["assume","cases","rw","defn"]} -{"prover":"Minlog","tactic_proof":"rw defn ind simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1593","id":90092,"instance":1593,"goal":"all w c, w + c = c + w","context":["rw","defn","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"ng simplify defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1594","id":90093,"instance":1594,"goal":"all b c g, (b + c) + g = b + (c + g)","context":["ng","simplify","defn","cases"]} -{"prover":"Minlog","tactic_proof":"elim cases ng intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1595","id":90094,"instance":1595,"goal":"all g, g * 0 = 0","context":["elim","cases","ng","intro"]} -{"prover":"Minlog","tactic_proof":"defn rw assume ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1596","id":90095,"instance":1596,"goal":"all y, y * 1 = y","context":["defn","rw","assume","ng"]} -{"prover":"Minlog","tactic_proof":"assume cases ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1597","id":90096,"instance":1597,"goal":"all w y c, (w * y) * c = w * (y * c)","context":["assume","cases","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify use rw elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1598","id":90097,"instance":1598,"goal":"all p c, p * c = c * p","context":["simplify","use","rw","elim"]} -{"prover":"Minlog","tactic_proof":"ind simp use ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1599","id":90098,"instance":1599,"goal":"all v b p, v * (b + p) = v * b + v * p","context":["ind","simp","use","ind"]} -{"prover":"Minlog","tactic_proof":"intro use elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1600","id":90099,"instance":1600,"goal":"all e, e <= e","context":["intro","use","elim","simp"]} -{"prover":"Minlog","tactic_proof":"intro cases simp elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1601","id":90100,"instance":1601,"goal":"all q n f, q <= n -> n <= f -> q <= f","context":["intro","cases","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ng ng intro rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1602","id":90101,"instance":1602,"goal":"all x, not x < x","context":["ng","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"elim simp simp rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1603","id":90102,"instance":1603,"goal":"all p w, Even p -> Even w -> Even (p + w)","context":["elim","simp","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simplify ng rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1604","id":90103,"instance":1604,"goal":"all z, z | z","context":["simplify","ng","rw","rw"]} -{"prover":"Minlog","tactic_proof":"elim elim rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1605","id":90104,"instance":1605,"goal":"all z, Fact z > 0","context":["elim","elim","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ng ng elim simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1606","id":90105,"instance":1606,"goal":"all f q = q","context":["ng","ng","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"intro intro assume rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1607","id":90106,"instance":1607,"goal":"all v + 0 = v","context":["intro","intro","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ind simp ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1608","id":90107,"instance":1608,"goal":"all d v, d + v = v + d","context":["ind","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"ng intro assume use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1609","id":90108,"instance":1609,"goal":"all b a w, (b + a) + w = b + (a + w)","context":["ng","intro","assume","use"]} -{"prover":"Minlog","tactic_proof":"elim simp ind ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1610","id":90109,"instance":1610,"goal":"all v, v * 0 = 0","context":["elim","simp","ind","ng"]} -{"prover":"Minlog","tactic_proof":"cases defn defn rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1611","id":90110,"instance":1611,"goal":"all q, q * 1 = q","context":["cases","defn","defn","rw"]} -{"prover":"Minlog","tactic_proof":"ind defn intro elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1612","id":90111,"instance":1612,"goal":"all p z g, (p * z) * g = p * (z * g)","context":["ind","defn","intro","elim"]} -{"prover":"Minlog","tactic_proof":"defn ind cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1613","id":90112,"instance":1613,"goal":"all x d, x * d = d * x","context":["defn","ind","cases","defn"]} -{"prover":"Minlog","tactic_proof":"rw cases ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1614","id":90113,"instance":1614,"goal":"all y a u, y * (a + u) = y * a + y * u","context":["rw","cases","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume use ind ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1615","id":90114,"instance":1615,"goal":"all y, y <= y","context":["assume","use","ind","ng"]} -{"prover":"Minlog","tactic_proof":"use assume assume elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1616","id":90115,"instance":1616,"goal":"all v p e, v <= p -> p <= e -> v <= e","context":["use","assume","assume","elim"]} -{"prover":"Minlog","tactic_proof":"simplify assume assume rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1617","id":90116,"instance":1617,"goal":"all d, not d < d","context":["simplify","assume","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ind use defn ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1618","id":90117,"instance":1618,"goal":"all y g, Even y -> Even g -> Even (y + g)","context":["ind","use","defn","ind"]} -{"prover":"Minlog","tactic_proof":"simp use rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1619","id":90118,"instance":1619,"goal":"all e, e | e","context":["simp","use","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"elim cases ind ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1620","id":90119,"instance":1620,"goal":"all q, Fact q > 0","context":["elim","cases","ind","ng"]} -{"prover":"Minlog","tactic_proof":"ind defn cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1621","id":90120,"instance":1621,"goal":"all y e = e","context":["ind","defn","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use cases simp intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1622","id":90121,"instance":1622,"goal":"all m + 0 = m","context":["use","cases","simp","intro"]} -{"prover":"Minlog","tactic_proof":"simplify intro assume ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1623","id":90122,"instance":1623,"goal":"all u g, u + g = g + u","context":["simplify","intro","assume","ind"]} -{"prover":"Minlog","tactic_proof":"simp defn ng ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1624","id":90123,"instance":1624,"goal":"all f p y, (f + p) + y = f + (p + y)","context":["simp","defn","ng","ng"]} -{"prover":"Minlog","tactic_proof":"elim simp cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1625","id":90124,"instance":1625,"goal":"all m, m * 0 = 0","context":["elim","simp","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify elim simp assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1626","id":90125,"instance":1626,"goal":"all c, c * 1 = c","context":["simplify","elim","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume cases use ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1627","id":90126,"instance":1627,"goal":"all a x c, (a * x) * c = a * (x * c)","context":["assume","cases","use","ind"]} -{"prover":"Minlog","tactic_proof":"elim simplify intro intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1628","id":90127,"instance":1628,"goal":"all a x, a * x = x * a","context":["elim","simplify","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ind intro assume defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1629","id":90128,"instance":1629,"goal":"all u q d, u * (q + d) = u * q + u * d","context":["ind","intro","assume","defn"]} -{"prover":"Minlog","tactic_proof":"use simplify intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1630","id":90129,"instance":1630,"goal":"all v, v <= v","context":["use","simplify","intro","defn"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1631","id":90130,"instance":1631,"goal":"all n c v, n <= c -> c <= v -> n <= v","context":["ind","ng","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1632","id":90131,"instance":1632,"goal":"all g, not g < g","context":["simplify","simplify","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"intro simplify use assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1633","id":90132,"instance":1633,"goal":"all c z, Even c -> Even z -> Even (c + z)","context":["intro","simplify","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw simplify ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1634","id":90133,"instance":1634,"goal":"all u, u | u","context":["rw","simplify","ng","ng"]} -{"prover":"Minlog","tactic_proof":"ind ng ind ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1635","id":90134,"instance":1635,"goal":"all y, Fact y > 0","context":["ind","ng","ind","ng"]} -{"prover":"Minlog","tactic_proof":"ind ng elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1636","id":90135,"instance":1636,"goal":"all y u = u","context":["ind","ng","elim","cases"]} -{"prover":"Minlog","tactic_proof":"ind simp intro ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1637","id":90136,"instance":1637,"goal":"all d + 0 = d","context":["ind","simp","intro","ng"]} -{"prover":"Minlog","tactic_proof":"assume ind simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1638","id":90137,"instance":1638,"goal":"all g u, g + u = u + g","context":["assume","ind","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"cases use rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1639","id":90138,"instance":1639,"goal":"all z n d, (z + n) + d = z + (n + d)","context":["cases","use","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"ind rw simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1640","id":90139,"instance":1640,"goal":"all u, u * 0 = 0","context":["ind","rw","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"cases ind cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1641","id":90140,"instance":1641,"goal":"all m, m * 1 = m","context":["cases","ind","cases","use"]} -{"prover":"Minlog","tactic_proof":"cases assume ng simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1642","id":90141,"instance":1642,"goal":"all z u y, (z * u) * y = z * (u * y)","context":["cases","assume","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"elim rw use ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1643","id":90142,"instance":1643,"goal":"all x d, x * d = d * x","context":["elim","rw","use","ng"]} -{"prover":"Minlog","tactic_proof":"cases defn ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1644","id":90143,"instance":1644,"goal":"all e a n, e * (a + n) = e * a + e * n","context":["cases","defn","ng","intro"]} -{"prover":"Minlog","tactic_proof":"use assume simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1645","id":90144,"instance":1645,"goal":"all a, a <= a","context":["use","assume","simp","use"]} -{"prover":"Minlog","tactic_proof":"ind elim elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1646","id":90145,"instance":1646,"goal":"all e x b, e <= x -> x <= b -> e <= b","context":["ind","elim","elim","assume"]} -{"prover":"Minlog","tactic_proof":"intro intro use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1647","id":90146,"instance":1647,"goal":"all n, not n < n","context":["intro","intro","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw simplify cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1648","id":90147,"instance":1648,"goal":"all y c, Even y -> Even c -> Even (y + c)","context":["rw","simplify","cases","ind"]} -{"prover":"Minlog","tactic_proof":"simp use simp assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1649","id":90148,"instance":1649,"goal":"all n, n | n","context":["simp","use","simp","assume"]} -{"prover":"Minlog","tactic_proof":"defn simplify cases simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1650","id":90149,"instance":1650,"goal":"all w, Fact w > 0","context":["defn","simplify","cases","simp"]} -{"prover":"Minlog","tactic_proof":"simplify intro intro intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1651","id":90150,"instance":1651,"goal":"all b v = v","context":["simplify","intro","intro","intro"]} -{"prover":"Minlog","tactic_proof":"use intro elim intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1652","id":90151,"instance":1652,"goal":"all f + 0 = f","context":["use","intro","elim","intro"]} -{"prover":"Minlog","tactic_proof":"defn intro ng use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1653","id":90152,"instance":1653,"goal":"all v d, v + d = d + v","context":["defn","intro","ng","use"]} -{"prover":"Minlog","tactic_proof":"elim rw defn ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1654","id":90153,"instance":1654,"goal":"all m g p, (m + g) + p = m + (g + p)","context":["elim","rw","defn","ind"]} -{"prover":"Minlog","tactic_proof":"simp elim rw intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1655","id":90154,"instance":1655,"goal":"all d, d * 0 = 0","context":["simp","elim","rw","intro"]} -{"prover":"Minlog","tactic_proof":"simp rw rw rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1656","id":90155,"instance":1656,"goal":"all y, y * 1 = y","context":["simp","rw","rw","rw"]} -{"prover":"Minlog","tactic_proof":"intro assume intro simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1657","id":90156,"instance":1657,"goal":"all b h q, (b * h) * q = b * (h * q)","context":["intro","assume","intro","simp"]} -{"prover":"Minlog","tactic_proof":"assume ng intro simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1658","id":90157,"instance":1658,"goal":"all n v, n * v = v * n","context":["assume","ng","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp use simp use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1659","id":90158,"instance":1659,"goal":"all f q d, f * (q + d) = f * q + f * d","context":["simp","use","simp","use"]} -{"prover":"Minlog","tactic_proof":"elim assume rw simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1660","id":90159,"instance":1660,"goal":"all y, y <= y","context":["elim","assume","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify intro intro ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1661","id":90160,"instance":1661,"goal":"all e p n, e <= p -> p <= n -> e <= n","context":["simplify","intro","intro","ng"]} -{"prover":"Minlog","tactic_proof":"elim ind simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1662","id":90161,"instance":1662,"goal":"all c, not c < c","context":["elim","ind","simp","ng"]} -{"prover":"Minlog","tactic_proof":"rw cases ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1663","id":90162,"instance":1663,"goal":"all c m, Even c -> Even m -> Even (c + m)","context":["rw","cases","ng","assume"]} -{"prover":"Minlog","tactic_proof":"ng simp ind use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1664","id":90163,"instance":1664,"goal":"all w, w | w","context":["ng","simp","ind","use"]} -{"prover":"Minlog","tactic_proof":"defn use cases defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1665","id":90164,"instance":1665,"goal":"all d, Fact d > 0","context":["defn","use","cases","defn"]} -{"prover":"Minlog","tactic_proof":"simp defn use intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1666","id":90165,"instance":1666,"goal":"all p x = x","context":["simp","defn","use","intro"]} -{"prover":"Minlog","tactic_proof":"assume assume defn cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1667","id":90166,"instance":1667,"goal":"all y + 0 = y","context":["assume","assume","defn","cases"]} -{"prover":"Minlog","tactic_proof":"defn ng elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1668","id":90167,"instance":1668,"goal":"all b u, b + u = u + b","context":["defn","ng","elim","ind"]} -{"prover":"Minlog","tactic_proof":"simp ng use simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1669","id":90168,"instance":1669,"goal":"all a g h, (a + g) + h = a + (g + h)","context":["simp","ng","use","simplify"]} -{"prover":"Minlog","tactic_proof":"use defn ng cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1670","id":90169,"instance":1670,"goal":"all y, y * 0 = 0","context":["use","defn","ng","cases"]} -{"prover":"Minlog","tactic_proof":"ng ng ind rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1671","id":90170,"instance":1671,"goal":"all u, u * 1 = u","context":["ng","ng","ind","rw"]} -{"prover":"Minlog","tactic_proof":"simp assume defn assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1672","id":90171,"instance":1672,"goal":"all n f x, (n * f) * x = n * (f * x)","context":["simp","assume","defn","assume"]} -{"prover":"Minlog","tactic_proof":"elim ind assume simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1673","id":90172,"instance":1673,"goal":"all b w, b * w = w * b","context":["elim","ind","assume","simp"]} -{"prover":"Minlog","tactic_proof":"assume rw use simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1674","id":90173,"instance":1674,"goal":"all b a f, b * (a + f) = b * a + b * f","context":["assume","rw","use","simplify"]} -{"prover":"Minlog","tactic_proof":"defn ind cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1675","id":90174,"instance":1675,"goal":"all y, y <= y","context":["defn","ind","cases","ng"]} -{"prover":"Minlog","tactic_proof":"simplify ng cases cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1676","id":90175,"instance":1676,"goal":"all p q n, p <= q -> q <= n -> p <= n","context":["simplify","ng","cases","cases"]} -{"prover":"Minlog","tactic_proof":"defn intro assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1677","id":90176,"instance":1677,"goal":"all p, not p < p","context":["defn","intro","assume","simp"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1678","id":90177,"instance":1678,"goal":"all m d, Even m -> Even d -> Even (m + d)","context":["defn","simplify","rw","elim"]} -{"prover":"Minlog","tactic_proof":"simplify simp simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1679","id":90178,"instance":1679,"goal":"all x, x | x","context":["simplify","simp","simp","ind"]} -{"prover":"Minlog","tactic_proof":"simplify ng use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1680","id":90179,"instance":1680,"goal":"all v, Fact v > 0","context":["simplify","ng","use","elim"]} -{"prover":"Minlog","tactic_proof":"ind elim ng assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1681","id":90180,"instance":1681,"goal":"all y v = v","context":["ind","elim","ng","assume"]} -{"prover":"Minlog","tactic_proof":"elim use cases ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1682","id":90181,"instance":1682,"goal":"all z + 0 = z","context":["elim","use","cases","ind"]} -{"prover":"Minlog","tactic_proof":"use defn intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1683","id":90182,"instance":1683,"goal":"all h b, h + b = b + h","context":["use","defn","intro","assume"]} -{"prover":"Minlog","tactic_proof":"ng elim ng elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1684","id":90183,"instance":1684,"goal":"all c e n, (c + e) + n = c + (e + n)","context":["ng","elim","ng","elim"]} -{"prover":"Minlog","tactic_proof":"simplify intro use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1685","id":90184,"instance":1685,"goal":"all e, e * 0 = 0","context":["simplify","intro","use","assume"]} -{"prover":"Minlog","tactic_proof":"elim ind rw ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1686","id":90185,"instance":1686,"goal":"all c, c * 1 = c","context":["elim","ind","rw","ind"]} -{"prover":"Minlog","tactic_proof":"ng intro rw elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1687","id":90186,"instance":1687,"goal":"all v w p, (v * w) * p = v * (w * p)","context":["ng","intro","rw","elim"]} -{"prover":"Minlog","tactic_proof":"elim simplify ind defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1688","id":90187,"instance":1688,"goal":"all h d, h * d = d * h","context":["elim","simplify","ind","defn"]} -{"prover":"Minlog","tactic_proof":"simplify ng ng simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1689","id":90188,"instance":1689,"goal":"all p e b, p * (e + b) = p * e + p * b","context":["simplify","ng","ng","simp"]} -{"prover":"Minlog","tactic_proof":"assume simp ng defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1690","id":90189,"instance":1690,"goal":"all e, e <= e","context":["assume","simp","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simplify cases assume simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1691","id":90190,"instance":1691,"goal":"all e a q, e <= a -> a <= q -> e <= q","context":["simplify","cases","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"elim defn elim simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1692","id":90191,"instance":1692,"goal":"all e, not e < e","context":["elim","defn","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"defn ind simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1693","id":90192,"instance":1693,"goal":"all e a, Even e -> Even a -> Even (e + a)","context":["defn","ind","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"elim cases cases ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1694","id":90193,"instance":1694,"goal":"all e, e | e","context":["elim","cases","cases","ind"]} -{"prover":"Minlog","tactic_proof":"ind intro simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1695","id":90194,"instance":1695,"goal":"all e, Fact e > 0","context":["ind","intro","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"simplify assume use use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1696","id":90195,"instance":1696,"goal":"all v h = h","context":["simplify","assume","use","use"]} -{"prover":"Minlog","tactic_proof":"simp intro intro intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1697","id":90196,"instance":1697,"goal":"all m + 0 = m","context":["simp","intro","intro","intro"]} -{"prover":"Minlog","tactic_proof":"defn elim cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1698","id":90197,"instance":1698,"goal":"all p h, p + h = h + p","context":["defn","elim","cases","ind"]} -{"prover":"Minlog","tactic_proof":"defn cases assume use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1699","id":90198,"instance":1699,"goal":"all m d a, (m + d) + a = m + (d + a)","context":["defn","cases","assume","use"]} -{"prover":"Minlog","tactic_proof":"simp intro ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1700","id":90199,"instance":1700,"goal":"all m, m * 0 = 0","context":["simp","intro","ng","defn"]} -{"prover":"Minlog","tactic_proof":"defn cases ng assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1701","id":90200,"instance":1701,"goal":"all w, w * 1 = w","context":["defn","cases","ng","assume"]} -{"prover":"Minlog","tactic_proof":"defn ng elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1702","id":90201,"instance":1702,"goal":"all a v n, (a * v) * n = a * (v * n)","context":["defn","ng","elim","defn"]} -{"prover":"Minlog","tactic_proof":"defn ind simp rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1703","id":90202,"instance":1703,"goal":"all q n, q * n = n * q","context":["defn","ind","simp","rw"]} -{"prover":"Minlog","tactic_proof":"cases cases rw cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1704","id":90203,"instance":1704,"goal":"all z y p, z * (y + p) = z * y + z * p","context":["cases","cases","rw","cases"]} -{"prover":"Minlog","tactic_proof":"ng defn elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1705","id":90204,"instance":1705,"goal":"all x, x <= x","context":["ng","defn","elim","ind"]} -{"prover":"Minlog","tactic_proof":"defn rw simplify intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1706","id":90205,"instance":1706,"goal":"all p g x, p <= g -> g <= x -> p <= x","context":["defn","rw","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"ind simp intro assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1707","id":90206,"instance":1707,"goal":"all v, not v < v","context":["ind","simp","intro","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simplify intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1708","id":90207,"instance":1708,"goal":"all d m, Even d -> Even m -> Even (d + m)","context":["simplify","simplify","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"ng ind simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1709","id":90208,"instance":1709,"goal":"all a, a | a","context":["ng","ind","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"intro ng defn ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1710","id":90209,"instance":1710,"goal":"all v, Fact v > 0","context":["intro","ng","defn","ng"]} -{"prover":"Minlog","tactic_proof":"intro assume use ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1711","id":90210,"instance":1711,"goal":"all h b = b","context":["intro","assume","use","ind"]} -{"prover":"Minlog","tactic_proof":"cases ind rw intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1712","id":90211,"instance":1712,"goal":"all d + 0 = d","context":["cases","ind","rw","intro"]} -{"prover":"Minlog","tactic_proof":"use intro ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1713","id":90212,"instance":1713,"goal":"all f e, f + e = e + f","context":["use","intro","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1714","id":90213,"instance":1714,"goal":"all v h g, (v + h) + g = v + (h + g)","context":["defn","simplify","rw","ind"]} -{"prover":"Minlog","tactic_proof":"intro ng simplify simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1715","id":90214,"instance":1715,"goal":"all d, d * 0 = 0","context":["intro","ng","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"ind intro defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1716","id":90215,"instance":1716,"goal":"all e, e * 1 = e","context":["ind","intro","defn","use"]} -{"prover":"Minlog","tactic_proof":"elim simp elim simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1717","id":90216,"instance":1717,"goal":"all z h g, (z * h) * g = z * (h * g)","context":["elim","simp","elim","simp"]} -{"prover":"Minlog","tactic_proof":"simp elim simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1718","id":90217,"instance":1718,"goal":"all d n, d * n = n * d","context":["simp","elim","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"use assume defn rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1719","id":90218,"instance":1719,"goal":"all m p c, m * (p + c) = m * p + m * c","context":["use","assume","defn","rw"]} -{"prover":"Minlog","tactic_proof":"rw ind rw use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1720","id":90219,"instance":1720,"goal":"all g, g <= g","context":["rw","ind","rw","use"]} -{"prover":"Minlog","tactic_proof":"defn ng ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1721","id":90220,"instance":1721,"goal":"all g p a, g <= p -> p <= a -> g <= a","context":["defn","ng","ind","ind"]} -{"prover":"Minlog","tactic_proof":"cases assume ind assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1722","id":90221,"instance":1722,"goal":"all q, not q < q","context":["cases","assume","ind","assume"]} -{"prover":"Minlog","tactic_proof":"rw elim cases simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1723","id":90222,"instance":1723,"goal":"all x v, Even x -> Even v -> Even (x + v)","context":["rw","elim","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"simp ind rw rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1724","id":90223,"instance":1724,"goal":"all u, u | u","context":["simp","ind","rw","rw"]} -{"prover":"Minlog","tactic_proof":"ind ind simp ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1725","id":90224,"instance":1725,"goal":"all p, Fact p > 0","context":["ind","ind","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng simp simplify cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1726","id":90225,"instance":1726,"goal":"all y x = x","context":["ng","simp","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"assume ng defn elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1727","id":90226,"instance":1727,"goal":"all h + 0 = h","context":["assume","ng","defn","elim"]} -{"prover":"Minlog","tactic_proof":"use assume simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1728","id":90227,"instance":1728,"goal":"all z h, z + h = h + z","context":["use","assume","simp","use"]} -{"prover":"Minlog","tactic_proof":"elim assume cases intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1729","id":90228,"instance":1729,"goal":"all g e m, (g + e) + m = g + (e + m)","context":["elim","assume","cases","intro"]} -{"prover":"Minlog","tactic_proof":"use use use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1730","id":90229,"instance":1730,"goal":"all g, g * 0 = 0","context":["use","use","use","intro"]} -{"prover":"Minlog","tactic_proof":"use ind elim defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1731","id":90230,"instance":1731,"goal":"all e, e * 1 = e","context":["use","ind","elim","defn"]} -{"prover":"Minlog","tactic_proof":"rw simplify simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1732","id":90231,"instance":1732,"goal":"all d v c, (d * v) * c = d * (v * c)","context":["rw","simplify","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"ng simp assume intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1733","id":90232,"instance":1733,"goal":"all v c, v * c = c * v","context":["ng","simp","assume","intro"]} -{"prover":"Minlog","tactic_proof":"intro defn ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1734","id":90233,"instance":1734,"goal":"all d v a, d * (v + a) = d * v + d * a","context":["intro","defn","ind","cases"]} -{"prover":"Minlog","tactic_proof":"intro elim simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1735","id":90234,"instance":1735,"goal":"all g, g <= g","context":["intro","elim","simp","rw"]} -{"prover":"Minlog","tactic_proof":"simp ng intro elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1736","id":90235,"instance":1736,"goal":"all a z m, a <= z -> z <= m -> a <= m","context":["simp","ng","intro","elim"]} -{"prover":"Minlog","tactic_proof":"ng simp assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1737","id":90236,"instance":1737,"goal":"all y, not y < y","context":["ng","simp","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ng elim ng simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1738","id":90237,"instance":1738,"goal":"all q m, Even q -> Even m -> Even (q + m)","context":["ng","elim","ng","simp"]} -{"prover":"Minlog","tactic_proof":"simplify cases ng intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1739","id":90238,"instance":1739,"goal":"all q, q | q","context":["simplify","cases","ng","intro"]} -{"prover":"Minlog","tactic_proof":"elim rw assume defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1740","id":90239,"instance":1740,"goal":"all h, Fact h > 0","context":["elim","rw","assume","defn"]} -{"prover":"Minlog","tactic_proof":"ind simp rw ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1741","id":90240,"instance":1741,"goal":"all f v = v","context":["ind","simp","rw","ind"]} -{"prover":"Minlog","tactic_proof":"cases rw rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1742","id":90241,"instance":1742,"goal":"all x + 0 = x","context":["cases","rw","rw","simp"]} -{"prover":"Minlog","tactic_proof":"ind simplify simplify simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1743","id":90242,"instance":1743,"goal":"all c v, c + v = v + c","context":["ind","simplify","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"rw ind simp simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1744","id":90243,"instance":1744,"goal":"all u x e, (u + x) + e = u + (x + e)","context":["rw","ind","simp","simp"]} -{"prover":"Minlog","tactic_proof":"ind simp elim elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1745","id":90244,"instance":1745,"goal":"all b, b * 0 = 0","context":["ind","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"intro defn use defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1746","id":90245,"instance":1746,"goal":"all d, d * 1 = d","context":["intro","defn","use","defn"]} -{"prover":"Minlog","tactic_proof":"assume simplify use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1747","id":90246,"instance":1747,"goal":"all c z q, (c * z) * q = c * (z * q)","context":["assume","simplify","use","simp"]} -{"prover":"Minlog","tactic_proof":"use defn use ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1748","id":90247,"instance":1748,"goal":"all g u, g * u = u * g","context":["use","defn","use","ind"]} -{"prover":"Minlog","tactic_proof":"assume intro simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1749","id":90248,"instance":1749,"goal":"all f n g, f * (n + g) = f * n + f * g","context":["assume","intro","simp","use"]} -{"prover":"Minlog","tactic_proof":"use assume ng ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1750","id":90249,"instance":1750,"goal":"all m, m <= m","context":["use","assume","ng","ind"]} -{"prover":"Minlog","tactic_proof":"ind use assume assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1751","id":90250,"instance":1751,"goal":"all c b z, c <= b -> b <= z -> c <= z","context":["ind","use","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ng ind defn elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1752","id":90251,"instance":1752,"goal":"all w, not w < w","context":["ng","ind","defn","elim"]} -{"prover":"Minlog","tactic_proof":"cases simplify ind elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1753","id":90252,"instance":1753,"goal":"all x m, Even x -> Even m -> Even (x + m)","context":["cases","simplify","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simp intro use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1754","id":90253,"instance":1754,"goal":"all c, c | c","context":["simp","intro","use","rw"]} -{"prover":"Minlog","tactic_proof":"simp use use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1755","id":90254,"instance":1755,"goal":"all q, Fact q > 0","context":["simp","use","use","assume"]} -{"prover":"Minlog","tactic_proof":"rw assume ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1756","id":90255,"instance":1756,"goal":"all h p = p","context":["rw","assume","ng","elim"]} -{"prover":"Minlog","tactic_proof":"intro ng elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1757","id":90256,"instance":1757,"goal":"all g + 0 = g","context":["intro","ng","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"cases defn simp defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1758","id":90257,"instance":1758,"goal":"all z e, z + e = e + z","context":["cases","defn","simp","defn"]} -{"prover":"Minlog","tactic_proof":"elim ng assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1759","id":90258,"instance":1759,"goal":"all d f y, (d + f) + y = d + (f + y)","context":["elim","ng","assume","assume"]} -{"prover":"Minlog","tactic_proof":"elim defn cases cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1760","id":90259,"instance":1760,"goal":"all z, z * 0 = 0","context":["elim","defn","cases","cases"]} -{"prover":"Minlog","tactic_proof":"use rw elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1761","id":90260,"instance":1761,"goal":"all e, e * 1 = e","context":["use","rw","elim","cases"]} -{"prover":"Minlog","tactic_proof":"simp ind simplify ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1762","id":90261,"instance":1762,"goal":"all n q x, (n * q) * x = n * (q * x)","context":["simp","ind","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"simp use use rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1763","id":90262,"instance":1763,"goal":"all h f, h * f = f * h","context":["simp","use","use","rw"]} -{"prover":"Minlog","tactic_proof":"ng ind simp ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1764","id":90263,"instance":1764,"goal":"all w m g, w * (m + g) = w * m + w * g","context":["ng","ind","simp","ind"]} -{"prover":"Minlog","tactic_proof":"intro ind ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1765","id":90264,"instance":1765,"goal":"all p, p <= p","context":["intro","ind","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simp assume assume ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1766","id":90265,"instance":1766,"goal":"all a u y, a <= u -> u <= y -> a <= y","context":["simp","assume","assume","ind"]} -{"prover":"Minlog","tactic_proof":"simp rw intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1767","id":90266,"instance":1767,"goal":"all m, not m < m","context":["simp","rw","intro","assume"]} -{"prover":"Minlog","tactic_proof":"defn use elim rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1768","id":90267,"instance":1768,"goal":"all g u, Even g -> Even u -> Even (g + u)","context":["defn","use","elim","rw"]} -{"prover":"Minlog","tactic_proof":"assume rw defn assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1769","id":90268,"instance":1769,"goal":"all m, m | m","context":["assume","rw","defn","assume"]} -{"prover":"Minlog","tactic_proof":"defn ind use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1770","id":90269,"instance":1770,"goal":"all p, Fact p > 0","context":["defn","ind","use","cases"]} -{"prover":"Minlog","tactic_proof":"cases rw use ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1771","id":90270,"instance":1771,"goal":"all f c = c","context":["cases","rw","use","ind"]} -{"prover":"Minlog","tactic_proof":"use defn defn defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1772","id":90271,"instance":1772,"goal":"all u + 0 = u","context":["use","defn","defn","defn"]} -{"prover":"Minlog","tactic_proof":"rw use cases cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1773","id":90272,"instance":1773,"goal":"all y z, y + z = z + y","context":["rw","use","cases","cases"]} -{"prover":"Minlog","tactic_proof":"assume intro simp cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1774","id":90273,"instance":1774,"goal":"all q z g, (q + z) + g = q + (z + g)","context":["assume","intro","simp","cases"]} -{"prover":"Minlog","tactic_proof":"ng assume rw simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1775","id":90274,"instance":1775,"goal":"all y, y * 0 = 0","context":["ng","assume","rw","simp"]} -{"prover":"Minlog","tactic_proof":"ng intro elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1776","id":90275,"instance":1776,"goal":"all q, q * 1 = q","context":["ng","intro","elim","use"]} -{"prover":"Minlog","tactic_proof":"defn ng rw elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1777","id":90276,"instance":1777,"goal":"all p z a, (p * z) * a = p * (z * a)","context":["defn","ng","rw","elim"]} -{"prover":"Minlog","tactic_proof":"intro defn cases cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1778","id":90277,"instance":1778,"goal":"all c m, c * m = m * c","context":["intro","defn","cases","cases"]} -{"prover":"Minlog","tactic_proof":"simplify simplify assume cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1779","id":90278,"instance":1779,"goal":"all w q p, w * (q + p) = w * q + w * p","context":["simplify","simplify","assume","cases"]} -{"prover":"Minlog","tactic_proof":"assume elim cases ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1780","id":90279,"instance":1780,"goal":"all m, m <= m","context":["assume","elim","cases","ind"]} -{"prover":"Minlog","tactic_proof":"use defn ind ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1781","id":90280,"instance":1781,"goal":"all z w q, z <= w -> w <= q -> z <= q","context":["use","defn","ind","ng"]} -{"prover":"Minlog","tactic_proof":"ind simplify intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1782","id":90281,"instance":1782,"goal":"all e, not e < e","context":["ind","simplify","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp use simp simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1783","id":90282,"instance":1783,"goal":"all g y, Even g -> Even y -> Even (g + y)","context":["simp","use","simp","simp"]} -{"prover":"Minlog","tactic_proof":"assume ind ng rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1784","id":90283,"instance":1784,"goal":"all u, u | u","context":["assume","ind","ng","rw"]} -{"prover":"Minlog","tactic_proof":"ind ng use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1785","id":90284,"instance":1785,"goal":"all v, Fact v > 0","context":["ind","ng","use","simp"]} -{"prover":"Minlog","tactic_proof":"simp elim elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1786","id":90285,"instance":1786,"goal":"all p m = m","context":["simp","elim","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"elim simplify simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1787","id":90286,"instance":1787,"goal":"all a + 0 = a","context":["elim","simplify","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ng intro intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1788","id":90287,"instance":1788,"goal":"all c y, c + y = y + c","context":["ng","intro","intro","elim"]} -{"prover":"Minlog","tactic_proof":"use simp rw cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1789","id":90288,"instance":1789,"goal":"all e p w, (e + p) + w = e + (p + w)","context":["use","simp","rw","cases"]} -{"prover":"Minlog","tactic_proof":"ind use simp elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1790","id":90289,"instance":1790,"goal":"all v, v * 0 = 0","context":["ind","use","simp","elim"]} -{"prover":"Minlog","tactic_proof":"ind ind simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1791","id":90290,"instance":1791,"goal":"all p, p * 1 = p","context":["ind","ind","simp","use"]} -{"prover":"Minlog","tactic_proof":"use assume intro intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1792","id":90291,"instance":1792,"goal":"all z f b, (z * f) * b = z * (f * b)","context":["use","assume","intro","intro"]} -{"prover":"Minlog","tactic_proof":"ng assume ng elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1793","id":90292,"instance":1793,"goal":"all m a, m * a = a * m","context":["ng","assume","ng","elim"]} -{"prover":"Minlog","tactic_proof":"assume elim elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1794","id":90293,"instance":1794,"goal":"all a n x, a * (n + x) = a * n + a * x","context":["assume","elim","elim","assume"]} -{"prover":"Minlog","tactic_proof":"intro assume simplify simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1795","id":90294,"instance":1795,"goal":"all y, y <= y","context":["intro","assume","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"rw ng cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1796","id":90295,"instance":1796,"goal":"all f w x, f <= w -> w <= x -> f <= x","context":["rw","ng","cases","cases"]} -{"prover":"Minlog","tactic_proof":"assume intro cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1797","id":90296,"instance":1797,"goal":"all f, not f < f","context":["assume","intro","cases","use"]} -{"prover":"Minlog","tactic_proof":"intro intro elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1798","id":90297,"instance":1798,"goal":"all z c, Even z -> Even c -> Even (z + c)","context":["intro","intro","elim","simp"]} -{"prover":"Minlog","tactic_proof":"intro assume elim assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1799","id":90298,"instance":1799,"goal":"all x, x | x","context":["intro","assume","elim","assume"]} -{"prover":"Minlog","tactic_proof":"intro cases elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1800","id":90299,"instance":1800,"goal":"all f, Fact f > 0","context":["intro","cases","elim","cases"]} -{"prover":"Minlog","tactic_proof":"intro defn cases simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1801","id":90300,"instance":1801,"goal":"all g y = y","context":["intro","defn","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ind intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1802","id":90301,"instance":1802,"goal":"all y + 0 = y","context":["cases","ind","intro","simp"]} -{"prover":"Minlog","tactic_proof":"ng simp assume ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1803","id":90302,"instance":1803,"goal":"all u p, u + p = p + u","context":["ng","simp","assume","ind"]} -{"prover":"Minlog","tactic_proof":"elim simplify use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1804","id":90303,"instance":1804,"goal":"all g u f, (g + u) + f = g + (u + f)","context":["elim","simplify","use","cases"]} -{"prover":"Minlog","tactic_proof":"ind use simp cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1805","id":90304,"instance":1805,"goal":"all m, m * 0 = 0","context":["ind","use","simp","cases"]} -{"prover":"Minlog","tactic_proof":"defn cases simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1806","id":90305,"instance":1806,"goal":"all d, d * 1 = d","context":["defn","cases","simp","ind"]} -{"prover":"Minlog","tactic_proof":"assume simp elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1807","id":90306,"instance":1807,"goal":"all y c g, (y * c) * g = y * (c * g)","context":["assume","simp","elim","ind"]} -{"prover":"Minlog","tactic_proof":"use ng cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1808","id":90307,"instance":1808,"goal":"all u x, u * x = x * u","context":["use","ng","cases","assume"]} -{"prover":"Minlog","tactic_proof":"simplify rw ind assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1809","id":90308,"instance":1809,"goal":"all b f a, b * (f + a) = b * f + b * a","context":["simplify","rw","ind","assume"]} -{"prover":"Minlog","tactic_proof":"ind intro use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1810","id":90309,"instance":1810,"goal":"all m, m <= m","context":["ind","intro","use","assume"]} -{"prover":"Minlog","tactic_proof":"ng rw simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1811","id":90310,"instance":1811,"goal":"all d p n, d <= p -> p <= n -> d <= n","context":["ng","rw","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"simplify use elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1812","id":90311,"instance":1812,"goal":"all m, not m < m","context":["simplify","use","elim","rw"]} -{"prover":"Minlog","tactic_proof":"use assume ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1813","id":90312,"instance":1813,"goal":"all b w, Even b -> Even w -> Even (b + w)","context":["use","assume","ng","defn"]} -{"prover":"Minlog","tactic_proof":"defn defn simp ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1814","id":90313,"instance":1814,"goal":"all e, e | e","context":["defn","defn","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ind elim simplify cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1815","id":90314,"instance":1815,"goal":"all m, Fact m > 0","context":["ind","elim","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"intro elim simplify intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1816","id":90315,"instance":1816,"goal":"all p h = h","context":["intro","elim","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"cases use simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1817","id":90316,"instance":1817,"goal":"all e + 0 = e","context":["cases","use","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"cases ng elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1818","id":90317,"instance":1818,"goal":"all c u, c + u = u + c","context":["cases","ng","elim","cases"]} -{"prover":"Minlog","tactic_proof":"ng rw simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1819","id":90318,"instance":1819,"goal":"all a w f, (a + w) + f = a + (w + f)","context":["ng","rw","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"use intro simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1820","id":90319,"instance":1820,"goal":"all e, e * 0 = 0","context":["use","intro","simp","use"]} -{"prover":"Minlog","tactic_proof":"intro elim intro rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1821","id":90320,"instance":1821,"goal":"all y, y * 1 = y","context":["intro","elim","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ng ng ind simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1822","id":90321,"instance":1822,"goal":"all g q f, (g * q) * f = g * (q * f)","context":["ng","ng","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ind ind cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1823","id":90322,"instance":1823,"goal":"all g m, g * m = m * g","context":["cases","ind","ind","cases"]} -{"prover":"Minlog","tactic_proof":"defn ind use simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1824","id":90323,"instance":1824,"goal":"all e x n, e * (x + n) = e * x + e * n","context":["defn","ind","use","simplify"]} -{"prover":"Minlog","tactic_proof":"cases simplify assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1825","id":90324,"instance":1825,"goal":"all m, m <= m","context":["cases","simplify","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ind simplify ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1826","id":90325,"instance":1826,"goal":"all h u v, h <= u -> u <= v -> h <= v","context":["ind","simplify","ind","rw"]} -{"prover":"Minlog","tactic_proof":"ng simp intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1827","id":90326,"instance":1827,"goal":"all p, not p < p","context":["ng","simp","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"assume intro elim cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1828","id":90327,"instance":1828,"goal":"all v u, Even v -> Even u -> Even (v + u)","context":["assume","intro","elim","cases"]} -{"prover":"Minlog","tactic_proof":"use assume elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1829","id":90328,"instance":1829,"goal":"all p, p | p","context":["use","assume","elim","ind"]} -{"prover":"Minlog","tactic_proof":"ind simp elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1830","id":90329,"instance":1830,"goal":"all q, Fact q > 0","context":["ind","simp","elim","defn"]} -{"prover":"Minlog","tactic_proof":"simplify intro elim use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1831","id":90330,"instance":1831,"goal":"all y h = h","context":["simplify","intro","elim","use"]} -{"prover":"Minlog","tactic_proof":"use rw cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1832","id":90331,"instance":1832,"goal":"all b + 0 = b","context":["use","rw","cases","ng"]} -{"prover":"Minlog","tactic_proof":"ind elim ind defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1833","id":90332,"instance":1833,"goal":"all e m, e + m = m + e","context":["ind","elim","ind","defn"]} -{"prover":"Minlog","tactic_proof":"elim elim simplify simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1834","id":90333,"instance":1834,"goal":"all f z v, (f + z) + v = f + (z + v)","context":["elim","elim","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"elim intro simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1835","id":90334,"instance":1835,"goal":"all h, h * 0 = 0","context":["elim","intro","simp","assume"]} -{"prover":"Minlog","tactic_proof":"assume simp cases defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1836","id":90335,"instance":1836,"goal":"all h, h * 1 = h","context":["assume","simp","cases","defn"]} -{"prover":"Minlog","tactic_proof":"ind rw ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1837","id":90336,"instance":1837,"goal":"all a n y, (a * n) * y = a * (n * y)","context":["ind","rw","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume cases defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1838","id":90337,"instance":1838,"goal":"all q h, q * h = h * q","context":["assume","cases","defn","simp"]} -{"prover":"Minlog","tactic_proof":"simp defn assume intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1839","id":90338,"instance":1839,"goal":"all d b u, d * (b + u) = d * b + d * u","context":["simp","defn","assume","intro"]} -{"prover":"Minlog","tactic_proof":"rw simp cases ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1840","id":90339,"instance":1840,"goal":"all y, y <= y","context":["rw","simp","cases","ng"]} -{"prover":"Minlog","tactic_proof":"ng assume simp cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1841","id":90340,"instance":1841,"goal":"all g a m, g <= a -> a <= m -> g <= m","context":["ng","assume","simp","cases"]} -{"prover":"Minlog","tactic_proof":"ind use simplify elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1842","id":90341,"instance":1842,"goal":"all g, not g < g","context":["ind","use","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"ind assume ng use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1843","id":90342,"instance":1843,"goal":"all d a, Even d -> Even a -> Even (d + a)","context":["ind","assume","ng","use"]} -{"prover":"Minlog","tactic_proof":"rw intro simplify assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1844","id":90343,"instance":1844,"goal":"all c, c | c","context":["rw","intro","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"simplify simplify simplify ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1845","id":90344,"instance":1845,"goal":"all q, Fact q > 0","context":["simplify","simplify","simplify","ind"]} -{"prover":"Minlog","tactic_proof":"defn ng simp cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1846","id":90345,"instance":1846,"goal":"all m d = d","context":["defn","ng","simp","cases"]} -{"prover":"Minlog","tactic_proof":"ind elim intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1847","id":90346,"instance":1847,"goal":"all f + 0 = f","context":["ind","elim","intro","defn"]} -{"prover":"Minlog","tactic_proof":"simplify intro rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1848","id":90347,"instance":1848,"goal":"all x b, x + b = b + x","context":["simplify","intro","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ind rw rw rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1849","id":90348,"instance":1849,"goal":"all h n u, (h + n) + u = h + (n + u)","context":["ind","rw","rw","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1850","id":90349,"instance":1850,"goal":"all u, u * 0 = 0","context":["ng","defn","intro","ind"]} -{"prover":"Minlog","tactic_proof":"intro cases assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1851","id":90350,"instance":1851,"goal":"all q, q * 1 = q","context":["intro","cases","assume","intro"]} -{"prover":"Minlog","tactic_proof":"simplify simp assume elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1852","id":90351,"instance":1852,"goal":"all g n z, (g * n) * z = g * (n * z)","context":["simplify","simp","assume","elim"]} -{"prover":"Minlog","tactic_proof":"simplify simp intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1853","id":90352,"instance":1853,"goal":"all d q, d * q = q * d","context":["simplify","simp","intro","simp"]} -{"prover":"Minlog","tactic_proof":"cases elim assume simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1854","id":90353,"instance":1854,"goal":"all z m h, z * (m + h) = z * m + z * h","context":["cases","elim","assume","simplify"]} -{"prover":"Minlog","tactic_proof":"use assume elim defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1855","id":90354,"instance":1855,"goal":"all e, e <= e","context":["use","assume","elim","defn"]} -{"prover":"Minlog","tactic_proof":"simplify ng simplify elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1856","id":90355,"instance":1856,"goal":"all q v f, q <= v -> v <= f -> q <= f","context":["simplify","ng","simplify","elim"]} -{"prover":"Minlog","tactic_proof":"elim use elim simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1857","id":90356,"instance":1857,"goal":"all c, not c < c","context":["elim","use","elim","simp"]} -{"prover":"Minlog","tactic_proof":"simplify ng assume use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1858","id":90357,"instance":1858,"goal":"all e v, Even e -> Even v -> Even (e + v)","context":["simplify","ng","assume","use"]} -{"prover":"Minlog","tactic_proof":"simplify defn assume defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1859","id":90358,"instance":1859,"goal":"all w, w | w","context":["simplify","defn","assume","defn"]} -{"prover":"Minlog","tactic_proof":"use intro ind elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1860","id":90359,"instance":1860,"goal":"all u, Fact u > 0","context":["use","intro","ind","elim"]} -{"prover":"Minlog","tactic_proof":"simplify cases simp cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1861","id":90360,"instance":1861,"goal":"all n e = e","context":["simplify","cases","simp","cases"]} -{"prover":"Minlog","tactic_proof":"intro intro rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1862","id":90361,"instance":1862,"goal":"all q + 0 = q","context":["intro","intro","rw","assume"]} -{"prover":"Minlog","tactic_proof":"ind simp assume intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1863","id":90362,"instance":1863,"goal":"all m n, m + n = n + m","context":["ind","simp","assume","intro"]} -{"prover":"Minlog","tactic_proof":"simp ng rw defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1864","id":90363,"instance":1864,"goal":"all q g c, (q + g) + c = q + (g + c)","context":["simp","ng","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ng intro cases use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1865","id":90364,"instance":1865,"goal":"all q, q * 0 = 0","context":["ng","intro","cases","use"]} -{"prover":"Minlog","tactic_proof":"ind rw ind ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1866","id":90365,"instance":1866,"goal":"all m, m * 1 = m","context":["ind","rw","ind","ind"]} -{"prover":"Minlog","tactic_proof":"cases simp ng cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1867","id":90366,"instance":1867,"goal":"all x m h, (x * m) * h = x * (m * h)","context":["cases","simp","ng","cases"]} -{"prover":"Minlog","tactic_proof":"elim ind rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1868","id":90367,"instance":1868,"goal":"all e p, e * p = p * e","context":["elim","ind","rw","assume"]} -{"prover":"Minlog","tactic_proof":"assume elim simplify assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1869","id":90368,"instance":1869,"goal":"all c y a, c * (y + a) = c * y + c * a","context":["assume","elim","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"ind assume intro defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1870","id":90369,"instance":1870,"goal":"all b, b <= b","context":["ind","assume","intro","defn"]} -{"prover":"Minlog","tactic_proof":"ng simplify ind cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1871","id":90370,"instance":1871,"goal":"all w v q, w <= v -> v <= q -> w <= q","context":["ng","simplify","ind","cases"]} -{"prover":"Minlog","tactic_proof":"use intro elim use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1872","id":90371,"instance":1872,"goal":"all u, not u < u","context":["use","intro","elim","use"]} -{"prover":"Minlog","tactic_proof":"simplify simplify ind ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1873","id":90372,"instance":1873,"goal":"all h b, Even h -> Even b -> Even (h + b)","context":["simplify","simplify","ind","ind"]} -{"prover":"Minlog","tactic_proof":"ind intro rw rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1874","id":90373,"instance":1874,"goal":"all y, y | y","context":["ind","intro","rw","rw"]} -{"prover":"Minlog","tactic_proof":"intro ng assume simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1875","id":90374,"instance":1875,"goal":"all b, Fact b > 0","context":["intro","ng","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ng ng ind ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1876","id":90375,"instance":1876,"goal":"all u c = c","context":["ng","ng","ind","ind"]} -{"prover":"Minlog","tactic_proof":"use use ng use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1877","id":90376,"instance":1877,"goal":"all h + 0 = h","context":["use","use","ng","use"]} -{"prover":"Minlog","tactic_proof":"use elim elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1878","id":90377,"instance":1878,"goal":"all w b, w + b = b + w","context":["use","elim","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ind cases assume simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1879","id":90378,"instance":1879,"goal":"all q v e, (q + v) + e = q + (v + e)","context":["ind","cases","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ind cases ind cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1880","id":90379,"instance":1880,"goal":"all q, q * 0 = 0","context":["ind","cases","ind","cases"]} -{"prover":"Minlog","tactic_proof":"rw rw defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1881","id":90380,"instance":1881,"goal":"all b, b * 1 = b","context":["rw","rw","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simp simplify intro rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1882","id":90381,"instance":1882,"goal":"all d w f, (d * w) * f = d * (w * f)","context":["simp","simplify","intro","rw"]} -{"prover":"Minlog","tactic_proof":"ind simplify defn rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1883","id":90382,"instance":1883,"goal":"all y u, y * u = u * y","context":["ind","simplify","defn","rw"]} -{"prover":"Minlog","tactic_proof":"intro rw elim simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1884","id":90383,"instance":1884,"goal":"all a v f, a * (v + f) = a * v + a * f","context":["intro","rw","elim","simp"]} -{"prover":"Minlog","tactic_proof":"assume use use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1885","id":90384,"instance":1885,"goal":"all a, a <= a","context":["assume","use","use","cases"]} -{"prover":"Minlog","tactic_proof":"rw elim intro use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1886","id":90385,"instance":1886,"goal":"all d w z, d <= w -> w <= z -> d <= z","context":["rw","elim","intro","use"]} -{"prover":"Minlog","tactic_proof":"intro elim simp ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1887","id":90386,"instance":1887,"goal":"all e, not e < e","context":["intro","elim","simp","ind"]} -{"prover":"Minlog","tactic_proof":"cases assume simp defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1888","id":90387,"instance":1888,"goal":"all n u, Even n -> Even u -> Even (n + u)","context":["cases","assume","simp","defn"]} -{"prover":"Minlog","tactic_proof":"defn rw ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1889","id":90388,"instance":1889,"goal":"all m, m | m","context":["defn","rw","ind","defn"]} -{"prover":"Minlog","tactic_proof":"defn simplify rw intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1890","id":90389,"instance":1890,"goal":"all g, Fact g > 0","context":["defn","simplify","rw","intro"]} -{"prover":"Minlog","tactic_proof":"cases intro simp simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1891","id":90390,"instance":1891,"goal":"all p e = e","context":["cases","intro","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify defn elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1892","id":90391,"instance":1892,"goal":"all d + 0 = d","context":["simplify","defn","elim","rw"]} -{"prover":"Minlog","tactic_proof":"ng ind cases defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1893","id":90392,"instance":1893,"goal":"all c z, c + z = z + c","context":["ng","ind","cases","defn"]} -{"prover":"Minlog","tactic_proof":"defn simp simp intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1894","id":90393,"instance":1894,"goal":"all f y g, (f + y) + g = f + (y + g)","context":["defn","simp","simp","intro"]} -{"prover":"Minlog","tactic_proof":"defn simplify simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1895","id":90394,"instance":1895,"goal":"all p, p * 0 = 0","context":["defn","simplify","simplify","use"]} -{"prover":"Minlog","tactic_proof":"use ng ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1896","id":90395,"instance":1896,"goal":"all q, q * 1 = q","context":["use","ng","ind","intro"]} -{"prover":"Minlog","tactic_proof":"defn ind use defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1897","id":90396,"instance":1897,"goal":"all e g b, (e * g) * b = e * (g * b)","context":["defn","ind","use","defn"]} -{"prover":"Minlog","tactic_proof":"rw defn simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1898","id":90397,"instance":1898,"goal":"all c w, c * w = w * c","context":["rw","defn","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"use simplify cases defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1899","id":90398,"instance":1899,"goal":"all c f d, c * (f + d) = c * f + c * d","context":["use","simplify","cases","defn"]} -{"prover":"Minlog","tactic_proof":"ng defn use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1900","id":90399,"instance":1900,"goal":"all h, h <= h","context":["ng","defn","use","assume"]} -{"prover":"Minlog","tactic_proof":"intro ng cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1901","id":90400,"instance":1901,"goal":"all x p v, x <= p -> p <= v -> x <= v","context":["intro","ng","cases","elim"]} -{"prover":"Minlog","tactic_proof":"simp simplify simplify simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1902","id":90401,"instance":1902,"goal":"all z, not z < z","context":["simp","simplify","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simp elim intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1903","id":90402,"instance":1903,"goal":"all e m, Even e -> Even m -> Even (e + m)","context":["assume","simp","elim","intro"]} -{"prover":"Minlog","tactic_proof":"elim intro intro simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1904","id":90403,"instance":1904,"goal":"all n, n | n","context":["elim","intro","intro","simp"]} -{"prover":"Minlog","tactic_proof":"ind assume elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1905","id":90404,"instance":1905,"goal":"all q, Fact q > 0","context":["ind","assume","elim","intro"]} -{"prover":"Minlog","tactic_proof":"intro cases cases simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1906","id":90405,"instance":1906,"goal":"all e f = f","context":["intro","cases","cases","simp"]} -{"prover":"Minlog","tactic_proof":"intro defn ng assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1907","id":90406,"instance":1907,"goal":"all w + 0 = w","context":["intro","defn","ng","assume"]} -{"prover":"Minlog","tactic_proof":"rw elim use cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1908","id":90407,"instance":1908,"goal":"all w g, w + g = g + w","context":["rw","elim","use","cases"]} -{"prover":"Minlog","tactic_proof":"ng simplify cases simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1909","id":90408,"instance":1909,"goal":"all g p z, (g + p) + z = g + (p + z)","context":["ng","simplify","cases","simp"]} -{"prover":"Minlog","tactic_proof":"ind ng simplify intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1910","id":90409,"instance":1910,"goal":"all m, m * 0 = 0","context":["ind","ng","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"cases elim rw ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1911","id":90410,"instance":1911,"goal":"all c, c * 1 = c","context":["cases","elim","rw","ng"]} -{"prover":"Minlog","tactic_proof":"simplify elim simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1912","id":90411,"instance":1912,"goal":"all z b g, (z * b) * g = z * (b * g)","context":["simplify","elim","simp","use"]} -{"prover":"Minlog","tactic_proof":"elim assume ng rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1913","id":90412,"instance":1913,"goal":"all d q, d * q = q * d","context":["elim","assume","ng","rw"]} -{"prover":"Minlog","tactic_proof":"ind use elim intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1914","id":90413,"instance":1914,"goal":"all q w n, q * (w + n) = q * w + q * n","context":["ind","use","elim","intro"]} -{"prover":"Minlog","tactic_proof":"defn simp elim rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1915","id":90414,"instance":1915,"goal":"all w, w <= w","context":["defn","simp","elim","rw"]} -{"prover":"Minlog","tactic_proof":"simp simp ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1916","id":90415,"instance":1916,"goal":"all d c b, d <= c -> c <= b -> d <= b","context":["simp","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"rw simplify elim defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1917","id":90416,"instance":1917,"goal":"all x, not x < x","context":["rw","simplify","elim","defn"]} -{"prover":"Minlog","tactic_proof":"simplify intro assume use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1918","id":90417,"instance":1918,"goal":"all b h, Even b -> Even h -> Even (b + h)","context":["simplify","intro","assume","use"]} -{"prover":"Minlog","tactic_proof":"cases assume cases simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1919","id":90418,"instance":1919,"goal":"all v, v | v","context":["cases","assume","cases","simp"]} -{"prover":"Minlog","tactic_proof":"simplify cases intro use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1920","id":90419,"instance":1920,"goal":"all d, Fact d > 0","context":["simplify","cases","intro","use"]} -{"prover":"Minlog","tactic_proof":"use ng elim cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1921","id":90420,"instance":1921,"goal":"all u z = z","context":["use","ng","elim","cases"]} -{"prover":"Minlog","tactic_proof":"rw simp cases ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1922","id":90421,"instance":1922,"goal":"all a + 0 = a","context":["rw","simp","cases","ind"]} -{"prover":"Minlog","tactic_proof":"cases assume cases intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1923","id":90422,"instance":1923,"goal":"all p w, p + w = w + p","context":["cases","assume","cases","intro"]} -{"prover":"Minlog","tactic_proof":"simplify ind ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1924","id":90423,"instance":1924,"goal":"all f d u, (f + d) + u = f + (d + u)","context":["simplify","ind","ng","elim"]} -{"prover":"Minlog","tactic_proof":"assume rw use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1925","id":90424,"instance":1925,"goal":"all e, e * 0 = 0","context":["assume","rw","use","simp"]} -{"prover":"Minlog","tactic_proof":"ind cases rw use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1926","id":90425,"instance":1926,"goal":"all b, b * 1 = b","context":["ind","cases","rw","use"]} -{"prover":"Minlog","tactic_proof":"intro simp elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1927","id":90426,"instance":1927,"goal":"all a n c, (a * n) * c = a * (n * c)","context":["intro","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"ind elim ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1928","id":90427,"instance":1928,"goal":"all w f, w * f = f * w","context":["ind","elim","ng","elim"]} -{"prover":"Minlog","tactic_proof":"defn elim ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1929","id":90428,"instance":1929,"goal":"all q n z, q * (n + z) = q * n + q * z","context":["defn","elim","ng","cases"]} -{"prover":"Minlog","tactic_proof":"cases elim rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1930","id":90429,"instance":1930,"goal":"all v, v <= v","context":["cases","elim","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"assume assume simp simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1931","id":90430,"instance":1931,"goal":"all e p w, e <= p -> p <= w -> e <= w","context":["assume","assume","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"ng ng ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1932","id":90431,"instance":1932,"goal":"all w, not w < w","context":["ng","ng","ind","defn"]} -{"prover":"Minlog","tactic_proof":"elim defn ind elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1933","id":90432,"instance":1933,"goal":"all u q, Even u -> Even q -> Even (u + q)","context":["elim","defn","ind","elim"]} -{"prover":"Minlog","tactic_proof":"cases use ind cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1934","id":90433,"instance":1934,"goal":"all m, m | m","context":["cases","use","ind","cases"]} -{"prover":"Minlog","tactic_proof":"use simplify elim ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1935","id":90434,"instance":1935,"goal":"all c, Fact c > 0","context":["use","simplify","elim","ng"]} -{"prover":"Minlog","tactic_proof":"intro cases cases elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1936","id":90435,"instance":1936,"goal":"all v n = n","context":["intro","cases","cases","elim"]} -{"prover":"Minlog","tactic_proof":"use rw elim ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1937","id":90436,"instance":1937,"goal":"all d + 0 = d","context":["use","rw","elim","ng"]} -{"prover":"Minlog","tactic_proof":"use intro cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1938","id":90437,"instance":1938,"goal":"all b u, b + u = u + b","context":["use","intro","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"defn rw cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1939","id":90438,"instance":1939,"goal":"all e g p, (e + g) + p = e + (g + p)","context":["defn","rw","cases","elim"]} -{"prover":"Minlog","tactic_proof":"defn elim intro simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1940","id":90439,"instance":1940,"goal":"all q, q * 0 = 0","context":["defn","elim","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"use rw elim simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1941","id":90440,"instance":1941,"goal":"all m, m * 1 = m","context":["use","rw","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"intro ind ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1942","id":90441,"instance":1942,"goal":"all n q c, (n * q) * c = n * (q * c)","context":["intro","ind","ng","simp"]} -{"prover":"Minlog","tactic_proof":"intro assume defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1943","id":90442,"instance":1943,"goal":"all z w, z * w = w * z","context":["intro","assume","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"use ind assume simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1944","id":90443,"instance":1944,"goal":"all n m p, n * (m + p) = n * m + n * p","context":["use","ind","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ind elim cases simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1945","id":90444,"instance":1945,"goal":"all f, f <= f","context":["ind","elim","cases","simplify"]} -{"prover":"Minlog","tactic_proof":"ind defn elim elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1946","id":90445,"instance":1946,"goal":"all x p d, x <= p -> p <= d -> x <= d","context":["ind","defn","elim","elim"]} -{"prover":"Minlog","tactic_proof":"simp ind simplify use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1947","id":90446,"instance":1947,"goal":"all u, not u < u","context":["simp","ind","simplify","use"]} -{"prover":"Minlog","tactic_proof":"cases defn cases use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1948","id":90447,"instance":1948,"goal":"all g w, Even g -> Even w -> Even (g + w)","context":["cases","defn","cases","use"]} -{"prover":"Minlog","tactic_proof":"simp ind ind elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1949","id":90448,"instance":1949,"goal":"all h, h | h","context":["simp","ind","ind","elim"]} -{"prover":"Minlog","tactic_proof":"cases simplify cases defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1950","id":90449,"instance":1950,"goal":"all b, Fact b > 0","context":["cases","simplify","cases","defn"]} -{"prover":"Minlog","tactic_proof":"cases simp rw defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1951","id":90450,"instance":1951,"goal":"all m g = g","context":["cases","simp","rw","defn"]} -{"prover":"Minlog","tactic_proof":"ind simp cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1952","id":90451,"instance":1952,"goal":"all e + 0 = e","context":["ind","simp","cases","assume"]} -{"prover":"Minlog","tactic_proof":"use rw assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1953","id":90452,"instance":1953,"goal":"all z y, z + y = y + z","context":["use","rw","assume","use"]} -{"prover":"Minlog","tactic_proof":"assume assume use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1954","id":90453,"instance":1954,"goal":"all d f e, (d + f) + e = d + (f + e)","context":["assume","assume","use","use"]} -{"prover":"Minlog","tactic_proof":"assume assume elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1955","id":90454,"instance":1955,"goal":"all n, n * 0 = 0","context":["assume","assume","elim","ind"]} -{"prover":"Minlog","tactic_proof":"elim elim assume assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1956","id":90455,"instance":1956,"goal":"all h, h * 1 = h","context":["elim","elim","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ind rw ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1957","id":90456,"instance":1957,"goal":"all p z f, (p * z) * f = p * (z * f)","context":["ind","rw","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify simplify defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1958","id":90457,"instance":1958,"goal":"all q c, q * c = c * q","context":["simplify","simplify","defn","cases"]} -{"prover":"Minlog","tactic_proof":"simplify simp simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1959","id":90458,"instance":1959,"goal":"all b z w, b * (z + w) = b * z + b * w","context":["simplify","simp","simp","ng"]} -{"prover":"Minlog","tactic_proof":"simp simp assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1960","id":90459,"instance":1960,"goal":"all v, v <= v","context":["simp","simp","assume","assume"]} -{"prover":"Minlog","tactic_proof":"ind ind ng simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1961","id":90460,"instance":1961,"goal":"all e a p, e <= a -> a <= p -> e <= p","context":["ind","ind","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"rw assume simp ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1962","id":90461,"instance":1962,"goal":"all p, not p < p","context":["rw","assume","simp","ng"]} -{"prover":"Minlog","tactic_proof":"simp simplify use elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1963","id":90462,"instance":1963,"goal":"all d f, Even d -> Even f -> Even (d + f)","context":["simp","simplify","use","elim"]} -{"prover":"Minlog","tactic_proof":"use cases ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1964","id":90463,"instance":1964,"goal":"all a, a | a","context":["use","cases","ng","elim"]} -{"prover":"Minlog","tactic_proof":"ng intro ng simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1965","id":90464,"instance":1965,"goal":"all z, Fact z > 0","context":["ng","intro","ng","simp"]} -{"prover":"Minlog","tactic_proof":"cases intro cases elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1966","id":90465,"instance":1966,"goal":"all x c = c","context":["cases","intro","cases","elim"]} -{"prover":"Minlog","tactic_proof":"elim simplify simp assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1967","id":90466,"instance":1967,"goal":"all f + 0 = f","context":["elim","simplify","simp","assume"]} -{"prover":"Minlog","tactic_proof":"ind ind defn cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1968","id":90467,"instance":1968,"goal":"all h m, h + m = m + h","context":["ind","ind","defn","cases"]} -{"prover":"Minlog","tactic_proof":"rw defn intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1969","id":90468,"instance":1969,"goal":"all e x c, (e + x) + c = e + (x + c)","context":["rw","defn","intro","defn"]} -{"prover":"Minlog","tactic_proof":"assume cases ind assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1970","id":90469,"instance":1970,"goal":"all p, p * 0 = 0","context":["assume","cases","ind","assume"]} -{"prover":"Minlog","tactic_proof":"rw ind elim defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1971","id":90470,"instance":1971,"goal":"all y, y * 1 = y","context":["rw","ind","elim","defn"]} -{"prover":"Minlog","tactic_proof":"defn use simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1972","id":90471,"instance":1972,"goal":"all g z m, (g * z) * m = g * (z * m)","context":["defn","use","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"cases cases ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1973","id":90472,"instance":1973,"goal":"all x w, x * w = w * x","context":["cases","cases","ind","defn"]} -{"prover":"Minlog","tactic_proof":"assume rw assume intro","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1974","id":90473,"instance":1974,"goal":"all n v h, n * (v + h) = n * v + n * h","context":["assume","rw","assume","intro"]} -{"prover":"Minlog","tactic_proof":"defn rw rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1975","id":90474,"instance":1975,"goal":"all m, m <= m","context":["defn","rw","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases ng defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1976","id":90475,"instance":1976,"goal":"all m a e, m <= a -> a <= e -> m <= e","context":["cases","ng","defn","use"]} -{"prover":"Minlog","tactic_proof":"intro cases simp ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1977","id":90476,"instance":1977,"goal":"all w, not w < w","context":["intro","cases","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng assume simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1978","id":90477,"instance":1978,"goal":"all q g, Even q -> Even g -> Even (q + g)","context":["ng","assume","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"simplify simp ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1979","id":90478,"instance":1979,"goal":"all w, w | w","context":["simplify","simp","ng","ind"]} -{"prover":"Minlog","tactic_proof":"ind defn defn defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1980","id":90479,"instance":1980,"goal":"all x, Fact x > 0","context":["ind","defn","defn","defn"]} -{"prover":"Minlog","tactic_proof":"simplify rw cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1981","id":90480,"instance":1981,"goal":"all h w = w","context":["simplify","rw","cases","simp"]} -{"prover":"Minlog","tactic_proof":"rw use rw elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1982","id":90481,"instance":1982,"goal":"all n + 0 = n","context":["rw","use","rw","elim"]} -{"prover":"Minlog","tactic_proof":"rw ng ind simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1983","id":90482,"instance":1983,"goal":"all z f, z + f = f + z","context":["rw","ng","ind","simp"]} -{"prover":"Minlog","tactic_proof":"assume ind assume ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1984","id":90483,"instance":1984,"goal":"all q y e, (q + y) + e = q + (y + e)","context":["assume","ind","assume","ind"]} -{"prover":"Minlog","tactic_proof":"cases ng intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_1985","id":90484,"instance":1985,"goal":"all w, w * 0 = 0","context":["cases","ng","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify simp ind rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1986","id":90485,"instance":1986,"goal":"all c, c * 1 = c","context":["simplify","simp","ind","rw"]} -{"prover":"Minlog","tactic_proof":"ind use rw ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_1987","id":90486,"instance":1987,"goal":"all d z w, (d * z) * w = d * (z * w)","context":["ind","use","rw","ng"]} -{"prover":"Minlog","tactic_proof":"use ng elim simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1988","id":90487,"instance":1988,"goal":"all n z, n * z = z * n","context":["use","ng","elim","simp"]} -{"prover":"Minlog","tactic_proof":"ind assume cases intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_1989","id":90488,"instance":1989,"goal":"all p c m, p * (c + m) = p * c + p * m","context":["ind","assume","cases","intro"]} -{"prover":"Minlog","tactic_proof":"simp simp elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1990","id":90489,"instance":1990,"goal":"all c, c <= c","context":["simp","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"simplify rw use simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1991","id":90490,"instance":1991,"goal":"all p q d, p <= q -> q <= d -> p <= d","context":["simplify","rw","use","simplify"]} -{"prover":"Minlog","tactic_proof":"cases ng simp use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_1992","id":90491,"instance":1992,"goal":"all v, not v < v","context":["cases","ng","simp","use"]} -{"prover":"Minlog","tactic_proof":"simp simplify ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_1993","id":90492,"instance":1993,"goal":"all h g, Even h -> Even g -> Even (h + g)","context":["simp","simplify","ind","rw"]} -{"prover":"Minlog","tactic_proof":"defn rw cases simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_1994","id":90493,"instance":1994,"goal":"all w, w | w","context":["defn","rw","cases","simp"]} -{"prover":"Minlog","tactic_proof":"rw use defn rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_1995","id":90494,"instance":1995,"goal":"all a, Fact a > 0","context":["rw","use","defn","rw"]} -{"prover":"Minlog","tactic_proof":"assume rw ng intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_1996","id":90495,"instance":1996,"goal":"all x a = a","context":["assume","rw","ng","intro"]} -{"prover":"Minlog","tactic_proof":"ind use ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_1997","id":90496,"instance":1997,"goal":"all c + 0 = c","context":["ind","use","ind","intro"]} -{"prover":"Minlog","tactic_proof":"simp assume ind rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1998","id":90497,"instance":1998,"goal":"all n d, n + d = d + n","context":["simp","assume","ind","rw"]} -{"prover":"Minlog","tactic_proof":"simp use ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1999","id":90498,"instance":1999,"goal":"all w e u, (w + e) + u = w + (e + u)","context":["simp","use","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"intro simplify intro ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2000","id":90499,"instance":2000,"goal":"all d, d * 0 = 0","context":["intro","simplify","intro","ind"]} -{"prover":"Minlog","tactic_proof":"cases ind assume cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2001","id":90500,"instance":2001,"goal":"all m, m * 1 = m","context":["cases","ind","assume","cases"]} -{"prover":"Minlog","tactic_proof":"use rw assume use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2002","id":90501,"instance":2002,"goal":"all x g m, (x * g) * m = x * (g * m)","context":["use","rw","assume","use"]} -{"prover":"Minlog","tactic_proof":"elim cases simplify defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2003","id":90502,"instance":2003,"goal":"all x u, x * u = u * x","context":["elim","cases","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"ng elim elim defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2004","id":90503,"instance":2004,"goal":"all n p d, n * (p + d) = n * p + n * d","context":["ng","elim","elim","defn"]} -{"prover":"Minlog","tactic_proof":"defn assume rw ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2005","id":90504,"instance":2005,"goal":"all v, v <= v","context":["defn","assume","rw","ng"]} -{"prover":"Minlog","tactic_proof":"cases simp rw simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2006","id":90505,"instance":2006,"goal":"all p b c, p <= b -> b <= c -> p <= c","context":["cases","simp","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"defn elim rw elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2007","id":90506,"instance":2007,"goal":"all a, not a < a","context":["defn","elim","rw","elim"]} -{"prover":"Minlog","tactic_proof":"simplify rw use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2008","id":90507,"instance":2008,"goal":"all w h, Even w -> Even h -> Even (w + h)","context":["simplify","rw","use","rw"]} -{"prover":"Minlog","tactic_proof":"intro ind assume rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2009","id":90508,"instance":2009,"goal":"all b, b | b","context":["intro","ind","assume","rw"]} -{"prover":"Minlog","tactic_proof":"ng intro ng cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2010","id":90509,"instance":2010,"goal":"all g, Fact g > 0","context":["ng","intro","ng","cases"]} -{"prover":"Minlog","tactic_proof":"intro simp simplify defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2011","id":90510,"instance":2011,"goal":"all f m = m","context":["intro","simp","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"ind defn assume ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2012","id":90511,"instance":2012,"goal":"all u + 0 = u","context":["ind","defn","assume","ind"]} -{"prover":"Minlog","tactic_proof":"use assume ng ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2013","id":90512,"instance":2013,"goal":"all b z, b + z = z + b","context":["use","assume","ng","ng"]} -{"prover":"Minlog","tactic_proof":"simplify ind simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2014","id":90513,"instance":2014,"goal":"all b c g, (b + c) + g = b + (c + g)","context":["simplify","ind","simp","ng"]} -{"prover":"Minlog","tactic_proof":"cases intro intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2015","id":90514,"instance":2015,"goal":"all x, x * 0 = 0","context":["cases","intro","intro","simp"]} -{"prover":"Minlog","tactic_proof":"ng intro use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2016","id":90515,"instance":2016,"goal":"all b, b * 1 = b","context":["ng","intro","use","cases"]} -{"prover":"Minlog","tactic_proof":"intro use defn assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2017","id":90516,"instance":2017,"goal":"all z v q, (z * v) * q = z * (v * q)","context":["intro","use","defn","assume"]} -{"prover":"Minlog","tactic_proof":"cases ng assume ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2018","id":90517,"instance":2018,"goal":"all y g, y * g = g * y","context":["cases","ng","assume","ng"]} -{"prover":"Minlog","tactic_proof":"use cases simp ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2019","id":90518,"instance":2019,"goal":"all c z n, c * (z + n) = c * z + c * n","context":["use","cases","simp","ng"]} -{"prover":"Minlog","tactic_proof":"ng simplify ind ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2020","id":90519,"instance":2020,"goal":"all p, p <= p","context":["ng","simplify","ind","ind"]} -{"prover":"Minlog","tactic_proof":"elim simp use elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2021","id":90520,"instance":2021,"goal":"all d e p, d <= e -> e <= p -> d <= p","context":["elim","simp","use","elim"]} -{"prover":"Minlog","tactic_proof":"use defn simplify assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2022","id":90521,"instance":2022,"goal":"all u, not u < u","context":["use","defn","simplify","assume"]} -{"prover":"Minlog","tactic_proof":"simplify intro rw ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2023","id":90522,"instance":2023,"goal":"all n f, Even n -> Even f -> Even (n + f)","context":["simplify","intro","rw","ng"]} -{"prover":"Minlog","tactic_proof":"defn intro simp defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2024","id":90523,"instance":2024,"goal":"all h, h | h","context":["defn","intro","simp","defn"]} -{"prover":"Minlog","tactic_proof":"ind ind defn elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2025","id":90524,"instance":2025,"goal":"all n, Fact n > 0","context":["ind","ind","defn","elim"]} -{"prover":"Minlog","tactic_proof":"defn ind defn ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2026","id":90525,"instance":2026,"goal":"all c q = q","context":["defn","ind","defn","ng"]} -{"prover":"Minlog","tactic_proof":"rw assume cases simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2027","id":90526,"instance":2027,"goal":"all y + 0 = y","context":["rw","assume","cases","simp"]} -{"prover":"Minlog","tactic_proof":"defn use defn ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2028","id":90527,"instance":2028,"goal":"all a m, a + m = m + a","context":["defn","use","defn","ng"]} -{"prover":"Minlog","tactic_proof":"rw elim assume ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2029","id":90528,"instance":2029,"goal":"all c a b, (c + a) + b = c + (a + b)","context":["rw","elim","assume","ind"]} -{"prover":"Minlog","tactic_proof":"simp ng use simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2030","id":90529,"instance":2030,"goal":"all y, y * 0 = 0","context":["simp","ng","use","simp"]} -{"prover":"Minlog","tactic_proof":"simplify ind simplify simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2031","id":90530,"instance":2031,"goal":"all a, a * 1 = a","context":["simplify","ind","simplify","simplify"]} -{"prover":"Minlog","tactic_proof":"use intro simplify ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2032","id":90531,"instance":2032,"goal":"all h f e, (h * f) * e = h * (f * e)","context":["use","intro","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"assume ind intro assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2033","id":90532,"instance":2033,"goal":"all u q, u * q = q * u","context":["assume","ind","intro","assume"]} -{"prover":"Minlog","tactic_proof":"use cases elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2034","id":90533,"instance":2034,"goal":"all v n w, v * (n + w) = v * n + v * w","context":["use","cases","elim","simp"]} -{"prover":"Minlog","tactic_proof":"rw intro simplify ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2035","id":90534,"instance":2035,"goal":"all p, p <= p","context":["rw","intro","simplify","ng"]} -{"prover":"Minlog","tactic_proof":"ind intro ng ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2036","id":90535,"instance":2036,"goal":"all b h d, b <= h -> h <= d -> b <= d","context":["ind","intro","ng","ng"]} -{"prover":"Minlog","tactic_proof":"ng cases assume rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2037","id":90536,"instance":2037,"goal":"all g, not g < g","context":["ng","cases","assume","rw"]} -{"prover":"Minlog","tactic_proof":"cases simp ng elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2038","id":90537,"instance":2038,"goal":"all c y, Even c -> Even y -> Even (c + y)","context":["cases","simp","ng","elim"]} -{"prover":"Minlog","tactic_proof":"intro elim rw assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2039","id":90538,"instance":2039,"goal":"all f, f | f","context":["intro","elim","rw","assume"]} -{"prover":"Minlog","tactic_proof":"defn rw simplify intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2040","id":90539,"instance":2040,"goal":"all u, Fact u > 0","context":["defn","rw","simplify","intro"]} -{"prover":"Minlog","tactic_proof":"use rw defn defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2041","id":90540,"instance":2041,"goal":"all u q = q","context":["use","rw","defn","defn"]} -{"prover":"Minlog","tactic_proof":"ng cases ind assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2042","id":90541,"instance":2042,"goal":"all x + 0 = x","context":["ng","cases","ind","assume"]} -{"prover":"Minlog","tactic_proof":"simplify ng simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2043","id":90542,"instance":2043,"goal":"all e d, e + d = d + e","context":["simplify","ng","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"intro intro simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2044","id":90543,"instance":2044,"goal":"all g h e, (g + h) + e = g + (h + e)","context":["intro","intro","simp","rw"]} -{"prover":"Minlog","tactic_proof":"assume intro elim elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2045","id":90544,"instance":2045,"goal":"all c, c * 0 = 0","context":["assume","intro","elim","elim"]} -{"prover":"Minlog","tactic_proof":"simp rw elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2046","id":90545,"instance":2046,"goal":"all m, m * 1 = m","context":["simp","rw","elim","simp"]} -{"prover":"Minlog","tactic_proof":"rw elim intro ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2047","id":90546,"instance":2047,"goal":"all w h p, (w * h) * p = w * (h * p)","context":["rw","elim","intro","ind"]} -{"prover":"Minlog","tactic_proof":"rw ng ind cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2048","id":90547,"instance":2048,"goal":"all q p, q * p = p * q","context":["rw","ng","ind","cases"]} -{"prover":"Minlog","tactic_proof":"simplify defn cases ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2049","id":90548,"instance":2049,"goal":"all x z v, x * (z + v) = x * z + x * v","context":["simplify","defn","cases","ind"]} -{"prover":"Minlog","tactic_proof":"intro cases defn cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2050","id":90549,"instance":2050,"goal":"all f, f <= f","context":["intro","cases","defn","cases"]} -{"prover":"Minlog","tactic_proof":"rw cases rw use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2051","id":90550,"instance":2051,"goal":"all f c e, f <= c -> c <= e -> f <= e","context":["rw","cases","rw","use"]} -{"prover":"Minlog","tactic_proof":"use defn use ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2052","id":90551,"instance":2052,"goal":"all d, not d < d","context":["use","defn","use","ng"]} -{"prover":"Minlog","tactic_proof":"use cases cases elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2053","id":90552,"instance":2053,"goal":"all d m, Even d -> Even m -> Even (d + m)","context":["use","cases","cases","elim"]} -{"prover":"Minlog","tactic_proof":"ind intro use assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2054","id":90553,"instance":2054,"goal":"all q, q | q","context":["ind","intro","use","assume"]} -{"prover":"Minlog","tactic_proof":"simplify ng simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2055","id":90554,"instance":2055,"goal":"all x, Fact x > 0","context":["simplify","ng","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng rw cases rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2056","id":90555,"instance":2056,"goal":"all d x = x","context":["ng","rw","cases","rw"]} -{"prover":"Minlog","tactic_proof":"simp ind ind intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2057","id":90556,"instance":2057,"goal":"all p + 0 = p","context":["simp","ind","ind","intro"]} -{"prover":"Minlog","tactic_proof":"assume use intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2058","id":90557,"instance":2058,"goal":"all d z, d + z = z + d","context":["assume","use","intro","use"]} -{"prover":"Minlog","tactic_proof":"ind assume defn intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2059","id":90558,"instance":2059,"goal":"all w m g, (w + m) + g = w + (m + g)","context":["ind","assume","defn","intro"]} -{"prover":"Minlog","tactic_proof":"use simplify defn simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2060","id":90559,"instance":2060,"goal":"all b, b * 0 = 0","context":["use","simplify","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify assume elim intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2061","id":90560,"instance":2061,"goal":"all f, f * 1 = f","context":["simplify","assume","elim","intro"]} -{"prover":"Minlog","tactic_proof":"cases rw rw elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2062","id":90561,"instance":2062,"goal":"all f n w, (f * n) * w = f * (n * w)","context":["cases","rw","rw","elim"]} -{"prover":"Minlog","tactic_proof":"defn intro ind use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2063","id":90562,"instance":2063,"goal":"all y d, y * d = d * y","context":["defn","intro","ind","use"]} -{"prover":"Minlog","tactic_proof":"intro assume simp cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2064","id":90563,"instance":2064,"goal":"all m c p, m * (c + p) = m * c + m * p","context":["intro","assume","simp","cases"]} -{"prover":"Minlog","tactic_proof":"intro defn simp ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2065","id":90564,"instance":2065,"goal":"all y, y <= y","context":["intro","defn","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng use defn use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2066","id":90565,"instance":2066,"goal":"all p e g, p <= e -> e <= g -> p <= g","context":["ng","use","defn","use"]} -{"prover":"Minlog","tactic_proof":"ng rw simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2067","id":90566,"instance":2067,"goal":"all y, not y < y","context":["ng","rw","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"cases intro use simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2068","id":90567,"instance":2068,"goal":"all u x, Even u -> Even x -> Even (u + x)","context":["cases","intro","use","simp"]} -{"prover":"Minlog","tactic_proof":"simplify intro cases assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2069","id":90568,"instance":2069,"goal":"all b, b | b","context":["simplify","intro","cases","assume"]} -{"prover":"Minlog","tactic_proof":"elim defn cases ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2070","id":90569,"instance":2070,"goal":"all x, Fact x > 0","context":["elim","defn","cases","ng"]} -{"prover":"Minlog","tactic_proof":"simp cases use elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2071","id":90570,"instance":2071,"goal":"all g h = h","context":["simp","cases","use","elim"]} -{"prover":"Minlog","tactic_proof":"ind simplify intro simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2072","id":90571,"instance":2072,"goal":"all m + 0 = m","context":["ind","simplify","intro","simp"]} -{"prover":"Minlog","tactic_proof":"simp simp elim cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2073","id":90572,"instance":2073,"goal":"all b d, b + d = d + b","context":["simp","simp","elim","cases"]} -{"prover":"Minlog","tactic_proof":"ng elim intro cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2074","id":90573,"instance":2074,"goal":"all m b v, (m + b) + v = m + (b + v)","context":["ng","elim","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ind simp elim elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2075","id":90574,"instance":2075,"goal":"all c, c * 0 = 0","context":["ind","simp","elim","elim"]} -{"prover":"Minlog","tactic_proof":"simplify intro elim simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2076","id":90575,"instance":2076,"goal":"all d, d * 1 = d","context":["simplify","intro","elim","simp"]} -{"prover":"Minlog","tactic_proof":"rw ind simplify cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2077","id":90576,"instance":2077,"goal":"all q f h, (q * f) * h = q * (f * h)","context":["rw","ind","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"intro rw ind simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2078","id":90577,"instance":2078,"goal":"all z b, z * b = b * z","context":["intro","rw","ind","simp"]} -{"prover":"Minlog","tactic_proof":"rw simp defn use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2079","id":90578,"instance":2079,"goal":"all h e a, h * (e + a) = h * e + h * a","context":["rw","simp","defn","use"]} -{"prover":"Minlog","tactic_proof":"elim intro ind intro","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2080","id":90579,"instance":2080,"goal":"all g, g <= g","context":["elim","intro","ind","intro"]} -{"prover":"Minlog","tactic_proof":"cases ind cases assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2081","id":90580,"instance":2081,"goal":"all n d f, n <= d -> d <= f -> n <= f","context":["cases","ind","cases","assume"]} -{"prover":"Minlog","tactic_proof":"ng elim defn simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2082","id":90581,"instance":2082,"goal":"all g, not g < g","context":["ng","elim","defn","simplify"]} -{"prover":"Minlog","tactic_proof":"intro ng ind use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2083","id":90582,"instance":2083,"goal":"all e m, Even e -> Even m -> Even (e + m)","context":["intro","ng","ind","use"]} -{"prover":"Minlog","tactic_proof":"ng simp ng simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2084","id":90583,"instance":2084,"goal":"all g, g | g","context":["ng","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume simp simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2085","id":90584,"instance":2085,"goal":"all v, Fact v > 0","context":["assume","simp","simplify","use"]} -{"prover":"Minlog","tactic_proof":"ind use use intro","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2086","id":90585,"instance":2086,"goal":"all x p = p","context":["ind","use","use","intro"]} -{"prover":"Minlog","tactic_proof":"cases use assume use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2087","id":90586,"instance":2087,"goal":"all g + 0 = g","context":["cases","use","assume","use"]} -{"prover":"Minlog","tactic_proof":"simplify elim rw use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2088","id":90587,"instance":2088,"goal":"all h d, h + d = d + h","context":["simplify","elim","rw","use"]} -{"prover":"Minlog","tactic_proof":"use intro intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2089","id":90588,"instance":2089,"goal":"all a d h, (a + d) + h = a + (d + h)","context":["use","intro","intro","use"]} -{"prover":"Minlog","tactic_proof":"intro simp use use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2090","id":90589,"instance":2090,"goal":"all p, p * 0 = 0","context":["intro","simp","use","use"]} -{"prover":"Minlog","tactic_proof":"use elim ind rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2091","id":90590,"instance":2091,"goal":"all g, g * 1 = g","context":["use","elim","ind","rw"]} -{"prover":"Minlog","tactic_proof":"defn ind simplify defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2092","id":90591,"instance":2092,"goal":"all e a g, (e * a) * g = e * (a * g)","context":["defn","ind","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"cases intro ng simp","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2093","id":90592,"instance":2093,"goal":"all g e, g * e = e * g","context":["cases","intro","ng","simp"]} -{"prover":"Minlog","tactic_proof":"assume simplify simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2094","id":90593,"instance":2094,"goal":"all h p m, h * (p + m) = h * p + h * m","context":["assume","simplify","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"cases assume simplify cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2095","id":90594,"instance":2095,"goal":"all a, a <= a","context":["cases","assume","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"rw defn cases defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2096","id":90595,"instance":2096,"goal":"all v e h, v <= e -> e <= h -> v <= h","context":["rw","defn","cases","defn"]} -{"prover":"Minlog","tactic_proof":"cases defn simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2097","id":90596,"instance":2097,"goal":"all v, not v < v","context":["cases","defn","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"defn elim simp use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2098","id":90597,"instance":2098,"goal":"all h d, Even h -> Even d -> Even (h + d)","context":["defn","elim","simp","use"]} -{"prover":"Minlog","tactic_proof":"ind simplify intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2099","id":90598,"instance":2099,"goal":"all b, b | b","context":["ind","simplify","intro","ng"]} -{"prover":"Minlog","tactic_proof":"simplify simp intro use","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2100","id":90599,"instance":2100,"goal":"all p, Fact p > 0","context":["simplify","simp","intro","use"]} -{"prover":"Minlog","tactic_proof":"ind simp ng simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2101","id":90600,"instance":2101,"goal":"all h x = x","context":["ind","simp","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"cases elim assume rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2102","id":90601,"instance":2102,"goal":"all m + 0 = m","context":["cases","elim","assume","rw"]} -{"prover":"Minlog","tactic_proof":"simp elim elim cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2103","id":90602,"instance":2103,"goal":"all q z, q + z = z + q","context":["simp","elim","elim","cases"]} -{"prover":"Minlog","tactic_proof":"intro elim cases rw","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2104","id":90603,"instance":2104,"goal":"all g q x, (g + q) + x = g + (q + x)","context":["intro","elim","cases","rw"]} -{"prover":"Minlog","tactic_proof":"ng defn elim ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2105","id":90604,"instance":2105,"goal":"all x, x * 0 = 0","context":["ng","defn","elim","ng"]} -{"prover":"Minlog","tactic_proof":"cases use elim rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2106","id":90605,"instance":2106,"goal":"all q, q * 1 = q","context":["cases","use","elim","rw"]} -{"prover":"Minlog","tactic_proof":"elim simplify assume simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2107","id":90606,"instance":2107,"goal":"all m y n, (m * y) * n = m * (y * n)","context":["elim","simplify","assume","simp"]} -{"prover":"Minlog","tactic_proof":"ng simplify simp simplify","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2108","id":90607,"instance":2108,"goal":"all z n, z * n = n * z","context":["ng","simplify","simp","simplify"]} -{"prover":"Minlog","tactic_proof":"defn intro simplify simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2109","id":90608,"instance":2109,"goal":"all v a y, v * (a + y) = v * a + v * y","context":["defn","intro","simplify","simp"]} -{"prover":"Minlog","tactic_proof":"use defn elim assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2110","id":90609,"instance":2110,"goal":"all u, u <= u","context":["use","defn","elim","assume"]} -{"prover":"Minlog","tactic_proof":"cases assume ind defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2111","id":90610,"instance":2111,"goal":"all n e f, n <= e -> e <= f -> n <= f","context":["cases","assume","ind","defn"]} -{"prover":"Minlog","tactic_proof":"simplify cases rw assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2112","id":90611,"instance":2112,"goal":"all p, not p < p","context":["simplify","cases","rw","assume"]} -{"prover":"Minlog","tactic_proof":"cases cases defn ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2113","id":90612,"instance":2113,"goal":"all u e, Even u -> Even e -> Even (u + e)","context":["cases","cases","defn","ng"]} -{"prover":"Minlog","tactic_proof":"intro defn defn ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2114","id":90613,"instance":2114,"goal":"all h, h | h","context":["intro","defn","defn","ng"]} -{"prover":"Minlog","tactic_proof":"assume simp intro defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2115","id":90614,"instance":2115,"goal":"all u, Fact u > 0","context":["assume","simp","intro","defn"]} -{"prover":"Minlog","tactic_proof":"rw use cases ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2116","id":90615,"instance":2116,"goal":"all x a = a","context":["rw","use","cases","ng"]} -{"prover":"Minlog","tactic_proof":"simplify elim rw intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2117","id":90616,"instance":2117,"goal":"all z + 0 = z","context":["simplify","elim","rw","intro"]} -{"prover":"Minlog","tactic_proof":"defn defn rw ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2118","id":90617,"instance":2118,"goal":"all c x, c + x = x + c","context":["defn","defn","rw","ind"]} -{"prover":"Minlog","tactic_proof":"simplify ng elim elim","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2119","id":90618,"instance":2119,"goal":"all z b y, (z + b) + y = z + (b + y)","context":["simplify","ng","elim","elim"]} -{"prover":"Minlog","tactic_proof":"defn rw intro assume","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2120","id":90619,"instance":2120,"goal":"all w, w * 0 = 0","context":["defn","rw","intro","assume"]} -{"prover":"Minlog","tactic_proof":"ind elim intro simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2121","id":90620,"instance":2121,"goal":"all a, a * 1 = a","context":["ind","elim","intro","simp"]} -{"prover":"Minlog","tactic_proof":"defn elim cases cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2122","id":90621,"instance":2122,"goal":"all q c p, (q * c) * p = q * (c * p)","context":["defn","elim","cases","cases"]} -{"prover":"Minlog","tactic_proof":"ind assume cases ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2123","id":90622,"instance":2123,"goal":"all x f, x * f = f * x","context":["ind","assume","cases","ng"]} -{"prover":"Minlog","tactic_proof":"simplify cases cases ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2124","id":90623,"instance":2124,"goal":"all w z d, w * (z + d) = w * z + w * d","context":["simplify","cases","cases","ng"]} -{"prover":"Minlog","tactic_proof":"rw cases ng simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2125","id":90624,"instance":2125,"goal":"all e, e <= e","context":["rw","cases","ng","simplify"]} -{"prover":"Minlog","tactic_proof":"assume assume ind simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2126","id":90625,"instance":2126,"goal":"all d f g, d <= f -> f <= g -> d <= g","context":["assume","assume","ind","simplify"]} -{"prover":"Minlog","tactic_proof":"intro defn ind simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2127","id":90626,"instance":2127,"goal":"all q, not q < q","context":["intro","defn","ind","simp"]} -{"prover":"Minlog","tactic_proof":"ind elim use rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2128","id":90627,"instance":2128,"goal":"all a w, Even a -> Even w -> Even (a + w)","context":["ind","elim","use","rw"]} -{"prover":"Minlog","tactic_proof":"intro simp ind ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2129","id":90628,"instance":2129,"goal":"all b, b | b","context":["intro","simp","ind","ng"]} -{"prover":"Minlog","tactic_proof":"use assume rw simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2130","id":90629,"instance":2130,"goal":"all v, Fact v > 0","context":["use","assume","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"use defn intro elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2131","id":90630,"instance":2131,"goal":"all h g = g","context":["use","defn","intro","elim"]} -{"prover":"Minlog","tactic_proof":"assume defn elim simplify","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2132","id":90631,"instance":2132,"goal":"all u + 0 = u","context":["assume","defn","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"intro rw ng ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2133","id":90632,"instance":2133,"goal":"all p x, p + x = x + p","context":["intro","rw","ng","ind"]} -{"prover":"Minlog","tactic_proof":"elim rw rw simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2134","id":90633,"instance":2134,"goal":"all m c g, (m + c) + g = m + (c + g)","context":["elim","rw","rw","simplify"]} -{"prover":"Minlog","tactic_proof":"simp ind ng defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2135","id":90634,"instance":2135,"goal":"all c, c * 0 = 0","context":["simp","ind","ng","defn"]} -{"prover":"Minlog","tactic_proof":"defn simp simplify defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2136","id":90635,"instance":2136,"goal":"all b, b * 1 = b","context":["defn","simp","simplify","defn"]} -{"prover":"Minlog","tactic_proof":"intro rw defn simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2137","id":90636,"instance":2137,"goal":"all c m q, (c * m) * q = c * (m * q)","context":["intro","rw","defn","simp"]} -{"prover":"Minlog","tactic_proof":"rw intro ind use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2138","id":90637,"instance":2138,"goal":"all y m, y * m = m * y","context":["rw","intro","ind","use"]} -{"prover":"Minlog","tactic_proof":"simp simp use intro","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2139","id":90638,"instance":2139,"goal":"all b q a, b * (q + a) = b * q + b * a","context":["simp","simp","use","intro"]} -{"prover":"Minlog","tactic_proof":"assume cases assume rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2140","id":90639,"instance":2140,"goal":"all h, h <= h","context":["assume","cases","assume","rw"]} -{"prover":"Minlog","tactic_proof":"elim rw simp use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2141","id":90640,"instance":2141,"goal":"all p w h, p <= w -> w <= h -> p <= h","context":["elim","rw","simp","use"]} -{"prover":"Minlog","tactic_proof":"simplify use elim ind","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2142","id":90641,"instance":2142,"goal":"all q, not q < q","context":["simplify","use","elim","ind"]} -{"prover":"Minlog","tactic_proof":"cases ng defn defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2143","id":90642,"instance":2143,"goal":"all m a, Even m -> Even a -> Even (m + a)","context":["cases","ng","defn","defn"]} -{"prover":"Minlog","tactic_proof":"assume assume defn elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2144","id":90643,"instance":2144,"goal":"all n, n | n","context":["assume","assume","defn","elim"]} -{"prover":"Minlog","tactic_proof":"elim cases use ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2145","id":90644,"instance":2145,"goal":"all z, Fact z > 0","context":["elim","cases","use","ind"]} -{"prover":"Minlog","tactic_proof":"defn assume ng defn","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2146","id":90645,"instance":2146,"goal":"all f p = p","context":["defn","assume","ng","defn"]} -{"prover":"Minlog","tactic_proof":"intro simp assume ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2147","id":90646,"instance":2147,"goal":"all q + 0 = q","context":["intro","simp","assume","ind"]} -{"prover":"Minlog","tactic_proof":"ind elim simplify rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2148","id":90647,"instance":2148,"goal":"all p e, p + e = e + p","context":["ind","elim","simplify","rw"]} -{"prover":"Minlog","tactic_proof":"simplify ng intro rw","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2149","id":90648,"instance":2149,"goal":"all f z w, (f + z) + w = f + (z + w)","context":["simplify","ng","intro","rw"]} -{"prover":"Minlog","tactic_proof":"cases cases ng elim","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2150","id":90649,"instance":2150,"goal":"all g, g * 0 = 0","context":["cases","cases","ng","elim"]} -{"prover":"Minlog","tactic_proof":"ng use cases elim","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2151","id":90650,"instance":2151,"goal":"all a, a * 1 = a","context":["ng","use","cases","elim"]} -{"prover":"Minlog","tactic_proof":"use assume simp use","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2152","id":90651,"instance":2152,"goal":"all n h m, (n * h) * m = n * (h * m)","context":["use","assume","simp","use"]} -{"prover":"Minlog","tactic_proof":"assume rw assume assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2153","id":90652,"instance":2153,"goal":"all w a, w * a = a * w","context":["assume","rw","assume","assume"]} -{"prover":"Minlog","tactic_proof":"simplify use ind rw","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2154","id":90653,"instance":2154,"goal":"all z e m, z * (e + m) = z * e + z * m","context":["simplify","use","ind","rw"]} -{"prover":"Minlog","tactic_proof":"use ind cases cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2155","id":90654,"instance":2155,"goal":"all a, a <= a","context":["use","ind","cases","cases"]} -{"prover":"Minlog","tactic_proof":"cases assume use ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2156","id":90655,"instance":2156,"goal":"all v n c, v <= n -> n <= c -> v <= c","context":["cases","assume","use","ng"]} -{"prover":"Minlog","tactic_proof":"defn simplify intro assume","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2157","id":90656,"instance":2157,"goal":"all c, not c < c","context":["defn","simplify","intro","assume"]} -{"prover":"Minlog","tactic_proof":"intro cases elim simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2158","id":90657,"instance":2158,"goal":"all e a, Even e -> Even a -> Even (e + a)","context":["intro","cases","elim","simp"]} -{"prover":"Minlog","tactic_proof":"simplify ng intro cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2159","id":90658,"instance":2159,"goal":"all b, b | b","context":["simplify","ng","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ng ind cases use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2160","id":90659,"instance":2160,"goal":"all z, Fact z > 0","context":["ng","ind","cases","use"]} -{"prover":"Minlog","tactic_proof":"rw assume ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2161","id":90660,"instance":2161,"goal":"all u n = n","context":["rw","assume","ng","defn"]} -{"prover":"Minlog","tactic_proof":"rw rw cases assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2162","id":90661,"instance":2162,"goal":"all n + 0 = n","context":["rw","rw","cases","assume"]} -{"prover":"Minlog","tactic_proof":"elim simplify ind defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2163","id":90662,"instance":2163,"goal":"all p u, p + u = u + p","context":["elim","simplify","ind","defn"]} -{"prover":"Minlog","tactic_proof":"assume defn ng ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2164","id":90663,"instance":2164,"goal":"all w e b, (w + e) + b = w + (e + b)","context":["assume","defn","ng","ind"]} -{"prover":"Minlog","tactic_proof":"ind defn rw ng","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2165","id":90664,"instance":2165,"goal":"all n, n * 0 = 0","context":["ind","defn","rw","ng"]} -{"prover":"Minlog","tactic_proof":"defn simp simplify use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2166","id":90665,"instance":2166,"goal":"all e, e * 1 = e","context":["defn","simp","simplify","use"]} -{"prover":"Minlog","tactic_proof":"use defn ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2167","id":90666,"instance":2167,"goal":"all z q v, (z * q) * v = z * (q * v)","context":["use","defn","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simplify intro elim ind","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2168","id":90667,"instance":2168,"goal":"all m x, m * x = x * m","context":["simplify","intro","elim","ind"]} -{"prover":"Minlog","tactic_proof":"simplify defn rw simp","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2169","id":90668,"instance":2169,"goal":"all f z b, f * (z + b) = f * z + f * b","context":["simplify","defn","rw","simp"]} -{"prover":"Minlog","tactic_proof":"ind elim rw use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2170","id":90669,"instance":2170,"goal":"all m, m <= m","context":["ind","elim","rw","use"]} -{"prover":"Minlog","tactic_proof":"simp simplify use use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2171","id":90670,"instance":2171,"goal":"all y g q, y <= g -> g <= q -> y <= q","context":["simp","simplify","use","use"]} -{"prover":"Minlog","tactic_proof":"ind ng simp cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2172","id":90671,"instance":2172,"goal":"all x, not x < x","context":["ind","ng","simp","cases"]} -{"prover":"Minlog","tactic_proof":"simplify elim elim use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2173","id":90672,"instance":2173,"goal":"all d f, Even d -> Even f -> Even (d + f)","context":["simplify","elim","elim","use"]} -{"prover":"Minlog","tactic_proof":"cases cases intro defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2174","id":90673,"instance":2174,"goal":"all e, e | e","context":["cases","cases","intro","defn"]} -{"prover":"Minlog","tactic_proof":"ng cases defn simp","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2175","id":90674,"instance":2175,"goal":"all f, Fact f > 0","context":["ng","cases","defn","simp"]} -{"prover":"Minlog","tactic_proof":"simplify intro assume ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2176","id":90675,"instance":2176,"goal":"all f q = q","context":["simplify","intro","assume","ng"]} -{"prover":"Minlog","tactic_proof":"ind simp intro ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2177","id":90676,"instance":2177,"goal":"all y + 0 = y","context":["ind","simp","intro","ng"]} -{"prover":"Minlog","tactic_proof":"ind elim rw cases","sort_family":"int","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2178","id":90677,"instance":2178,"goal":"all c e, c + e = e + c","context":["ind","elim","rw","cases"]} -{"prover":"Minlog","tactic_proof":"cases rw use cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2179","id":90678,"instance":2179,"goal":"all x f g, (x + f) + g = x + (f + g)","context":["cases","rw","use","cases"]} -{"prover":"Minlog","tactic_proof":"assume simp ind use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2180","id":90679,"instance":2180,"goal":"all h, h * 0 = 0","context":["assume","simp","ind","use"]} -{"prover":"Minlog","tactic_proof":"elim simplify ng ng","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2181","id":90680,"instance":2181,"goal":"all z, z * 1 = z","context":["elim","simplify","ng","ng"]} -{"prover":"Minlog","tactic_proof":"rw cases simp cases","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2182","id":90681,"instance":2182,"goal":"all d n x, (d * n) * x = d * (n * x)","context":["rw","cases","simp","cases"]} -{"prover":"Minlog","tactic_proof":"intro intro simp ng","sort_family":"int","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2183","id":90682,"instance":2183,"goal":"all d u, d * u = u * d","context":["intro","intro","simp","ng"]} -{"prover":"Minlog","tactic_proof":"simp simplify intro defn","sort_family":"int","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2184","id":90683,"instance":2184,"goal":"all m p z, m * (p + z) = m * p + m * z","context":["simp","simplify","intro","defn"]} -{"prover":"Minlog","tactic_proof":"simplify elim intro cases","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2185","id":90684,"instance":2185,"goal":"all c, c <= c","context":["simplify","elim","intro","cases"]} -{"prover":"Minlog","tactic_proof":"ind use ng defn","sort_family":"list","source":"minlog_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2186","id":90685,"instance":2186,"goal":"all p u c, p <= u -> u <= c -> p <= c","context":["ind","use","ng","defn"]} -{"prover":"Minlog","tactic_proof":"simplify intro simp rw","sort_family":"list","source":"minlog_synthetic/scaled","variant":"lt_irrefl","theorem":"lt_irrefl_2187","id":90686,"instance":2187,"goal":"all d, not d < d","context":["simplify","intro","simp","rw"]} -{"prover":"Minlog","tactic_proof":"ng rw elim simplify","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"even_sum","theorem":"even_sum_2188","id":90687,"instance":2188,"goal":"all h q, Even h -> Even q -> Even (h + q)","context":["ng","rw","elim","simplify"]} -{"prover":"Minlog","tactic_proof":"simplify ng use assume","sort_family":"list","source":"minlog_synthetic/scaled","variant":"div_refl","theorem":"div_refl_2189","id":90688,"instance":2189,"goal":"all g, g | g","context":["simplify","ng","use","assume"]} -{"prover":"Minlog","tactic_proof":"ind rw ng use","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"fact_pos","theorem":"fact_pos_2190","id":90689,"instance":2190,"goal":"all c, Fact c > 0","context":["ind","rw","ng","use"]} -{"prover":"Minlog","tactic_proof":"intro rw intro use","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"add_zero_left","theorem":"add_zero_left_2191","id":90690,"instance":2191,"goal":"all z d = d","context":["intro","rw","intro","use"]} -{"prover":"Minlog","tactic_proof":"cases ind simp ng","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_zero_right","theorem":"add_zero_right_2192","id":90691,"instance":2192,"goal":"all b + 0 = b","context":["cases","ind","simp","ng"]} -{"prover":"Minlog","tactic_proof":"defn use simplify cases","sort_family":"list","source":"minlog_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2193","id":90692,"instance":2193,"goal":"all h n, h + n = n + h","context":["defn","use","simplify","cases"]} -{"prover":"Minlog","tactic_proof":"rw use simp defn","sort_family":"nat","source":"minlog_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2194","id":90693,"instance":2194,"goal":"all e q f, (e + q) + f = e + (q + f)","context":["rw","use","simp","defn"]} -{"prover":"Minlog","tactic_proof":"intro intro use simp","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_zero","theorem":"mul_zero_2195","id":90694,"instance":2195,"goal":"all q, q * 0 = 0","context":["intro","intro","use","simp"]} -{"prover":"Minlog","tactic_proof":"rw intro simp ind","sort_family":"list","source":"minlog_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2196","id":90695,"instance":2196,"goal":"all m, m * 1 = m","context":["rw","intro","simp","ind"]} -{"prover":"Minlog","tactic_proof":"ng defn rw ind","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_assoc","theorem":"mul_assoc_2197","id":90696,"instance":2197,"goal":"all p u d, (p * u) * d = p * (u * d)","context":["ng","defn","rw","ind"]} -{"prover":"Minlog","tactic_proof":"ng cases rw assume","sort_family":"rat","source":"minlog_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2198","id":90697,"instance":2198,"goal":"all a v, a * v = v * a","context":["ng","cases","rw","assume"]} -{"prover":"Minlog","tactic_proof":"simp simplify intro simplify","sort_family":"list","source":"minlog_synthetic/scaled","variant":"distrib_l","theorem":"distrib_l_2199","id":90698,"instance":2199,"goal":"all a y g, a * (y + g) = a * y + a * g","context":["simp","simplify","intro","simplify"]} -{"prover":"Minlog","tactic_proof":"rw assume defn elim","sort_family":"int","source":"minlog_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2200","id":90699,"instance":2200,"goal":"all m, m <= m","context":["rw","assume","defn","elim"]} +{"id":88500,"prover":"Minlog","theorem":"imp_refl","goal":"A -> A","context":["assume A","exact A"],"tactic_proof":"assume A; exact A","source":"minlog_synthetic/natural_deduction"} +{"id":88501,"prover":"Minlog","theorem":"imp_trans","goal":"(A -> B) -> (B -> C) -> A -> C","context":["assume f","assume g","assume a","exact (g (f a))"],"tactic_proof":"assume f; assume g; assume a; exact (g (f a))","source":"minlog_synthetic/natural_deduction"} +{"id":88502,"prover":"Minlog","theorem":"and_intro","goal":"A -> B -> A /\\ B","context":["assume a","assume b","split","exact a","exact b"],"tactic_proof":"assume a; assume b; split; [exact a | exact b]","source":"minlog_synthetic/natural_deduction"} +{"id":88503,"prover":"Minlog","theorem":"and_elim_l","goal":"A /\\ B -> A","context":["assume h","elim h","assume a","assume b","exact a"],"tactic_proof":"assume h; elim h; assume a; assume b; exact a","source":"minlog_synthetic/natural_deduction"} +{"id":88504,"prover":"Minlog","theorem":"and_elim_r","goal":"A /\\ B -> B","context":["assume h","elim h","assume a","assume b","exact b"],"tactic_proof":"assume h; elim h; assume a; assume b; exact b","source":"minlog_synthetic/natural_deduction"} +{"id":88505,"prover":"Minlog","theorem":"or_intro_l","goal":"A -> A \\/ B","context":["assume a","left","exact a"],"tactic_proof":"assume a; left; exact a","source":"minlog_synthetic/natural_deduction"} +{"id":88506,"prover":"Minlog","theorem":"or_intro_r","goal":"B -> A \\/ B","context":["assume b","right","exact b"],"tactic_proof":"assume b; right; exact b","source":"minlog_synthetic/natural_deduction"} +{"id":88507,"prover":"Minlog","theorem":"or_elim","goal":"(A -> C) -> (B -> C) -> A \\/ B -> C","context":["assume f","assume g","assume h","elim h","assume a","exact (f a)","assume b","exact (g b)"],"tactic_proof":"assume f; assume g; assume h; elim h; [assume a; exact (f a) | assume b; exact (g b)]","source":"minlog_synthetic/natural_deduction"} +{"id":88508,"prover":"Minlog","theorem":"false_elim","goal":"F -> A","context":["assume h","elim h"],"tactic_proof":"assume h; elim h","source":"minlog_synthetic/natural_deduction"} +{"id":88509,"prover":"Minlog","theorem":"not_intro","goal":"(A -> F) -> ~A","context":["assume f","intro","exact f"],"tactic_proof":"assume f; intro; exact f","source":"minlog_synthetic/natural_deduction"} +{"id":88510,"prover":"Minlog","theorem":"double_neg_intro","goal":"A -> ~~A","context":["assume a","assume h","exact (h a)"],"tactic_proof":"assume a; assume h; exact (h a)","source":"minlog_synthetic/natural_deduction"} +{"id":88511,"prover":"Minlog","theorem":"contrapositive","goal":"(A -> B) -> ~B -> ~A","context":["assume f","assume nb","assume a","exact (nb (f a))"],"tactic_proof":"assume f; assume nb; assume a; exact (nb (f a))","source":"minlog_synthetic/natural_deduction"} +{"id":88512,"prover":"Minlog","theorem":"and_comm","goal":"A /\\ B -> B /\\ A","context":["assume h","elim h","assume a","assume b","split","exact b","exact a"],"tactic_proof":"assume h; elim h; assume a; assume b; split; [exact b | exact a]","source":"minlog_synthetic/natural_deduction"} +{"id":88513,"prover":"Minlog","theorem":"or_comm","goal":"A \\/ B -> B \\/ A","context":["assume h","elim h","assume a","right","exact a","assume b","left","exact b"],"tactic_proof":"assume h; elim h; [assume a; right; exact a | assume b; left; exact b]","source":"minlog_synthetic/natural_deduction"} +{"id":88514,"prover":"Minlog","theorem":"curry","goal":"(A /\\ B -> C) -> A -> B -> C","context":["assume f","assume a","assume b","exact (f (a, b))"],"tactic_proof":"assume f; assume a; assume b; exact (f (a, b))","source":"minlog_synthetic/natural_deduction"} +{"id":88515,"prover":"Minlog","theorem":"add_zero","goal":"all n. n + 0 = n","context":["ind on n","refl","assume IH","simp","exact IH"],"tactic_proof":"ind on n; [refl | assume IH; simp; exact IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88516,"prover":"Minlog","theorem":"add_succ","goal":"all m n. m + S(n) = S(m + n)","context":["ind on m","refl","assume IH","simp","use IH"],"tactic_proof":"ind on m; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88517,"prover":"Minlog","theorem":"add_comm","goal":"all m n. m + n = n + m","context":["ind on m","use add_zero","assume IH","simp","use add_succ","use IH"],"tactic_proof":"ind on m; [use add_zero | assume IH; simp; use add_succ; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88518,"prover":"Minlog","theorem":"add_assoc","goal":"all m n k. (m + n) + k = m + (n + k)","context":["ind on m","refl","assume IH","simp","use IH"],"tactic_proof":"ind on m; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88519,"prover":"Minlog","theorem":"mul_zero","goal":"all n. n * 0 = 0","context":["ind on n","refl","assume IH","simp","use IH"],"tactic_proof":"ind on n; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88520,"prover":"Minlog","theorem":"mul_succ","goal":"all m n. m * S(n) = m + m * n","context":["ind on m","refl","assume IH","simp","use IH","use add_assoc","use add_comm"],"tactic_proof":"ind on m; [refl | assume IH; simp; use IH; use add_assoc; use add_comm]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88521,"prover":"Minlog","theorem":"mul_comm","goal":"all m n. m * n = n * m","context":["ind on m","use mul_zero","assume IH","simp","use mul_succ","use IH"],"tactic_proof":"ind on m; [use mul_zero | assume IH; simp; use mul_succ; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88522,"prover":"Minlog","theorem":"sub_self","goal":"all n. n - n = 0","context":["ind on n","refl","assume IH","simp","use IH"],"tactic_proof":"ind on n; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88523,"prover":"Minlog","theorem":"max_comm","goal":"all m n. max(m, n) = max(n, m)","context":["cases","auto"],"tactic_proof":"cases; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88524,"prover":"Minlog","theorem":"min_comm","goal":"all m n. min(m, n) = min(n, m)","context":["cases","auto"],"tactic_proof":"cases; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88525,"prover":"Minlog","theorem":"max_assoc","goal":"all l m n. max(l, max(m, n)) = max(max(l, m), n)","context":["cases","auto"],"tactic_proof":"cases; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88526,"prover":"Minlog","theorem":"le_refl","goal":"all n. n <= n","context":["intro","refl"],"tactic_proof":"intro; refl","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88527,"prover":"Minlog","theorem":"le_trans","goal":"all l m n. l <= m -> m <= n -> l <= n","context":["assume h1","assume h2","elim h1","elim h2","auto"],"tactic_proof":"assume h1; assume h2; elim h1; elim h2; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88528,"prover":"Minlog","theorem":"le_antisym","goal":"all m n. m <= n -> n <= m -> m = n","context":["assume h1","assume h2","elim h1","elim h2","auto"],"tactic_proof":"assume h1; assume h2; elim h1; elim h2; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88529,"prover":"Minlog","theorem":"lt_succ","goal":"all n. n < S(n)","context":["intro","auto"],"tactic_proof":"intro; auto","source":"minlog_synthetic/arithmetic_extraction"} +{"id":88530,"prover":"Minlog","theorem":"sqrt_extraction","goal":"all n. ex m. m * m <= n /\\ n < (m+1) * (m+1)","context":["ind on n","ex 0","auto","assume IH","elim IH","assume m","cases","ex m","ex (S m)"],"tactic_proof":"ind on n; [ex 0; auto | assume IH; elim IH; assume m; cases; [ex m; auto | ex (S m); auto]]","source":"minlog_synthetic/program_extraction"} +{"id":88531,"prover":"Minlog","theorem":"gcd_extraction","goal":"all a b. b > 0 -> ex d. divides(d, a) /\\ divides(d, b) /\\ all c. divides(c, a) -> divides(c, b) -> divides(c, d)","context":["strong_ind on b","cases","auto"],"tactic_proof":"strong_ind on b; cases; auto","source":"minlog_synthetic/program_extraction"} +{"id":88532,"prover":"Minlog","theorem":"div_extraction","goal":"all a b. b > 0 -> ex q r. a = b * q + r /\\ r < b","context":["strong_ind on a","cases","auto"],"tactic_proof":"strong_ind on a; cases; auto","source":"minlog_synthetic/program_extraction"} +{"id":88533,"prover":"Minlog","theorem":"prime_test","goal":"all n. n >= 2 -> prime(n) \\/ ex d. 1 < d /\\ d < n /\\ divides(d, n)","context":["bounded_search","auto"],"tactic_proof":"bounded_search; auto","source":"minlog_synthetic/program_extraction"} +{"id":88534,"prover":"Minlog","theorem":"list_sort","goal":"all l:list(nat). ex l'. sorted(l') /\\ perm(l, l')","context":["ind on l","ex nil","auto","assume IH","elim IH","assume l'","insert"],"tactic_proof":"ind on l; [ex nil; auto | assume IH; elim IH; assume l'; insert; auto]","source":"minlog_synthetic/program_extraction"} +{"id":88535,"prover":"Minlog","theorem":"binary_search_extract","goal":"all a:array(nat) v. sorted(a) -> (ex i. a[i] = v) \\/ (all i. a[i] <> v)","context":["log_search","auto"],"tactic_proof":"log_search; auto","source":"minlog_synthetic/program_extraction"} +{"id":88536,"prover":"Minlog","theorem":"min_element","goal":"all l:list(nat). l <> nil -> ex m. mem(m, l) /\\ all x. mem(x, l) -> m <= x","context":["ind on l","auto"],"tactic_proof":"ind on l; auto","source":"minlog_synthetic/program_extraction"} +{"id":88537,"prover":"Minlog","theorem":"majority_element","goal":"all l:list(nat). length(l) > 0 -> ex m. count(m, l) > length(l) / 2 \\/ all x. count(x, l) <= length(l) / 2","context":["boyer_moore","auto"],"tactic_proof":"boyer_moore; auto","source":"minlog_synthetic/program_extraction"} +{"id":88538,"prover":"Minlog","theorem":"realizer_identity","goal":"[r : A -> A] = lambda x. x","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88539,"prover":"Minlog","theorem":"realizer_compose","goal":"[r : (A -> B) -> (B -> C) -> A -> C] = lambda f g x. g (f x)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88540,"prover":"Minlog","theorem":"realizer_pair","goal":"[r : A -> B -> A /\\ B] = lambda a b. (a, b)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88541,"prover":"Minlog","theorem":"realizer_fst","goal":"[r : A /\\ B -> A] = lambda p. fst(p)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88542,"prover":"Minlog","theorem":"realizer_snd","goal":"[r : A /\\ B -> B] = lambda p. snd(p)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88543,"prover":"Minlog","theorem":"realizer_inl","goal":"[r : A -> A \\/ B] = lambda a. inl(a)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88544,"prover":"Minlog","theorem":"realizer_inr","goal":"[r : B -> A \\/ B] = lambda b. inr(b)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88545,"prover":"Minlog","theorem":"realizer_case","goal":"[r : (A -> C) -> (B -> C) -> A \\/ B -> C] = lambda f g s. case s of inl(a) => f a | inr(b) => g b","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88546,"prover":"Minlog","theorem":"realizer_forall","goal":"[r : (all x:N. A(x))] = lambda x. r_x","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88547,"prover":"Minlog","theorem":"realizer_exists","goal":"[r : (ex x:N. A(x))] = (n, proof_of_A(n))","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88548,"prover":"Minlog","theorem":"realizer_induction","goal":"[r : nat_rec] = lambda base step n. (rec base step n)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizability"} +{"id":88549,"prover":"Minlog","theorem":"list_nil_append","goal":"all l:list. nil ++ l = l","context":["intro","refl"],"tactic_proof":"intro; refl","source":"minlog_synthetic/lists_minlog"} +{"id":88550,"prover":"Minlog","theorem":"list_append_nil","goal":"all l:list. l ++ nil = l","context":["ind on l","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88551,"prover":"Minlog","theorem":"list_append_assoc","goal":"all l1 l2 l3. (l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3)","context":["ind on l1","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l1; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88552,"prover":"Minlog","theorem":"list_length_append","goal":"all l1 l2. length(l1 ++ l2) = length(l1) + length(l2)","context":["ind on l1","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l1; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88553,"prover":"Minlog","theorem":"list_rev_involutive","goal":"all l. rev(rev(l)) = l","context":["ind on l","refl","assume IH","simp","use rev_append","use IH"],"tactic_proof":"ind on l; [refl | assume IH; simp; use rev_append; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88554,"prover":"Minlog","theorem":"list_rev_append","goal":"all l1 l2. rev(l1 ++ l2) = rev(l2) ++ rev(l1)","context":["ind on l1","simp","use list_append_nil","assume IH","use IH","use list_append_assoc"],"tactic_proof":"ind on l1; [simp; use list_append_nil | assume IH; simp; use IH; use list_append_assoc]","source":"minlog_synthetic/lists_minlog"} +{"id":88555,"prover":"Minlog","theorem":"list_map_id","goal":"all l. map(lambda x. x, l) = l","context":["ind on l","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88556,"prover":"Minlog","theorem":"list_map_compose","goal":"all f g l. map(f, map(g, l)) = map(lambda x. f (g x), l)","context":["ind on l","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88557,"prover":"Minlog","theorem":"list_filter_length","goal":"all p l. length(filter(p, l)) <= length(l)","context":["ind on l","auto","assume IH","cases p"],"tactic_proof":"ind on l; [auto | assume IH; cases p; auto]","source":"minlog_synthetic/lists_minlog"} +{"id":88558,"prover":"Minlog","theorem":"list_foldr_append","goal":"all f z l1 l2. foldr(f, z, l1 ++ l2) = foldr(f, foldr(f, z, l2), l1)","context":["ind on l1","refl","assume IH","simp","use IH"],"tactic_proof":"ind on l1; [refl | assume IH; simp; use IH]","source":"minlog_synthetic/lists_minlog"} +{"id":88559,"prover":"Minlog","theorem":"list_all_forall","goal":"all p l. (all x. mem(x, l) -> p(x)) -> list_all(p, l)","context":["ind on l","auto","assume IH","cases"],"tactic_proof":"ind on l; [auto | assume IH; cases; auto]","source":"minlog_synthetic/lists_minlog"} +{"id":88560,"prover":"Minlog","theorem":"tree_height_nonneg","goal":"all t. height(t) >= 0","context":["ind on t","auto"],"tactic_proof":"ind on t; auto","source":"minlog_synthetic/trees"} +{"id":88561,"prover":"Minlog","theorem":"tree_size_nonneg","goal":"all t. size(t) >= 1","context":["ind on t","auto"],"tactic_proof":"ind on t; auto","source":"minlog_synthetic/trees"} +{"id":88562,"prover":"Minlog","theorem":"tree_leaves_ge_1","goal":"all t. leaves(t) >= 1","context":["ind on t","auto"],"tactic_proof":"ind on t; auto","source":"minlog_synthetic/trees"} +{"id":88563,"prover":"Minlog","theorem":"tree_mirror_involutive","goal":"all t. mirror(mirror(t)) = t","context":["ind on t","refl","assume IH1 IH2","simp","use IH1","use IH2"],"tactic_proof":"ind on t; [refl | assume IH1 IH2; simp; use IH1; use IH2]","source":"minlog_synthetic/trees"} +{"id":88564,"prover":"Minlog","theorem":"tree_size_height","goal":"all t. size(t) <= 2^(height(t)+1) - 1","context":["ind on t","auto","assume IH1 IH2","simp","use IH1","use IH2"],"tactic_proof":"ind on t; [auto | assume IH1 IH2; simp; use IH1; use IH2]","source":"minlog_synthetic/trees"} +{"id":88565,"prover":"Minlog","theorem":"bst_inorder_sorted","goal":"all t. is_bst(t) -> sorted(inorder(t))","context":["ind on t","auto","assume IH","split"],"tactic_proof":"ind on t; [auto | assume IH; split; auto]","source":"minlog_synthetic/trees"} +{"id":88566,"prover":"Minlog","theorem":"tree_flatten_append","goal":"all t1 t2. flatten(Node(t1, x, t2)) = flatten(t1) ++ [x] ++ flatten(t2)","context":["intro","refl"],"tactic_proof":"intro; refl","source":"minlog_synthetic/trees"} +{"id":88567,"prover":"Minlog","theorem":"tree_count_sum","goal":"all x t. count(x, flatten(t)) = count_tree(x, t)","context":["ind on t","auto"],"tactic_proof":"ind on t; auto","source":"minlog_synthetic/trees"} +{"id":88568,"prover":"Minlog","theorem":"tnd","goal":"all A. A \\/ ~A","context":["use classical","auto"],"tactic_proof":"use classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88569,"prover":"Minlog","theorem":"dne","goal":"all A. ~~A -> A","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88570,"prover":"Minlog","theorem":"peirce_law","goal":"all A B. ((A -> B) -> A) -> A","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88571,"prover":"Minlog","theorem":"contrapos_classical","goal":"all A B. (~B -> ~A) -> (A -> B)","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88572,"prover":"Minlog","theorem":"or_impl","goal":"all A B. (A \\/ B) -> (~A -> B)","context":["assume h nA","elim h","assume a","absurd","assume b","exact b"],"tactic_proof":"assume h nA; elim h; [assume a; absurd | assume b; exact b]","source":"minlog_synthetic/classical_logic"} +{"id":88573,"prover":"Minlog","theorem":"impl_or","goal":"all A B. (A -> B) -> (~A \\/ B)","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88574,"prover":"Minlog","theorem":"and_not_or","goal":"all A B. ~A /\\ ~B -> ~(A \\/ B)","context":["assume h","elim h","assume nA nB","intro","elim","auto"],"tactic_proof":"assume h; elim h; assume nA nB; intro; elim; auto","source":"minlog_synthetic/classical_logic"} +{"id":88575,"prover":"Minlog","theorem":"demorgan_classical_1","goal":"all A B. ~(A /\\ B) -> (~A \\/ ~B)","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88576,"prover":"Minlog","theorem":"demorgan_classical_2","goal":"all A B. ~(A \\/ B) <-> (~A /\\ ~B)","context":["split","intros","auto"],"tactic_proof":"split; intros; split; intros; auto","source":"minlog_synthetic/classical_logic"} +{"id":88577,"prover":"Minlog","theorem":"classical_exists","goal":"all P. ~(all x. ~P(x)) -> ex x. P(x)","context":["assume h","classical","auto"],"tactic_proof":"assume h; classical; auto","source":"minlog_synthetic/classical_logic"} +{"id":88578,"prover":"Minlog","theorem":"cut_elim","goal":"all A B C. (Gamma |- A) -> (A, Gamma |- B) -> (Gamma |- B)","context":["intros","cut_admissible","auto"],"tactic_proof":"intros; cut_admissible; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88579,"prover":"Minlog","theorem":"structural_weaken","goal":"all A B. (Gamma |- B) -> (A, Gamma |- B)","context":["intros","weakening","auto"],"tactic_proof":"intros; weakening; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88580,"prover":"Minlog","theorem":"structural_contract","goal":"all A B. (A, A, Gamma |- B) -> (A, Gamma |- B)","context":["intros","contraction","auto"],"tactic_proof":"intros; contraction; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88581,"prover":"Minlog","theorem":"structural_exchange","goal":"all A B C. (A, B, Gamma |- C) -> (B, A, Gamma |- C)","context":["intros","exchange","auto"],"tactic_proof":"intros; exchange; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88582,"prover":"Minlog","theorem":"identity_axiom","goal":"all A. (A |- A)","context":["intro","axiom"],"tactic_proof":"intro; axiom","source":"minlog_synthetic/sequent_calculus"} +{"id":88583,"prover":"Minlog","theorem":"and_right","goal":"all A B. (Gamma |- A) -> (Gamma |- B) -> (Gamma |- A /\\ B)","context":["intros","andR","auto"],"tactic_proof":"intros; andR; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88584,"prover":"Minlog","theorem":"and_left_1","goal":"all A B C. (A, Gamma |- C) -> (A /\\ B, Gamma |- C)","context":["intros","andL1","auto"],"tactic_proof":"intros; andL1; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88585,"prover":"Minlog","theorem":"impl_right","goal":"all A B. (A, Gamma |- B) -> (Gamma |- A -> B)","context":["intros","implR","auto"],"tactic_proof":"intros; implR; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88586,"prover":"Minlog","theorem":"impl_left","goal":"all A B C. (Gamma |- A) -> (B, Gamma |- C) -> (A -> B, Gamma |- C)","context":["intros","implL","auto"],"tactic_proof":"intros; implL; auto","source":"minlog_synthetic/sequent_calculus"} +{"id":88587,"prover":"Minlog","theorem":"strong_ind","goal":"all P. (all n. (all m. m < n -> P(m)) -> P(n)) -> all n. P(n)","context":["assume h","strong_ind","auto"],"tactic_proof":"assume h; strong_ind; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88588,"prover":"Minlog","theorem":"course_of_values","goal":"all P. P(0) -> (all n. P(n) -> P(n+1)) -> all n. P(n)","context":["assume p0 step","nat_rec","auto"],"tactic_proof":"assume p0 step; nat_rec; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88589,"prover":"Minlog","theorem":"well_founded_acc","goal":"all R. well_founded(R) -> all P. (all x. (all y. R(y,x) -> P(y)) -> P(x)) -> all x. P(x)","context":["assume wf ih","wf_rec","auto"],"tactic_proof":"assume wf ih; wf_rec; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88590,"prover":"Minlog","theorem":"mutual_ind_evenodd","goal":"(all n. even(n) -> P(n)) /\\ (all n. odd(n) -> Q(n))","context":["mutual_ind","auto"],"tactic_proof":"mutual_ind; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88591,"prover":"Minlog","theorem":"struct_ind_list","goal":"all P. P(nil) -> (all x xs. P(xs) -> P(cons x xs)) -> all l. P(l)","context":["assume p_nil p_cons","list_rec","auto"],"tactic_proof":"assume p_nil p_cons; list_rec; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88592,"prover":"Minlog","theorem":"struct_ind_tree","goal":"all P. P(leaf) -> (all t1 x t2. P(t1) -> P(t2) -> P(Node t1 x t2)) -> all t. P(t)","context":["assume p_leaf p_node","tree_rec","auto"],"tactic_proof":"assume p_leaf p_node; tree_rec; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88593,"prover":"Minlog","theorem":"transfinite_ind","goal":"all P. (all beta. beta < alpha -> P(beta)) -> P(alpha)","context":["assume ih","transfinite","auto"],"tactic_proof":"assume ih; transfinite; auto","source":"minlog_synthetic/induction_advanced"} +{"id":88594,"prover":"Minlog","theorem":"higher_order_app","goal":"all f x. (lambda y. f y) x = f x","context":["beta_reduce","refl"],"tactic_proof":"beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88595,"prover":"Minlog","theorem":"fun_ext_ho","goal":"all f g. (all x. f x = g x) -> f = g","context":["assume h","fun_ext","auto"],"tactic_proof":"assume h; fun_ext; auto","source":"minlog_synthetic/higher_order"} +{"id":88596,"prover":"Minlog","theorem":"curry_uncurry","goal":"all f x y. curry (uncurry f) x y = f x y","context":["intros","beta_reduce","refl"],"tactic_proof":"intros; beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88597,"prover":"Minlog","theorem":"uncurry_curry","goal":"all f p. uncurry (curry f) p = f p","context":["intro","cases p","beta_reduce","refl"],"tactic_proof":"intro; cases p; beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88598,"prover":"Minlog","theorem":"compose_assoc","goal":"all f g h. compose (compose f g) h = compose f (compose g h)","context":["fun_ext","beta_reduce","refl"],"tactic_proof":"fun_ext; beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88599,"prover":"Minlog","theorem":"id_left","goal":"all f. compose id f = f","context":["fun_ext","beta_reduce","refl"],"tactic_proof":"fun_ext; beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88600,"prover":"Minlog","theorem":"id_right","goal":"all f. compose f id = f","context":["fun_ext","beta_reduce","refl"],"tactic_proof":"fun_ext; beta_reduce; refl","source":"minlog_synthetic/higher_order"} +{"id":88601,"prover":"Minlog","theorem":"ho_induction_schema","goal":"all P. (all x. x = 0 \\/ (ex y. x = y+1 /\\ P(y))) -> P(0) -> all x. P(x)","context":["assume h p0","induct","auto"],"tactic_proof":"assume h p0; induct; auto","source":"minlog_synthetic/higher_order"} +{"id":88602,"prover":"Minlog","theorem":"factorial_correct","goal":"all n. factorial(n) >= 1","context":["ind on n","simp","auto","assume IH","use IH"],"tactic_proof":"ind on n; [simp; auto | assume IH; simp; use IH]","source":"minlog_synthetic/program_verification"} +{"id":88603,"prover":"Minlog","theorem":"fibonacci_monotone","goal":"all m n. m <= n -> fib(m) <= fib(n)","context":["assume h","ind on n","auto"],"tactic_proof":"assume h; ind on n; auto","source":"minlog_synthetic/program_verification"} +{"id":88604,"prover":"Minlog","theorem":"sum_range_formula","goal":"all n. sum(0..n, lambda i. i) = n * (n+1) / 2","context":["ind on n","auto"],"tactic_proof":"ind on n; auto","source":"minlog_synthetic/program_verification"} +{"id":88605,"prover":"Minlog","theorem":"sort_preserves_elements","goal":"all l. mset(sort l) = mset(l)","context":["ind on l","auto"],"tactic_proof":"ind on l; auto","source":"minlog_synthetic/program_verification"} +{"id":88606,"prover":"Minlog","theorem":"reverse_length","goal":"all l. length(reverse l) = length l","context":["ind on l","refl","assume IH","simp","use length_append","use IH"],"tactic_proof":"ind on l; [refl | assume IH; simp; use length_append; use IH]","source":"minlog_synthetic/program_verification"} +{"id":88607,"prover":"Minlog","theorem":"gcd_correct","goal":"all a b. b > 0 -> divides(gcd(a,b), a) /\\ divides(gcd(a,b), b)","context":["strong_ind on b","cases","auto"],"tactic_proof":"strong_ind on b; cases; auto","source":"minlog_synthetic/program_verification"} +{"id":88608,"prover":"Minlog","theorem":"is_sorted_cons","goal":"all x xs. is_sorted(cons x xs) -> is_sorted xs","context":["intros","cases","auto"],"tactic_proof":"intros; cases; auto","source":"minlog_synthetic/program_verification"} +{"id":88609,"prover":"Minlog","theorem":"search_correct","goal":"all x l. linear_search(x, l) = true <-> mem(x, l)","context":["ind on l","split","auto"],"tactic_proof":"ind on l; split; auto","source":"minlog_synthetic/program_verification"} +{"id":88610,"prover":"Minlog","theorem":"modal_k","goal":"all A B. box(A -> B) -> box(A) -> box(B)","context":["intros","box_elim","auto"],"tactic_proof":"intros; box_elim; auto","source":"minlog_synthetic/modal_logic"} +{"id":88611,"prover":"Minlog","theorem":"modal_t","goal":"all A. box(A) -> A","context":["intro","reflexive_frame","auto"],"tactic_proof":"intro; reflexive_frame; auto","source":"minlog_synthetic/modal_logic"} +{"id":88612,"prover":"Minlog","theorem":"modal_4","goal":"all A. box(A) -> box(box(A))","context":["intro","transitive_frame","auto"],"tactic_proof":"intro; transitive_frame; auto","source":"minlog_synthetic/modal_logic"} +{"id":88613,"prover":"Minlog","theorem":"modal_5","goal":"all A. dia(A) -> box(dia(A))","context":["intro","euclidean_frame","auto"],"tactic_proof":"intro; euclidean_frame; auto","source":"minlog_synthetic/modal_logic"} +{"id":88614,"prover":"Minlog","theorem":"modal_dual","goal":"all A. dia(A) <-> ~box(~A)","context":["split","intros","auto"],"tactic_proof":"split; intros; auto","source":"minlog_synthetic/modal_logic"} +{"id":88615,"prover":"Minlog","theorem":"necessitation","goal":"all A. |- A -> |- box(A)","context":["intro","necessitate","auto"],"tactic_proof":"intro; necessitate; auto","source":"minlog_synthetic/modal_logic"} +{"id":88616,"prover":"Minlog","theorem":"distribution","goal":"all A B. box(A /\\ B) <-> box(A) /\\ box(B)","context":["split","intros","box_elim","auto"],"tactic_proof":"split; intros; split; box_elim; auto","source":"minlog_synthetic/modal_logic"} +{"id":88617,"prover":"Minlog","theorem":"disjunction_property","goal":"all A B. (|- A \\/ B) -> (|- A) \\/ (|- B)","context":["assume h","intuit","auto"],"tactic_proof":"assume h; intuit; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88618,"prover":"Minlog","theorem":"existence_property","goal":"all P. (|- ex x. P(x)) -> ex witness. |- P(witness)","context":["assume h","intuit","auto"],"tactic_proof":"assume h; intuit; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88619,"prover":"Minlog","theorem":"double_neg_shift","goal":"all A. ~~(A \\/ ~A)","context":["intro","assume h","apply h","right","left","auto"],"tactic_proof":"intro; assume h; apply h; right; intro; apply h; left; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88620,"prover":"Minlog","theorem":"glivenko","goal":"all A. (|- ~~A classically) -> (|- ~~A intuitionistically)","context":["assume h","glivenko","auto"],"tactic_proof":"assume h; glivenko; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88621,"prover":"Minlog","theorem":"kripke_model","goal":"all w:World, A. forces(w, A) <-> satisfies(w, A)","context":["intro","split","kripke","auto"],"tactic_proof":"intro; split; kripke; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88622,"prover":"Minlog","theorem":"negation_translation","goal":"all A. (~~A)_neg = ~~A_neg","context":["intro","neg_translation","auto"],"tactic_proof":"intro; neg_translation; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88623,"prover":"Minlog","theorem":"godel_neg","goal":"all A. valid(A) <-> valid(~~A_neg)","context":["intro","split","godel_negative","auto"],"tactic_proof":"intro; split; godel_negative; auto","source":"minlog_synthetic/intuitionistic"} +{"id":88624,"prover":"Minlog","theorem":"game_det","goal":"all G:Game. open(G) -> determined(G)","context":["assume h","gale_stewart","auto"],"tactic_proof":"assume h; gale_stewart; auto","source":"minlog_synthetic/games"} +{"id":88625,"prover":"Minlog","theorem":"strategy_winning","goal":"all G:Game, sigma:Strategy. winning(sigma, G) -> all play. consistent(play, sigma) -> G_wins(play)","context":["assume w","intro con","auto"],"tactic_proof":"assume w; intro con; auto","source":"minlog_synthetic/games"} +{"id":88626,"prover":"Minlog","theorem":"zermelo","goal":"all G:FiniteGame. determined(G)","context":["intro","backward_induction","auto"],"tactic_proof":"intro; backward_induction; auto","source":"minlog_synthetic/games"} +{"id":88627,"prover":"Minlog","theorem":"minimax","goal":"all G:ZeroSum, v:Value. v = max(min(payoff))","context":["intro","minimax_theorem","auto"],"tactic_proof":"intro; minimax_theorem; auto","source":"minlog_synthetic/games"} +{"id":88628,"prover":"Minlog","theorem":"nash_existence","goal":"all G:FiniteGame. ex sigma. nash_equilibrium(sigma, G)","context":["intro","brouwer","auto"],"tactic_proof":"intro; brouwer; auto","source":"minlog_synthetic/games"} +{"id":88629,"prover":"Minlog","theorem":"realizer_nat_rec","goal":"[r : nat_rec] = lambda base step n. (fix F lambda k. if k=0 then base else step k (F (k-1))) n","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88630,"prover":"Minlog","theorem":"realizer_list_rec","goal":"[r : list_rec] = lambda nil_case cons_case l. (fix F lambda xs. match xs with nil => nil_case | x::xs' => cons_case x (F xs')) l","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88631,"prover":"Minlog","theorem":"realizer_bounded_search","goal":"[r : bounded_search_lemma] = lambda p bound. search_loop p 0 bound","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88632,"prover":"Minlog","theorem":"realizer_minimization","goal":"[r : minimization] = lambda f. mu (lambda n. f n = 0)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88633,"prover":"Minlog","theorem":"realizer_fixpoint","goal":"[r : fix_point_thm] = lambda f. fix f","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88634,"prover":"Minlog","theorem":"realizer_choice","goal":"[r : countable_choice] = lambda R. lambda n. choose (R n)","context":["normalize","refl"],"tactic_proof":"normalize; refl","source":"minlog_synthetic/realizers_advanced"} +{"id":88635,"prover":"Minlog","theorem":"bit_length","goal":"all n. n > 0 -> bit_length(n) = floor(log2(n)) + 1","context":["strong_ind on n","auto"],"tactic_proof":"strong_ind on n; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88636,"prover":"Minlog","theorem":"parity","goal":"all n. parity(n) = n mod 2","context":["ind on n","auto"],"tactic_proof":"ind on n; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88637,"prover":"Minlog","theorem":"binary_rep","goal":"all n. binary_decode(binary_encode(n)) = n","context":["ind on n","auto"],"tactic_proof":"ind on n; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88638,"prover":"Minlog","theorem":"pow_two_le","goal":"all n. 2^n > n","context":["ind on n","auto"],"tactic_proof":"ind on n; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88639,"prover":"Minlog","theorem":"pigeon_bounded","goal":"all f:fin(n)->fin(m). n > m -> ex i j. i <> j /\\ f(i) = f(j)","context":["assume","pigeonhole","auto"],"tactic_proof":"assume; pigeonhole; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88640,"prover":"Minlog","theorem":"bounded_search","goal":"all p:nat->bool. (ex n. n <= N /\\ p n = true) -> ex k. k <= N /\\ p k = true /\\ (all j. j < k -> p j = false)","context":["assume","bounded_search","auto"],"tactic_proof":"assume; bounded_search; auto","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88641,"prover":"Minlog","theorem":"pspace_pspace","goal":"pspace = pspace","context":["refl"],"tactic_proof":"refl","source":"minlog_synthetic/bounded_arithmetic"} +{"id":88642,"prover":"Minlog","theorem":"fin_sat_decidable","goal":"all phi:Formula. decidable(sat(phi))","context":["intro","propositional_sat","auto"],"tactic_proof":"intro; propositional_sat; auto","source":"minlog_synthetic/finite_logic"} +{"id":88643,"prover":"Minlog","theorem":"resolution_sound","goal":"all phi. res_derives(empty, phi) -> unsat(phi)","context":["assume h","resolution_soundness","auto"],"tactic_proof":"assume h; resolution_soundness; auto","source":"minlog_synthetic/finite_logic"} +{"id":88644,"prover":"Minlog","theorem":"resolution_complete","goal":"all phi. unsat(phi) -> res_derives(empty, phi)","context":["assume h","resolution_completeness","auto"],"tactic_proof":"assume h; resolution_completeness; auto","source":"minlog_synthetic/finite_logic"} +{"id":88645,"prover":"Minlog","theorem":"dpll_terminates","goal":"all phi. terminates(dpll(phi))","context":["intro","wf_induct","auto"],"tactic_proof":"intro; wf_induct; auto","source":"minlog_synthetic/finite_logic"} +{"id":88646,"prover":"Minlog","theorem":"cnf_equivalent","goal":"all phi. equivalent(phi, to_cnf(phi))","context":["intro","tseitin","auto"],"tactic_proof":"intro; tseitin; auto","source":"minlog_synthetic/finite_logic"} +{"id":88647,"prover":"Minlog","theorem":"horn_sat_poly","goal":"all phi:HornFormula. poly_time_decidable(sat(phi))","context":["intro","unit_propagation","auto"],"tactic_proof":"intro; unit_propagation; auto","source":"minlog_synthetic/finite_logic"} diff --git a/training_data/proof_states_nuprl.jsonl b/training_data/proof_states_nuprl.jsonl index b13ac959..3d3e4cb6 100644 --- a/training_data/proof_states_nuprl.jsonl +++ b/training_data/proof_states_nuprl.jsonl @@ -1,2200 +1,152 @@ -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1","id":88000,"instance":1,"goal":"all a:Int. all b:Int. a + b = b + a","type_family":"Int","context":["Complete Auto","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2","id":88001,"instance":2,"goal":"all b:Nat. all e:Nat. all y:Nat. (b+e)+y = b+(e+y)","type_family":"Nat","context":["Thin","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_3","id":88002,"instance":3,"goal":"all a:Real. a + 0 = a","type_family":"Real","context":["Complete Auto","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_4","id":88003,"instance":4,"goal":"all z:Atom. all f:Atom. z * f = f * z","type_family":"Atom","context":["SplitOn","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_5","id":88004,"instance":5,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["D 0","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_6","id":88005,"instance":6,"goal":"all y:List(Int). all v:List(Int). all n:List(Int). y*(v+n) = y*v + y*n","type_family":"List(Int)","context":["D -1","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_7","id":88006,"instance":7,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["Lemma","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_8","id":88007,"instance":8,"goal":"all x:List(Atom). all f:List(Atom). all n:List(Atom). (x <= f) => (f <= n) => (x <= n)","type_family":"List(Atom)","context":["Thin","D -1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_9","id":88008,"instance":9,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["Reduce","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_10","id":88009,"instance":10,"goal":"all m:Nat -> Nat. all d:Nat -> Nat. (m = d) => (d = m)","type_family":"Nat -> Nat","context":["Assert","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_11","id":88010,"instance":11,"goal":"(gamma) => (kappa) => (gamma)","type_family":"Int","context":["Assert","SplitOn","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_12","id":88011,"instance":12,"goal":"((a) & (y)) iff ((y) & (a))","type_family":"Nat","context":["DoSubst","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_13","id":88012,"instance":13,"goal":"((g) | (p)) iff ((p) | (g))","type_family":"Real","context":["Thin","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_14","id":88013,"instance":14,"goal":"not (not (z)) => (z)","type_family":"Atom","context":["Lemma","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_15","id":88014,"instance":15,"goal":"all u:Bool. all w:Bool. all p:Bool. append(u, append(w, p)) = append(append(u, w), p)","type_family":"Bool","context":["Assert","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_16","id":88015,"instance":16,"goal":"all gamma:List(Int). all phi:List(Int). gamma + phi = phi + gamma","type_family":"List(Int)","context":["Auto","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_17","id":88016,"instance":17,"goal":"all n:List(Nat). all h:List(Nat). all w:List(Nat). (n+h)+w = n+(h+w)","type_family":"List(Nat)","context":["D -1","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_18","id":88017,"instance":18,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["BackThru","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_19","id":88018,"instance":19,"goal":"all z:Int -> Int. all w:Int -> Int. z * w = w * z","type_family":"Int -> Int","context":["RepUR","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_20","id":88019,"instance":20,"goal":"all alpha:Nat -> Nat. alpha * 1 = alpha","type_family":"Nat -> Nat","context":["Thin","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_21","id":88020,"instance":21,"goal":"all m:Int. all x:Int. all c:Int. m*(x+c) = m*x + m*c","type_family":"Int","context":["Thin","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_22","id":88021,"instance":22,"goal":"all beta:Nat. beta <= beta","type_family":"Nat","context":["Reduce","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_23","id":88022,"instance":23,"goal":"all u:Real. all m:Real. all d:Real. (u <= m) => (m <= d) => (u <= d)","type_family":"Real","context":["Auto","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_24","id":88023,"instance":24,"goal":"all x:Atom. x = x","type_family":"Atom","context":["RepUR","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_25","id":88024,"instance":25,"goal":"all e:Bool. all u:Bool. (e = u) => (u = e)","type_family":"Bool","context":["D -1","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_26","id":88025,"instance":26,"goal":"(g) => (f) => (g)","type_family":"List(Int)","context":["BackThru","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_27","id":88026,"instance":27,"goal":"((e) & (m)) iff ((m) & (e))","type_family":"List(Nat)","context":["InstLemma","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_28","id":88027,"instance":28,"goal":"((g) | (y)) iff ((y) | (g))","type_family":"List(Atom)","context":["Reduce","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_29","id":88028,"instance":29,"goal":"not (not (b)) => (b)","type_family":"Int -> Int","context":["Lemma","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_30","id":88029,"instance":30,"goal":"all x:Nat -> Nat. all q:Nat -> Nat. all w:Nat -> Nat. append(x, append(q, w)) = append(append(x, q), w)","type_family":"Nat -> Nat","context":["InstLemma","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_31","id":88030,"instance":31,"goal":"all w:Int. all h:Int. w + h = h + w","type_family":"Int","context":["DoSubst","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_32","id":88031,"instance":32,"goal":"all z:Nat. all p:Nat. all u:Nat. (z+p)+u = z+(p+u)","type_family":"Nat","context":["Thin","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_33","id":88032,"instance":33,"goal":"all chi:Real. chi + 0 = chi","type_family":"Real","context":["D 1","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_34","id":88033,"instance":34,"goal":"all alpha:Atom. all phi:Atom. alpha * phi = phi * alpha","type_family":"Atom","context":["Lemma","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_35","id":88034,"instance":35,"goal":"all y:Bool. y * 1 = y","type_family":"Bool","context":["BackThru","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_36","id":88035,"instance":36,"goal":"all b:List(Int). all n:List(Int). all y:List(Int). b*(n+y) = b*n + b*y","type_family":"List(Int)","context":["InstLemma","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_37","id":88036,"instance":37,"goal":"all e:List(Nat). e <= e","type_family":"List(Nat)","context":["RepUR","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_38","id":88037,"instance":38,"goal":"all w:List(Atom). all v:List(Atom). all a:List(Atom). (w <= v) => (v <= a) => (w <= a)","type_family":"List(Atom)","context":["Auto","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_39","id":88038,"instance":39,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["BackThru","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_40","id":88039,"instance":40,"goal":"all u:Nat -> Nat. all g:Nat -> Nat. (u = g) => (g = u)","type_family":"Nat -> Nat","context":["InstLemma","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_41","id":88040,"instance":41,"goal":"(n) => (d) => (n)","type_family":"Int","context":["DoSubst","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_42","id":88041,"instance":42,"goal":"((q) & (g)) iff ((g) & (q))","type_family":"Nat","context":["D 0","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_43","id":88042,"instance":43,"goal":"((f) | (h)) iff ((h) | (f))","type_family":"Real","context":["D 1","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_44","id":88043,"instance":44,"goal":"not (not (e)) => (e)","type_family":"Atom","context":["SplitOn","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_45","id":88044,"instance":45,"goal":"all a:Bool. all p:Bool. all c:Bool. append(a, append(p, c)) = append(append(a, p), c)","type_family":"Bool","context":["Thin","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_46","id":88045,"instance":46,"goal":"all gamma:List(Int). all beta:List(Int). gamma + beta = beta + gamma","type_family":"List(Int)","context":["D 1","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_47","id":88046,"instance":47,"goal":"all h:List(Nat). all n:List(Nat). all e:List(Nat). (h+n)+e = h+(n+e)","type_family":"List(Nat)","context":["SplitOn","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_48","id":88047,"instance":48,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["Reduce","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_49","id":88048,"instance":49,"goal":"all q:Int -> Int. all z:Int -> Int. q * z = z * q","type_family":"Int -> Int","context":["D 0","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_50","id":88049,"instance":50,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["SplitOn","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_51","id":88050,"instance":51,"goal":"all y:Int. all z:Int. all a:Int. y*(z+a) = y*z + y*a","type_family":"Int","context":["Auto","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_52","id":88051,"instance":52,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["Assert","D 1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_53","id":88052,"instance":53,"goal":"all e:Real. all g:Real. all n:Real. (e <= g) => (g <= n) => (e <= n)","type_family":"Real","context":["SplitOn","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_54","id":88053,"instance":54,"goal":"all v:Atom. v = v","type_family":"Atom","context":["RepUR","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_55","id":88054,"instance":55,"goal":"all p:Bool. all c:Bool. (p = c) => (c = p)","type_family":"Bool","context":["D -1","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_56","id":88055,"instance":56,"goal":"(w) => (d) => (w)","type_family":"List(Int)","context":["D 0","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_57","id":88056,"instance":57,"goal":"((z) & (n)) iff ((n) & (z))","type_family":"List(Nat)","context":["SplitOn","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_58","id":88057,"instance":58,"goal":"((q) | (u)) iff ((u) | (q))","type_family":"List(Atom)","context":["RepUR","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_59","id":88058,"instance":59,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["SplitOn","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_60","id":88059,"instance":60,"goal":"all c:Nat -> Nat. all u:Nat -> Nat. all w:Nat -> Nat. append(c, append(u, w)) = append(append(c, u), w)","type_family":"Nat -> Nat","context":["Auto","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_61","id":88060,"instance":61,"goal":"all w:Int. all d:Int. w + d = d + w","type_family":"Int","context":["InstLemma","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_62","id":88061,"instance":62,"goal":"all m:Nat. all a:Nat. all w:Nat. (m+a)+w = m+(a+w)","type_family":"Nat","context":["InstLemma","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_63","id":88062,"instance":63,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["RepUR","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_64","id":88063,"instance":64,"goal":"all p:Atom. all w:Atom. p * w = w * p","type_family":"Atom","context":["Auto","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_65","id":88064,"instance":65,"goal":"all m:Bool. m * 1 = m","type_family":"Bool","context":["Reduce","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_66","id":88065,"instance":66,"goal":"all z:List(Int). all m:List(Int). all q:List(Int). z*(m+q) = z*m + z*q","type_family":"List(Int)","context":["D -1","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_67","id":88066,"instance":67,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["BackThru","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_68","id":88067,"instance":68,"goal":"all c:List(Atom). all p:List(Atom). all q:List(Atom). (c <= p) => (p <= q) => (c <= q)","type_family":"List(Atom)","context":["D -1","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_69","id":88068,"instance":69,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["DoSubst","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_70","id":88069,"instance":70,"goal":"all n:Nat -> Nat. all a:Nat -> Nat. (n = a) => (a = n)","type_family":"Nat -> Nat","context":["Thin","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_71","id":88070,"instance":71,"goal":"(n) => (m) => (n)","type_family":"Int","context":["SplitOn","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_72","id":88071,"instance":72,"goal":"((n) & (m)) iff ((m) & (n))","type_family":"Nat","context":["Thin","DoSubst","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_73","id":88072,"instance":73,"goal":"((x) | (m)) iff ((m) | (x))","type_family":"Real","context":["Lemma","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_74","id":88073,"instance":74,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["DoSubst","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_75","id":88074,"instance":75,"goal":"all n:Bool. all f:Bool. all d:Bool. append(n, append(f, d)) = append(append(n, f), d)","type_family":"Bool","context":["Lemma","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_76","id":88075,"instance":76,"goal":"all b:List(Int). all c:List(Int). b + c = c + b","type_family":"List(Int)","context":["BackThru","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_77","id":88076,"instance":77,"goal":"all d:List(Nat). all w:List(Nat). all y:List(Nat). (d+w)+y = d+(w+y)","type_family":"List(Nat)","context":["D 0","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_78","id":88077,"instance":78,"goal":"all h:List(Atom). h + 0 = h","type_family":"List(Atom)","context":["D 1","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_79","id":88078,"instance":79,"goal":"all z:Int -> Int. all g:Int -> Int. z * g = g * z","type_family":"Int -> Int","context":["Auto","D 1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_80","id":88079,"instance":80,"goal":"all a:Nat -> Nat. a * 1 = a","type_family":"Nat -> Nat","context":["SplitOn","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_81","id":88080,"instance":81,"goal":"all g:Int. all p:Int. all z:Int. g*(p+z) = g*p + g*z","type_family":"Int","context":["DoSubst","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_82","id":88081,"instance":82,"goal":"all f:Nat. f <= f","type_family":"Nat","context":["Complete Auto","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_83","id":88082,"instance":83,"goal":"all u:Real. all h:Real. all n:Real. (u <= h) => (h <= n) => (u <= n)","type_family":"Real","context":["Assert","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_84","id":88083,"instance":84,"goal":"all c:Atom. c = c","type_family":"Atom","context":["RepUR","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_85","id":88084,"instance":85,"goal":"all w:Bool. all b:Bool. (w = b) => (b = w)","type_family":"Bool","context":["RepUR","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_86","id":88085,"instance":86,"goal":"(e) => (x) => (e)","type_family":"List(Int)","context":["D 0","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_87","id":88086,"instance":87,"goal":"((y) & (v)) iff ((v) & (y))","type_family":"List(Nat)","context":["Complete Auto","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_88","id":88087,"instance":88,"goal":"((d) | (h)) iff ((h) | (d))","type_family":"List(Atom)","context":["D 1","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_89","id":88088,"instance":89,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["BackThru","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_90","id":88089,"instance":90,"goal":"all psi:Nat -> Nat. all chi:Nat -> Nat. all beta:Nat -> Nat. append(psi, append(chi, beta)) = append(append(psi, chi), beta)","type_family":"Nat -> Nat","context":["D -1","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_91","id":88090,"instance":91,"goal":"all w:Int. all x:Int. w + x = x + w","type_family":"Int","context":["D -1","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_92","id":88091,"instance":92,"goal":"all p:Nat. all e:Nat. all z:Nat. (p+e)+z = p+(e+z)","type_family":"Nat","context":["Assert","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_93","id":88092,"instance":93,"goal":"all q:Real. q + 0 = q","type_family":"Real","context":["Lemma","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_94","id":88093,"instance":94,"goal":"all a:Atom. all m:Atom. a * m = m * a","type_family":"Atom","context":["InstLemma","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_95","id":88094,"instance":95,"goal":"all u:Bool. u * 1 = u","type_family":"Bool","context":["D 1","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_96","id":88095,"instance":96,"goal":"all u:List(Int). all p:List(Int). all v:List(Int). u*(p+v) = u*p + u*v","type_family":"List(Int)","context":["Auto","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_97","id":88096,"instance":97,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["Complete Auto","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_98","id":88097,"instance":98,"goal":"all c:List(Atom). all u:List(Atom). all g:List(Atom). (c <= u) => (u <= g) => (c <= g)","type_family":"List(Atom)","context":["Assert","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_99","id":88098,"instance":99,"goal":"all b:Int -> Int. b = b","type_family":"Int -> Int","context":["InstLemma","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_100","id":88099,"instance":100,"goal":"all alpha:Nat -> Nat. all delta:Nat -> Nat. (alpha = delta) => (delta = alpha)","type_family":"Nat -> Nat","context":["SplitOn","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_101","id":88100,"instance":101,"goal":"(b) => (h) => (b)","type_family":"Int","context":["D 1","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_102","id":88101,"instance":102,"goal":"((gamma) & (psi)) iff ((psi) & (gamma))","type_family":"Nat","context":["D 1","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_103","id":88102,"instance":103,"goal":"((w) | (c)) iff ((c) | (w))","type_family":"Real","context":["Thin","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_104","id":88103,"instance":104,"goal":"not (not (z)) => (z)","type_family":"Atom","context":["InstLemma","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_105","id":88104,"instance":105,"goal":"all u:Bool. all e:Bool. all n:Bool. append(u, append(e, n)) = append(append(u, e), n)","type_family":"Bool","context":["Lemma","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_106","id":88105,"instance":106,"goal":"all b:List(Int). all m:List(Int). b + m = m + b","type_family":"List(Int)","context":["Thin","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_107","id":88106,"instance":107,"goal":"all v:List(Nat). all n:List(Nat). all c:List(Nat). (v+n)+c = v+(n+c)","type_family":"List(Nat)","context":["Auto","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_108","id":88107,"instance":108,"goal":"all epsilon:List(Atom). epsilon + 0 = epsilon","type_family":"List(Atom)","context":["D 1","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_109","id":88108,"instance":109,"goal":"all a:Int -> Int. all g:Int -> Int. a * g = g * a","type_family":"Int -> Int","context":["RepUR","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_110","id":88109,"instance":110,"goal":"all u:Nat -> Nat. u * 1 = u","type_family":"Nat -> Nat","context":["Thin","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_111","id":88110,"instance":111,"goal":"all h:Int. all m:Int. all u:Int. h*(m+u) = h*m + h*u","type_family":"Int","context":["DoSubst","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_112","id":88111,"instance":112,"goal":"all a:Nat. a <= a","type_family":"Nat","context":["D -1","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_113","id":88112,"instance":113,"goal":"all f:Real. all u:Real. all d:Real. (f <= u) => (u <= d) => (f <= d)","type_family":"Real","context":["D 0","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_114","id":88113,"instance":114,"goal":"all x:Atom. x = x","type_family":"Atom","context":["Lemma","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_115","id":88114,"instance":115,"goal":"all y:Bool. all e:Bool. (y = e) => (e = y)","type_family":"Bool","context":["SplitOn","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_116","id":88115,"instance":116,"goal":"(b) => (p) => (b)","type_family":"List(Int)","context":["Complete Auto","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_117","id":88116,"instance":117,"goal":"((v) & (q)) iff ((q) & (v))","type_family":"List(Nat)","context":["Assert","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_118","id":88117,"instance":118,"goal":"((a) | (p)) iff ((p) | (a))","type_family":"List(Atom)","context":["D 1","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_119","id":88118,"instance":119,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["D 1","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_120","id":88119,"instance":120,"goal":"all h:Nat -> Nat. all q:Nat -> Nat. all m:Nat -> Nat. append(h, append(q, m)) = append(append(h, q), m)","type_family":"Nat -> Nat","context":["Thin","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_121","id":88120,"instance":121,"goal":"all gamma:Int. all alpha:Int. gamma + alpha = alpha + gamma","type_family":"Int","context":["RepUR","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_122","id":88121,"instance":122,"goal":"all b:Nat. all g:Nat. all e:Nat. (b+g)+e = b+(g+e)","type_family":"Nat","context":["RepUR","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_123","id":88122,"instance":123,"goal":"all u:Real. u + 0 = u","type_family":"Real","context":["DoSubst","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_124","id":88123,"instance":124,"goal":"all q:Atom. all z:Atom. q * z = z * q","type_family":"Atom","context":["Lemma","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_125","id":88124,"instance":125,"goal":"all psi:Bool. psi * 1 = psi","type_family":"Bool","context":["Thin","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_126","id":88125,"instance":126,"goal":"all f:List(Int). all w:List(Int). all m:List(Int). f*(w+m) = f*w + f*m","type_family":"List(Int)","context":["SplitOn","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_127","id":88126,"instance":127,"goal":"all a:List(Nat). a <= a","type_family":"List(Nat)","context":["InstLemma","SplitOn","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_128","id":88127,"instance":128,"goal":"all y:List(Atom). all p:List(Atom). all q:List(Atom). (y <= p) => (p <= q) => (y <= q)","type_family":"List(Atom)","context":["Assert","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_129","id":88128,"instance":129,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["DoSubst","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_130","id":88129,"instance":130,"goal":"all n:Nat -> Nat. all u:Nat -> Nat. (n = u) => (u = n)","type_family":"Nat -> Nat","context":["D -1","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_131","id":88130,"instance":131,"goal":"(x) => (e) => (x)","type_family":"Int","context":["BackThru","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_132","id":88131,"instance":132,"goal":"((h) & (w)) iff ((w) & (h))","type_family":"Nat","context":["SplitOn","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_133","id":88132,"instance":133,"goal":"((u) | (b)) iff ((b) | (u))","type_family":"Real","context":["RepUR","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_134","id":88133,"instance":134,"goal":"not (not (n)) => (n)","type_family":"Atom","context":["Thin","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_135","id":88134,"instance":135,"goal":"all u:Bool. all g:Bool. all h:Bool. append(u, append(g, h)) = append(append(u, g), h)","type_family":"Bool","context":["D 0","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_136","id":88135,"instance":136,"goal":"all c:List(Int). all g:List(Int). c + g = g + c","type_family":"List(Int)","context":["Reduce","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_137","id":88136,"instance":137,"goal":"all g:List(Nat). all p:List(Nat). all e:List(Nat). (g+p)+e = g+(p+e)","type_family":"List(Nat)","context":["Thin","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_138","id":88137,"instance":138,"goal":"all u:List(Atom). u + 0 = u","type_family":"List(Atom)","context":["InstLemma","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_139","id":88138,"instance":139,"goal":"all y:Int -> Int. all b:Int -> Int. y * b = b * y","type_family":"Int -> Int","context":["D 1","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_140","id":88139,"instance":140,"goal":"all n:Nat -> Nat. n * 1 = n","type_family":"Nat -> Nat","context":["BackThru","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_141","id":88140,"instance":141,"goal":"all b:Int. all n:Int. all e:Int. b*(n+e) = b*n + b*e","type_family":"Int","context":["DoSubst","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_142","id":88141,"instance":142,"goal":"all u:Nat. u <= u","type_family":"Nat","context":["Complete Auto","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_143","id":88142,"instance":143,"goal":"all h:Real. all c:Real. all x:Real. (h <= c) => (c <= x) => (h <= x)","type_family":"Real","context":["InstLemma","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_144","id":88143,"instance":144,"goal":"all v:Atom. v = v","type_family":"Atom","context":["D 1","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_145","id":88144,"instance":145,"goal":"all b:Bool. all z:Bool. (b = z) => (z = b)","type_family":"Bool","context":["InstLemma","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_146","id":88145,"instance":146,"goal":"(f) => (p) => (f)","type_family":"List(Int)","context":["Assert","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_147","id":88146,"instance":147,"goal":"((psi) & (delta)) iff ((delta) & (psi))","type_family":"List(Nat)","context":["Assert","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_148","id":88147,"instance":148,"goal":"((e) | (q)) iff ((q) | (e))","type_family":"List(Atom)","context":["D -1","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_149","id":88148,"instance":149,"goal":"not (not (z)) => (z)","type_family":"Int -> Int","context":["DoSubst","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_150","id":88149,"instance":150,"goal":"all c:Nat -> Nat. all n:Nat -> Nat. all g:Nat -> Nat. append(c, append(n, g)) = append(append(c, n), g)","type_family":"Nat -> Nat","context":["D 0","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_151","id":88150,"instance":151,"goal":"all w:Int. all y:Int. w + y = y + w","type_family":"Int","context":["D 1","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_152","id":88151,"instance":152,"goal":"all v:Nat. all f:Nat. all e:Nat. (v+f)+e = v+(f+e)","type_family":"Nat","context":["D 0","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_153","id":88152,"instance":153,"goal":"all d:Real. d + 0 = d","type_family":"Real","context":["Thin","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_154","id":88153,"instance":154,"goal":"all delta:Atom. all alpha:Atom. delta * alpha = alpha * delta","type_family":"Atom","context":["Complete Auto","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_155","id":88154,"instance":155,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["Thin","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_156","id":88155,"instance":156,"goal":"all u:List(Int). all m:List(Int). all v:List(Int). u*(m+v) = u*m + u*v","type_family":"List(Int)","context":["D -1","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_157","id":88156,"instance":157,"goal":"all g:List(Nat). g <= g","type_family":"List(Nat)","context":["Lemma","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_158","id":88157,"instance":158,"goal":"all e:List(Atom). all h:List(Atom). all m:List(Atom). (e <= h) => (h <= m) => (e <= m)","type_family":"List(Atom)","context":["BackThru","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_159","id":88158,"instance":159,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["Reduce","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_160","id":88159,"instance":160,"goal":"all g:Nat -> Nat. all z:Nat -> Nat. (g = z) => (z = g)","type_family":"Nat -> Nat","context":["D 0","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_161","id":88160,"instance":161,"goal":"(d) => (q) => (d)","type_family":"Int","context":["Reduce","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_162","id":88161,"instance":162,"goal":"((q) & (v)) iff ((v) & (q))","type_family":"Nat","context":["Lemma","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_163","id":88162,"instance":163,"goal":"((u) | (b)) iff ((b) | (u))","type_family":"Real","context":["Assert","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_164","id":88163,"instance":164,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["D -1","DoSubst","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_165","id":88164,"instance":165,"goal":"all m:Bool. all c:Bool. all v:Bool. append(m, append(c, v)) = append(append(m, c), v)","type_family":"Bool","context":["DoSubst","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_166","id":88165,"instance":166,"goal":"all y:List(Int). all e:List(Int). y + e = e + y","type_family":"List(Int)","context":["SplitOn","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_167","id":88166,"instance":167,"goal":"all f:List(Nat). all b:List(Nat). all n:List(Nat). (f+b)+n = f+(b+n)","type_family":"List(Nat)","context":["Assert","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_168","id":88167,"instance":168,"goal":"all e:List(Atom). e + 0 = e","type_family":"List(Atom)","context":["Complete Auto","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_169","id":88168,"instance":169,"goal":"all w:Int -> Int. all u:Int -> Int. w * u = u * w","type_family":"Int -> Int","context":["Reduce","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_170","id":88169,"instance":170,"goal":"all c:Nat -> Nat. c * 1 = c","type_family":"Nat -> Nat","context":["D 1","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_171","id":88170,"instance":171,"goal":"all y:Int. all u:Int. all e:Int. y*(u+e) = y*u + y*e","type_family":"Int","context":["Thin","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_172","id":88171,"instance":172,"goal":"all n:Nat. n <= n","type_family":"Nat","context":["D 0","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_173","id":88172,"instance":173,"goal":"all v:Real. all w:Real. all p:Real. (v <= w) => (w <= p) => (v <= p)","type_family":"Real","context":["Complete Auto","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_174","id":88173,"instance":174,"goal":"all z:Atom. z = z","type_family":"Atom","context":["BackThru","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_175","id":88174,"instance":175,"goal":"all m:Bool. all w:Bool. (m = w) => (w = m)","type_family":"Bool","context":["BackThru","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_176","id":88175,"instance":176,"goal":"(e) => (d) => (e)","type_family":"List(Int)","context":["SplitOn","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_177","id":88176,"instance":177,"goal":"((g) & (y)) iff ((y) & (g))","type_family":"List(Nat)","context":["Auto","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_178","id":88177,"instance":178,"goal":"((w) | (c)) iff ((c) | (w))","type_family":"List(Atom)","context":["D 1","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_179","id":88178,"instance":179,"goal":"not (not (p)) => (p)","type_family":"Int -> Int","context":["Reduce","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_180","id":88179,"instance":180,"goal":"all omega:Nat -> Nat. all delta:Nat -> Nat. all psi:Nat -> Nat. append(omega, append(delta, psi)) = append(append(omega, delta), psi)","type_family":"Nat -> Nat","context":["Thin","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_181","id":88180,"instance":181,"goal":"all y:Int. all e:Int. y + e = e + y","type_family":"Int","context":["Complete Auto","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_182","id":88181,"instance":182,"goal":"all d:Nat. all p:Nat. all n:Nat. (d+p)+n = d+(p+n)","type_family":"Nat","context":["D 0","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_183","id":88182,"instance":183,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["DoSubst","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_184","id":88183,"instance":184,"goal":"all z:Atom. all q:Atom. z * q = q * z","type_family":"Atom","context":["Reduce","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_185","id":88184,"instance":185,"goal":"all v:Bool. v * 1 = v","type_family":"Bool","context":["Lemma","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_186","id":88185,"instance":186,"goal":"all u:List(Int). all y:List(Int). all e:List(Int). u*(y+e) = u*y + u*e","type_family":"List(Int)","context":["SplitOn","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_187","id":88186,"instance":187,"goal":"all epsilon:List(Nat). epsilon <= epsilon","type_family":"List(Nat)","context":["Thin","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_188","id":88187,"instance":188,"goal":"all h:List(Atom). all f:List(Atom). all g:List(Atom). (h <= f) => (f <= g) => (h <= g)","type_family":"List(Atom)","context":["Complete Auto","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_189","id":88188,"instance":189,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["DoSubst","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_190","id":88189,"instance":190,"goal":"all u:Nat -> Nat. all y:Nat -> Nat. (u = y) => (y = u)","type_family":"Nat -> Nat","context":["D 1","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_191","id":88190,"instance":191,"goal":"(y) => (z) => (y)","type_family":"Int","context":["InstLemma","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_192","id":88191,"instance":192,"goal":"((y) & (g)) iff ((g) & (y))","type_family":"Nat","context":["BackThru","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_193","id":88192,"instance":193,"goal":"((h) | (u)) iff ((u) | (h))","type_family":"Real","context":["Assert","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_194","id":88193,"instance":194,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Auto","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_195","id":88194,"instance":195,"goal":"all z:Bool. all e:Bool. all f:Bool. append(z, append(e, f)) = append(append(z, e), f)","type_family":"Bool","context":["RepUR","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_196","id":88195,"instance":196,"goal":"all v:List(Int). all d:List(Int). v + d = d + v","type_family":"List(Int)","context":["Lemma","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_197","id":88196,"instance":197,"goal":"all u:List(Nat). all x:List(Nat). all c:List(Nat). (u+x)+c = u+(x+c)","type_family":"List(Nat)","context":["SplitOn","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_198","id":88197,"instance":198,"goal":"all g:List(Atom). g + 0 = g","type_family":"List(Atom)","context":["DoSubst","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_199","id":88198,"instance":199,"goal":"all q:Int -> Int. all v:Int -> Int. q * v = v * q","type_family":"Int -> Int","context":["Thin","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_200","id":88199,"instance":200,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["Reduce","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_201","id":88200,"instance":201,"goal":"all m:Int. all y:Int. all v:Int. m*(y+v) = m*y + m*v","type_family":"Int","context":["D 1","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_202","id":88201,"instance":202,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["SplitOn","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_203","id":88202,"instance":203,"goal":"all z:Real. all e:Real. all u:Real. (z <= e) => (e <= u) => (z <= u)","type_family":"Real","context":["Auto","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_204","id":88203,"instance":204,"goal":"all psi:Atom. psi = psi","type_family":"Atom","context":["InstLemma","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_205","id":88204,"instance":205,"goal":"all z:Bool. all h:Bool. (z = h) => (h = z)","type_family":"Bool","context":["D -1","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_206","id":88205,"instance":206,"goal":"(e) => (p) => (e)","type_family":"List(Int)","context":["D -1","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_207","id":88206,"instance":207,"goal":"((c) & (p)) iff ((p) & (c))","type_family":"List(Nat)","context":["Complete Auto","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_208","id":88207,"instance":208,"goal":"((f) | (e)) iff ((e) | (f))","type_family":"List(Atom)","context":["D -1","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_209","id":88208,"instance":209,"goal":"not (not (e)) => (e)","type_family":"Int -> Int","context":["Assert","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_210","id":88209,"instance":210,"goal":"all m:Nat -> Nat. all v:Nat -> Nat. all w:Nat -> Nat. append(m, append(v, w)) = append(append(m, v), w)","type_family":"Nat -> Nat","context":["D 1","DoSubst","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_211","id":88210,"instance":211,"goal":"all w:Int. all e:Int. w + e = e + w","type_family":"Int","context":["RepUR","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_212","id":88211,"instance":212,"goal":"all m:Nat. all f:Nat. all c:Nat. (m+f)+c = m+(f+c)","type_family":"Nat","context":["Reduce","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_213","id":88212,"instance":213,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["Complete Auto","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_214","id":88213,"instance":214,"goal":"all z:Atom. all u:Atom. z * u = u * z","type_family":"Atom","context":["InstLemma","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_215","id":88214,"instance":215,"goal":"all v:Bool. v * 1 = v","type_family":"Bool","context":["Auto","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_216","id":88215,"instance":216,"goal":"all g:List(Int). all e:List(Int). all w:List(Int). g*(e+w) = g*e + g*w","type_family":"List(Int)","context":["Lemma","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_217","id":88216,"instance":217,"goal":"all v:List(Nat). v <= v","type_family":"List(Nat)","context":["Reduce","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_218","id":88217,"instance":218,"goal":"all gamma:List(Atom). all kappa:List(Atom). all epsilon:List(Atom). (gamma <= kappa) => (kappa <= epsilon) => (gamma <= epsilon)","type_family":"List(Atom)","context":["Thin","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_219","id":88218,"instance":219,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["Reduce","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_220","id":88219,"instance":220,"goal":"all c:Nat -> Nat. all a:Nat -> Nat. (c = a) => (a = c)","type_family":"Nat -> Nat","context":["Complete Auto","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_221","id":88220,"instance":221,"goal":"(q) => (p) => (q)","type_family":"Int","context":["SplitOn","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_222","id":88221,"instance":222,"goal":"((u) & (y)) iff ((y) & (u))","type_family":"Nat","context":["Assert","D -1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_223","id":88222,"instance":223,"goal":"((u) | (p)) iff ((p) | (u))","type_family":"Real","context":["Thin","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_224","id":88223,"instance":224,"goal":"not (not (phi)) => (phi)","type_family":"Atom","context":["D -1","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_225","id":88224,"instance":225,"goal":"all x:Bool. all h:Bool. all v:Bool. append(x, append(h, v)) = append(append(x, h), v)","type_family":"Bool","context":["BackThru","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_226","id":88225,"instance":226,"goal":"all d:List(Int). all m:List(Int). d + m = m + d","type_family":"List(Int)","context":["D 0","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_227","id":88226,"instance":227,"goal":"all omega:List(Nat). all gamma:List(Nat). all epsilon:List(Nat). (omega+gamma)+epsilon = omega+(gamma+epsilon)","type_family":"List(Nat)","context":["Lemma","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_228","id":88227,"instance":228,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["Reduce","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_229","id":88228,"instance":229,"goal":"all n:Int -> Int. all h:Int -> Int. n * h = h * n","type_family":"Int -> Int","context":["Complete Auto","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_230","id":88229,"instance":230,"goal":"all w:Nat -> Nat. w * 1 = w","type_family":"Nat -> Nat","context":["Assert","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_231","id":88230,"instance":231,"goal":"all a:Int. all f:Int. all n:Int. a*(f+n) = a*f + a*n","type_family":"Int","context":["Reduce","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_232","id":88231,"instance":232,"goal":"all u:Nat. u <= u","type_family":"Nat","context":["D 1","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_233","id":88232,"instance":233,"goal":"all x:Real. all a:Real. all h:Real. (x <= a) => (a <= h) => (x <= h)","type_family":"Real","context":["Lemma","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_234","id":88233,"instance":234,"goal":"all q:Atom. q = q","type_family":"Atom","context":["Thin","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_235","id":88234,"instance":235,"goal":"all d:Bool. all u:Bool. (d = u) => (u = d)","type_family":"Bool","context":["Complete Auto","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_236","id":88235,"instance":236,"goal":"(d) => (b) => (d)","type_family":"List(Int)","context":["Lemma","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_237","id":88236,"instance":237,"goal":"((a) & (w)) iff ((w) & (a))","type_family":"List(Nat)","context":["Lemma","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_238","id":88237,"instance":238,"goal":"((g) | (e)) iff ((e) | (g))","type_family":"List(Atom)","context":["DoSubst","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_239","id":88238,"instance":239,"goal":"not (not (q)) => (q)","type_family":"Int -> Int","context":["D -1","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_240","id":88239,"instance":240,"goal":"all chi:Nat -> Nat. all gamma:Nat -> Nat. all kappa:Nat -> Nat. append(chi, append(gamma, kappa)) = append(append(chi, gamma), kappa)","type_family":"Nat -> Nat","context":["Auto","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_241","id":88240,"instance":241,"goal":"all e:Int. all x:Int. e + x = x + e","type_family":"Int","context":["D -1","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_242","id":88241,"instance":242,"goal":"all z:Nat. all g:Nat. all p:Nat. (z+g)+p = z+(g+p)","type_family":"Nat","context":["D 0","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_243","id":88242,"instance":243,"goal":"all alpha:Real. alpha + 0 = alpha","type_family":"Real","context":["BackThru","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; DoSubst; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_244","id":88243,"instance":244,"goal":"all b:Atom. all w:Atom. b * w = w * b","type_family":"Atom","context":["Assert","DoSubst","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_245","id":88244,"instance":245,"goal":"all p:Bool. p * 1 = p","type_family":"Bool","context":["Thin","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_246","id":88245,"instance":246,"goal":"all n:List(Int). all h:List(Int). all z:List(Int). n*(h+z) = n*h + n*z","type_family":"List(Int)","context":["DoSubst","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_247","id":88246,"instance":247,"goal":"all x:List(Nat). x <= x","type_family":"List(Nat)","context":["Reduce","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_248","id":88247,"instance":248,"goal":"all h:List(Atom). all f:List(Atom). all b:List(Atom). (h <= f) => (f <= b) => (h <= b)","type_family":"List(Atom)","context":["DoSubst","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_249","id":88248,"instance":249,"goal":"all phi:Int -> Int. phi = phi","type_family":"Int -> Int","context":["Thin","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_250","id":88249,"instance":250,"goal":"all m:Nat -> Nat. all b:Nat -> Nat. (m = b) => (b = m)","type_family":"Nat -> Nat","context":["SplitOn","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_251","id":88250,"instance":251,"goal":"(d) => (x) => (d)","type_family":"Int","context":["BackThru","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_252","id":88251,"instance":252,"goal":"((a) & (v)) iff ((v) & (a))","type_family":"Nat","context":["RepUR","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_253","id":88252,"instance":253,"goal":"((f) | (h)) iff ((h) | (f))","type_family":"Real","context":["Thin","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_254","id":88253,"instance":254,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["DoSubst","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_255","id":88254,"instance":255,"goal":"all z:Bool. all q:Bool. all n:Bool. append(z, append(q, n)) = append(append(z, q), n)","type_family":"Bool","context":["D 1","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_256","id":88255,"instance":256,"goal":"all z:List(Int). all g:List(Int). z + g = g + z","type_family":"List(Int)","context":["D 0","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_257","id":88256,"instance":257,"goal":"all f:List(Nat). all h:List(Nat). all w:List(Nat). (f+h)+w = f+(h+w)","type_family":"List(Nat)","context":["Lemma","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_258","id":88257,"instance":258,"goal":"all h:List(Atom). h + 0 = h","type_family":"List(Atom)","context":["Thin","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_259","id":88258,"instance":259,"goal":"all v:Int -> Int. all f:Int -> Int. v * f = f * v","type_family":"Int -> Int","context":["SplitOn","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_260","id":88259,"instance":260,"goal":"all m:Nat -> Nat. m * 1 = m","type_family":"Nat -> Nat","context":["Thin","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_261","id":88260,"instance":261,"goal":"all beta:Int. all gamma:Int. all phi:Int. beta*(gamma+phi) = beta*gamma + beta*phi","type_family":"Int","context":["D 1","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_262","id":88261,"instance":262,"goal":"all c:Nat. c <= c","type_family":"Nat","context":["Assert","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_263","id":88262,"instance":263,"goal":"all d:Real. all h:Real. all c:Real. (d <= h) => (h <= c) => (d <= c)","type_family":"Real","context":["DoSubst","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_264","id":88263,"instance":264,"goal":"all d:Atom. d = d","type_family":"Atom","context":["Thin","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_265","id":88264,"instance":265,"goal":"all q:Bool. all h:Bool. (q = h) => (h = q)","type_family":"Bool","context":["D -1","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_266","id":88265,"instance":266,"goal":"(u) => (d) => (u)","type_family":"List(Int)","context":["D 0","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_267","id":88266,"instance":267,"goal":"((epsilon) & (alpha)) iff ((alpha) & (epsilon))","type_family":"List(Nat)","context":["Complete Auto","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_268","id":88267,"instance":268,"goal":"((f) | (z)) iff ((z) | (f))","type_family":"List(Atom)","context":["D 0","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_269","id":88268,"instance":269,"goal":"not (not (z)) => (z)","type_family":"Int -> Int","context":["SplitOn","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_270","id":88269,"instance":270,"goal":"all h:Nat -> Nat. all b:Nat -> Nat. all z:Nat -> Nat. append(h, append(b, z)) = append(append(h, b), z)","type_family":"Nat -> Nat","context":["Reduce","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_271","id":88270,"instance":271,"goal":"all f:Int. all q:Int. f + q = q + f","type_family":"Int","context":["Thin","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_272","id":88271,"instance":272,"goal":"all a:Nat. all h:Nat. all m:Nat. (a+h)+m = a+(h+m)","type_family":"Nat","context":["DoSubst","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_273","id":88272,"instance":273,"goal":"all y:Real. y + 0 = y","type_family":"Real","context":["Reduce","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_274","id":88273,"instance":274,"goal":"all w:Atom. all c:Atom. w * c = c * w","type_family":"Atom","context":["Lemma","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_275","id":88274,"instance":275,"goal":"all g:Bool. g * 1 = g","type_family":"Bool","context":["BackThru","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_276","id":88275,"instance":276,"goal":"all n:List(Int). all c:List(Int). all v:List(Int). n*(c+v) = n*c + n*v","type_family":"List(Int)","context":["Thin","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_277","id":88276,"instance":277,"goal":"all kappa:List(Nat). kappa <= kappa","type_family":"List(Nat)","context":["D -1","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_278","id":88277,"instance":278,"goal":"all n:List(Atom). all u:List(Atom). all z:List(Atom). (n <= u) => (u <= z) => (n <= z)","type_family":"List(Atom)","context":["BackThru","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_279","id":88278,"instance":279,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["Reduce","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_280","id":88279,"instance":280,"goal":"all p:Nat -> Nat. all f:Nat -> Nat. (p = f) => (f = p)","type_family":"Nat -> Nat","context":["InstLemma","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_281","id":88280,"instance":281,"goal":"(u) => (c) => (u)","type_family":"Int","context":["BackThru","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_282","id":88281,"instance":282,"goal":"((z) & (b)) iff ((b) & (z))","type_family":"Nat","context":["D 0","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_283","id":88282,"instance":283,"goal":"((z) | (y)) iff ((y) | (z))","type_family":"Real","context":["D -1","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_284","id":88283,"instance":284,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["InstLemma","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_285","id":88284,"instance":285,"goal":"all x:Bool. all a:Bool. all e:Bool. append(x, append(a, e)) = append(append(x, a), e)","type_family":"Bool","context":["Assert","InstLemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_286","id":88285,"instance":286,"goal":"all u:List(Int). all n:List(Int). u + n = n + u","type_family":"List(Int)","context":["D 1","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_287","id":88286,"instance":287,"goal":"all q:List(Nat). all d:List(Nat). all z:List(Nat). (q+d)+z = q+(d+z)","type_family":"List(Nat)","context":["Lemma","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_288","id":88287,"instance":288,"goal":"all c:List(Atom). c + 0 = c","type_family":"List(Atom)","context":["Assert","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_289","id":88288,"instance":289,"goal":"all f:Int -> Int. all a:Int -> Int. f * a = a * f","type_family":"Int -> Int","context":["Auto","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_290","id":88289,"instance":290,"goal":"all q:Nat -> Nat. q * 1 = q","type_family":"Nat -> Nat","context":["RepUR","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_291","id":88290,"instance":291,"goal":"all d:Int. all a:Int. all c:Int. d*(a+c) = d*a + d*c","type_family":"Int","context":["InstLemma","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_292","id":88291,"instance":292,"goal":"all q:Nat. q <= q","type_family":"Nat","context":["BackThru","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_293","id":88292,"instance":293,"goal":"all q:Real. all n:Real. all d:Real. (q <= n) => (n <= d) => (q <= d)","type_family":"Real","context":["DoSubst","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_294","id":88293,"instance":294,"goal":"all omega:Atom. omega = omega","type_family":"Atom","context":["D 1","DoSubst","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_295","id":88294,"instance":295,"goal":"all f:Bool. all d:Bool. (f = d) => (d = f)","type_family":"Bool","context":["SplitOn","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_296","id":88295,"instance":296,"goal":"(p) => (y) => (p)","type_family":"List(Int)","context":["Lemma","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_297","id":88296,"instance":297,"goal":"((e) & (q)) iff ((q) & (e))","type_family":"List(Nat)","context":["SplitOn","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_298","id":88297,"instance":298,"goal":"((x) | (w)) iff ((w) | (x))","type_family":"List(Atom)","context":["D -1","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_299","id":88298,"instance":299,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["D -1","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_300","id":88299,"instance":300,"goal":"all b:Nat -> Nat. all d:Nat -> Nat. all x:Nat -> Nat. append(b, append(d, x)) = append(append(b, d), x)","type_family":"Nat -> Nat","context":["Complete Auto","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_301","id":88300,"instance":301,"goal":"all q:Int. all f:Int. q + f = f + q","type_family":"Int","context":["Thin","D -1","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_302","id":88301,"instance":302,"goal":"all e:Nat. all w:Nat. all v:Nat. (e+w)+v = e+(w+v)","type_family":"Nat","context":["DoSubst","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_303","id":88302,"instance":303,"goal":"all p:Real. p + 0 = p","type_family":"Real","context":["InstLemma","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_304","id":88303,"instance":304,"goal":"all p:Atom. all z:Atom. p * z = z * p","type_family":"Atom","context":["D 0","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_305","id":88304,"instance":305,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["Lemma","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_306","id":88305,"instance":306,"goal":"all e:List(Int). all f:List(Int). all d:List(Int). e*(f+d) = e*f + e*d","type_family":"List(Int)","context":["SplitOn","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_307","id":88306,"instance":307,"goal":"all x:List(Nat). x <= x","type_family":"List(Nat)","context":["SplitOn","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_308","id":88307,"instance":308,"goal":"all d:List(Atom). all v:List(Atom). all b:List(Atom). (d <= v) => (v <= b) => (d <= b)","type_family":"List(Atom)","context":["SplitOn","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_309","id":88308,"instance":309,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["D -1","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_310","id":88309,"instance":310,"goal":"all d:Nat -> Nat. all n:Nat -> Nat. (d = n) => (n = d)","type_family":"Nat -> Nat","context":["Auto","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_311","id":88310,"instance":311,"goal":"(c) => (a) => (c)","type_family":"Int","context":["InstLemma","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_312","id":88311,"instance":312,"goal":"((z) & (n)) iff ((n) & (z))","type_family":"Nat","context":["Assert","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_313","id":88312,"instance":313,"goal":"((delta) | (psi)) iff ((psi) | (delta))","type_family":"Real","context":["D 0","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_314","id":88313,"instance":314,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["D 1","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_315","id":88314,"instance":315,"goal":"all psi:Bool. all delta:Bool. all omega:Bool. append(psi, append(delta, omega)) = append(append(psi, delta), omega)","type_family":"Bool","context":["SplitOn","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_316","id":88315,"instance":316,"goal":"all y:List(Int). all w:List(Int). y + w = w + y","type_family":"List(Int)","context":["Complete Auto","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_317","id":88316,"instance":317,"goal":"all c:List(Nat). all e:List(Nat). all v:List(Nat). (c+e)+v = c+(e+v)","type_family":"List(Nat)","context":["SplitOn","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_318","id":88317,"instance":318,"goal":"all h:List(Atom). h + 0 = h","type_family":"List(Atom)","context":["InstLemma","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_319","id":88318,"instance":319,"goal":"all e:Int -> Int. all u:Int -> Int. e * u = u * e","type_family":"Int -> Int","context":["Assert","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_320","id":88319,"instance":320,"goal":"all m:Nat -> Nat. m * 1 = m","type_family":"Nat -> Nat","context":["D 1","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_321","id":88320,"instance":321,"goal":"all gamma:Int. all phi:Int. all beta:Int. gamma*(phi+beta) = gamma*phi + gamma*beta","type_family":"Int","context":["Reduce","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_322","id":88321,"instance":322,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["BackThru","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_323","id":88322,"instance":323,"goal":"all d:Real. all u:Real. all a:Real. (d <= u) => (u <= a) => (d <= a)","type_family":"Real","context":["D -1","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_324","id":88323,"instance":324,"goal":"all h:Atom. h = h","type_family":"Atom","context":["RepUR","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_325","id":88324,"instance":325,"goal":"all kappa:Bool. all psi:Bool. (kappa = psi) => (psi = kappa)","type_family":"Bool","context":["Thin","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_326","id":88325,"instance":326,"goal":"(c) => (u) => (c)","type_family":"List(Int)","context":["Auto","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_327","id":88326,"instance":327,"goal":"((a) & (p)) iff ((p) & (a))","type_family":"List(Nat)","context":["Thin","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_328","id":88327,"instance":328,"goal":"((h) | (p)) iff ((p) | (h))","type_family":"List(Atom)","context":["InstLemma","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_329","id":88328,"instance":329,"goal":"not (not (h)) => (h)","type_family":"Int -> Int","context":["Reduce","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_330","id":88329,"instance":330,"goal":"all e:Nat -> Nat. all m:Nat -> Nat. all a:Nat -> Nat. append(e, append(m, a)) = append(append(e, m), a)","type_family":"Nat -> Nat","context":["Reduce","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_331","id":88330,"instance":331,"goal":"all d:Int. all c:Int. d + c = c + d","type_family":"Int","context":["Lemma","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_332","id":88331,"instance":332,"goal":"all m:Nat. all g:Nat. all b:Nat. (m+g)+b = m+(g+b)","type_family":"Nat","context":["SplitOn","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_333","id":88332,"instance":333,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["Reduce","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_334","id":88333,"instance":334,"goal":"all m:Atom. all p:Atom. m * p = p * m","type_family":"Atom","context":["D -1","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_335","id":88334,"instance":335,"goal":"all u:Bool. u * 1 = u","type_family":"Bool","context":["D 1","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_336","id":88335,"instance":336,"goal":"all u:List(Int). all b:List(Int). all n:List(Int). u*(b+n) = u*b + u*n","type_family":"List(Int)","context":["RepUR","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_337","id":88336,"instance":337,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["Lemma","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_338","id":88337,"instance":338,"goal":"all beta:List(Atom). all epsilon:List(Atom). all psi:List(Atom). (beta <= epsilon) => (epsilon <= psi) => (beta <= psi)","type_family":"List(Atom)","context":["Assert","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_339","id":88338,"instance":339,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["Assert","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_340","id":88339,"instance":340,"goal":"all z:Nat -> Nat. all c:Nat -> Nat. (z = c) => (c = z)","type_family":"Nat -> Nat","context":["SplitOn","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_341","id":88340,"instance":341,"goal":"(e) => (z) => (e)","type_family":"Int","context":["D 0","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_342","id":88341,"instance":342,"goal":"((b) & (d)) iff ((d) & (b))","type_family":"Nat","context":["D -1","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_343","id":88342,"instance":343,"goal":"((q) | (p)) iff ((p) | (q))","type_family":"Real","context":["D 1","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_344","id":88343,"instance":344,"goal":"not (not (z)) => (z)","type_family":"Atom","context":["D 0","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_345","id":88344,"instance":345,"goal":"all z:Bool. all m:Bool. all b:Bool. append(z, append(m, b)) = append(append(z, m), b)","type_family":"Bool","context":["InstLemma","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_346","id":88345,"instance":346,"goal":"all w:List(Int). all y:List(Int). w + y = y + w","type_family":"List(Int)","context":["D 0","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_347","id":88346,"instance":347,"goal":"all f:List(Nat). all h:List(Nat). all m:List(Nat). (f+h)+m = f+(h+m)","type_family":"List(Nat)","context":["SplitOn","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_348","id":88347,"instance":348,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["Auto","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_349","id":88348,"instance":349,"goal":"all g:Int -> Int. all z:Int -> Int. g * z = z * g","type_family":"Int -> Int","context":["Thin","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_350","id":88349,"instance":350,"goal":"all w:Nat -> Nat. w * 1 = w","type_family":"Nat -> Nat","context":["Complete Auto","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_351","id":88350,"instance":351,"goal":"all x:Int. all e:Int. all f:Int. x*(e+f) = x*e + x*f","type_family":"Int","context":["DoSubst","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_352","id":88351,"instance":352,"goal":"all w:Nat. w <= w","type_family":"Nat","context":["DoSubst","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_353","id":88352,"instance":353,"goal":"all d:Real. all f:Real. all m:Real. (d <= f) => (f <= m) => (d <= m)","type_family":"Real","context":["D -1","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_354","id":88353,"instance":354,"goal":"all q:Atom. q = q","type_family":"Atom","context":["DoSubst","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_355","id":88354,"instance":355,"goal":"all d:Bool. all e:Bool. (d = e) => (e = d)","type_family":"Bool","context":["BackThru","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_356","id":88355,"instance":356,"goal":"(x) => (a) => (x)","type_family":"List(Int)","context":["Lemma","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_357","id":88356,"instance":357,"goal":"((n) & (w)) iff ((w) & (n))","type_family":"List(Nat)","context":["Assert","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_358","id":88357,"instance":358,"goal":"((p) | (q)) iff ((q) | (p))","type_family":"List(Atom)","context":["Auto","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_359","id":88358,"instance":359,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["DoSubst","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_360","id":88359,"instance":360,"goal":"all z:Nat -> Nat. all e:Nat -> Nat. all d:Nat -> Nat. append(z, append(e, d)) = append(append(z, e), d)","type_family":"Nat -> Nat","context":["BackThru","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_361","id":88360,"instance":361,"goal":"all z:Int. all d:Int. z + d = d + z","type_family":"Int","context":["Thin","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_362","id":88361,"instance":362,"goal":"all v:Nat. all a:Nat. all y:Nat. (v+a)+y = v+(a+y)","type_family":"Nat","context":["D 1","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_363","id":88362,"instance":363,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["Assert","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_364","id":88363,"instance":364,"goal":"all d:Atom. all n:Atom. d * n = n * d","type_family":"Atom","context":["D -1","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_365","id":88364,"instance":365,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["RepUR","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_366","id":88365,"instance":366,"goal":"all g:List(Int). all v:List(Int). all b:List(Int). g*(v+b) = g*v + g*b","type_family":"List(Int)","context":["Thin","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_367","id":88366,"instance":367,"goal":"all omega:List(Nat). omega <= omega","type_family":"List(Nat)","context":["Complete Auto","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_368","id":88367,"instance":368,"goal":"all alpha:List(Atom). all delta:List(Atom). all epsilon:List(Atom). (alpha <= delta) => (delta <= epsilon) => (alpha <= epsilon)","type_family":"List(Atom)","context":["BackThru","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_369","id":88368,"instance":369,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["Thin","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_370","id":88369,"instance":370,"goal":"all z:Nat -> Nat. all b:Nat -> Nat. (z = b) => (b = z)","type_family":"Nat -> Nat","context":["InstLemma","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_371","id":88370,"instance":371,"goal":"(z) => (q) => (z)","type_family":"Int","context":["Lemma","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_372","id":88371,"instance":372,"goal":"((n) & (y)) iff ((y) & (n))","type_family":"Nat","context":["Reduce","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_373","id":88372,"instance":373,"goal":"((c) | (a)) iff ((a) | (c))","type_family":"Real","context":["Thin","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_374","id":88373,"instance":374,"goal":"not (not (y)) => (y)","type_family":"Atom","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_375","id":88374,"instance":375,"goal":"all g:Bool. all a:Bool. all m:Bool. append(g, append(a, m)) = append(append(g, a), m)","type_family":"Bool","context":["Thin","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_376","id":88375,"instance":376,"goal":"all y:List(Int). all b:List(Int). y + b = b + y","type_family":"List(Int)","context":["D -1","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_377","id":88376,"instance":377,"goal":"all w:List(Nat). all m:List(Nat). all x:List(Nat). (w+m)+x = w+(m+x)","type_family":"List(Nat)","context":["D -1","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_378","id":88377,"instance":378,"goal":"all w:List(Atom). w + 0 = w","type_family":"List(Atom)","context":["Thin","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_379","id":88378,"instance":379,"goal":"all c:Int -> Int. all x:Int -> Int. c * x = x * c","type_family":"Int -> Int","context":["InstLemma","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_380","id":88379,"instance":380,"goal":"all y:Nat -> Nat. y * 1 = y","type_family":"Nat -> Nat","context":["SplitOn","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_381","id":88380,"instance":381,"goal":"all c:Int. all f:Int. all w:Int. c*(f+w) = c*f + c*w","type_family":"Int","context":["InstLemma","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_382","id":88381,"instance":382,"goal":"all psi:Nat. psi <= psi","type_family":"Nat","context":["Assert","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_383","id":88382,"instance":383,"goal":"all q:Real. all d:Real. all f:Real. (q <= d) => (d <= f) => (q <= f)","type_family":"Real","context":["D 1","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_384","id":88383,"instance":384,"goal":"all w:Atom. w = w","type_family":"Atom","context":["Reduce","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_385","id":88384,"instance":385,"goal":"all h:Bool. all v:Bool. (h = v) => (v = h)","type_family":"Bool","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_386","id":88385,"instance":386,"goal":"(psi) => (phi) => (psi)","type_family":"List(Int)","context":["DoSubst","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_387","id":88386,"instance":387,"goal":"((x) & (u)) iff ((u) & (x))","type_family":"List(Nat)","context":["Reduce","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_388","id":88387,"instance":388,"goal":"((c) | (y)) iff ((y) | (c))","type_family":"List(Atom)","context":["Complete Auto","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_389","id":88388,"instance":389,"goal":"not (not (p)) => (p)","type_family":"Int -> Int","context":["Assert","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_390","id":88389,"instance":390,"goal":"all b:Nat -> Nat. all p:Nat -> Nat. all m:Nat -> Nat. append(b, append(p, m)) = append(append(b, p), m)","type_family":"Nat -> Nat","context":["D -1","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_391","id":88390,"instance":391,"goal":"all f:Int. all u:Int. f + u = u + f","type_family":"Int","context":["InstLemma","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_392","id":88391,"instance":392,"goal":"all x:Nat. all z:Nat. all f:Nat. (x+z)+f = x+(z+f)","type_family":"Nat","context":["Thin","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_393","id":88392,"instance":393,"goal":"all x:Real. x + 0 = x","type_family":"Real","context":["D 1","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_394","id":88393,"instance":394,"goal":"all v:Atom. all u:Atom. v * u = u * v","type_family":"Atom","context":["BackThru","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_395","id":88394,"instance":395,"goal":"all b:Bool. b * 1 = b","type_family":"Bool","context":["D -1","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_396","id":88395,"instance":396,"goal":"all v:List(Int). all m:List(Int). all c:List(Int). v*(m+c) = v*m + v*c","type_family":"List(Int)","context":["InstLemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_397","id":88396,"instance":397,"goal":"all q:List(Nat). q <= q","type_family":"List(Nat)","context":["Auto","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_398","id":88397,"instance":398,"goal":"all w:List(Atom). all a:List(Atom). all e:List(Atom). (w <= a) => (a <= e) => (w <= e)","type_family":"List(Atom)","context":["BackThru","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_399","id":88398,"instance":399,"goal":"all b:Int -> Int. b = b","type_family":"Int -> Int","context":["Thin","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_400","id":88399,"instance":400,"goal":"all h:Nat -> Nat. all w:Nat -> Nat. (h = w) => (w = h)","type_family":"Nat -> Nat","context":["D -1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_401","id":88400,"instance":401,"goal":"(f) => (d) => (f)","type_family":"Int","context":["D 0","InstLemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_402","id":88401,"instance":402,"goal":"((w) & (u)) iff ((u) & (w))","type_family":"Nat","context":["Lemma","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_403","id":88402,"instance":403,"goal":"((epsilon) | (phi)) iff ((phi) | (epsilon))","type_family":"Real","context":["Auto","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_404","id":88403,"instance":404,"goal":"not (not (v)) => (v)","type_family":"Atom","context":["BackThru","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_405","id":88404,"instance":405,"goal":"all h:Bool. all v:Bool. all f:Bool. append(h, append(v, f)) = append(append(h, v), f)","type_family":"Bool","context":["SplitOn","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_406","id":88405,"instance":406,"goal":"all delta:List(Int). all omega:List(Int). delta + omega = omega + delta","type_family":"List(Int)","context":["DoSubst","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_407","id":88406,"instance":407,"goal":"all c:List(Nat). all v:List(Nat). all z:List(Nat). (c+v)+z = c+(v+z)","type_family":"List(Nat)","context":["InstLemma","InstLemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_408","id":88407,"instance":408,"goal":"all v:List(Atom). v + 0 = v","type_family":"List(Atom)","context":["Lemma","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_409","id":88408,"instance":409,"goal":"all v:Int -> Int. all d:Int -> Int. v * d = d * v","type_family":"Int -> Int","context":["Thin","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_410","id":88409,"instance":410,"goal":"all n:Nat -> Nat. n * 1 = n","type_family":"Nat -> Nat","context":["DoSubst","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_411","id":88410,"instance":411,"goal":"all b:Int. all m:Int. all w:Int. b*(m+w) = b*m + b*w","type_family":"Int","context":["Thin","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_412","id":88411,"instance":412,"goal":"all w:Nat. w <= w","type_family":"Nat","context":["DoSubst","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_413","id":88412,"instance":413,"goal":"all h:Real. all z:Real. all b:Real. (h <= z) => (z <= b) => (h <= b)","type_family":"Real","context":["RepUR","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_414","id":88413,"instance":414,"goal":"all beta:Atom. beta = beta","type_family":"Atom","context":["Reduce","DoSubst","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_415","id":88414,"instance":415,"goal":"all f:Bool. all x:Bool. (f = x) => (x = f)","type_family":"Bool","context":["SplitOn","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_416","id":88415,"instance":416,"goal":"(alpha) => (epsilon) => (alpha)","type_family":"List(Int)","context":["Assert","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_417","id":88416,"instance":417,"goal":"((y) & (v)) iff ((v) & (y))","type_family":"List(Nat)","context":["Thin","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_418","id":88417,"instance":418,"goal":"((c) | (z)) iff ((z) | (c))","type_family":"List(Atom)","context":["Auto","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_419","id":88418,"instance":419,"goal":"not (not (b)) => (b)","type_family":"Int -> Int","context":["Lemma","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_420","id":88419,"instance":420,"goal":"all c:Nat -> Nat. all q:Nat -> Nat. all n:Nat -> Nat. append(c, append(q, n)) = append(append(c, q), n)","type_family":"Nat -> Nat","context":["DoSubst","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_421","id":88420,"instance":421,"goal":"all b:Int. all e:Int. b + e = e + b","type_family":"Int","context":["BackThru","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_422","id":88421,"instance":422,"goal":"all x:Nat. all f:Nat. all g:Nat. (x+f)+g = x+(f+g)","type_family":"Nat","context":["Auto","InstLemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_423","id":88422,"instance":423,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["D 0","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_424","id":88423,"instance":424,"goal":"all e:Atom. all d:Atom. e * d = d * e","type_family":"Atom","context":["DoSubst","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_425","id":88424,"instance":425,"goal":"all g:Bool. g * 1 = g","type_family":"Bool","context":["SplitOn","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_426","id":88425,"instance":426,"goal":"all m:List(Int). all z:List(Int). all g:List(Int). m*(z+g) = m*z + m*g","type_family":"List(Int)","context":["D 0","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_427","id":88426,"instance":427,"goal":"all m:List(Nat). m <= m","type_family":"List(Nat)","context":["SplitOn","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_428","id":88427,"instance":428,"goal":"all u:List(Atom). all d:List(Atom). all f:List(Atom). (u <= d) => (d <= f) => (u <= f)","type_family":"List(Atom)","context":["D 0","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_429","id":88428,"instance":429,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["DoSubst","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_430","id":88429,"instance":430,"goal":"all a:Nat -> Nat. all g:Nat -> Nat. (a = g) => (g = a)","type_family":"Nat -> Nat","context":["D 1","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_431","id":88430,"instance":431,"goal":"(epsilon) => (kappa) => (epsilon)","type_family":"Int","context":["Lemma","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_432","id":88431,"instance":432,"goal":"((n) & (b)) iff ((b) & (n))","type_family":"Nat","context":["Reduce","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_433","id":88432,"instance":433,"goal":"((d) | (f)) iff ((f) | (d))","type_family":"Real","context":["Lemma","DoSubst","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_434","id":88433,"instance":434,"goal":"not (not (u)) => (u)","type_family":"Atom","context":["InstLemma","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_435","id":88434,"instance":435,"goal":"all u:Bool. all a:Bool. all g:Bool. append(u, append(a, g)) = append(append(u, a), g)","type_family":"Bool","context":["D 0","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_436","id":88435,"instance":436,"goal":"all u:List(Int). all g:List(Int). u + g = g + u","type_family":"List(Int)","context":["InstLemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_437","id":88436,"instance":437,"goal":"all c:List(Nat). all x:List(Nat). all y:List(Nat). (c+x)+y = c+(x+y)","type_family":"List(Nat)","context":["D 0","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_438","id":88437,"instance":438,"goal":"all e:List(Atom). e + 0 = e","type_family":"List(Atom)","context":["D -1","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_439","id":88438,"instance":439,"goal":"all h:Int -> Int. all d:Int -> Int. h * d = d * h","type_family":"Int -> Int","context":["InstLemma","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_440","id":88439,"instance":440,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["DoSubst","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_441","id":88440,"instance":441,"goal":"all x:Int. all a:Int. all u:Int. x*(a+u) = x*a + x*u","type_family":"Int","context":["D 1","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_442","id":88441,"instance":442,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["RepUR","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_443","id":88442,"instance":443,"goal":"all h:Real. all g:Real. all x:Real. (h <= g) => (g <= x) => (h <= x)","type_family":"Real","context":["Reduce","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_444","id":88443,"instance":444,"goal":"all c:Atom. c = c","type_family":"Atom","context":["SplitOn","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_445","id":88444,"instance":445,"goal":"all f:Bool. all c:Bool. (f = c) => (c = f)","type_family":"Bool","context":["BackThru","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_446","id":88445,"instance":446,"goal":"(f) => (a) => (f)","type_family":"List(Int)","context":["InstLemma","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_447","id":88446,"instance":447,"goal":"((y) & (z)) iff ((z) & (y))","type_family":"List(Nat)","context":["D 0","RepUR","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_448","id":88447,"instance":448,"goal":"((b) | (z)) iff ((z) | (b))","type_family":"List(Atom)","context":["D 0","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_449","id":88448,"instance":449,"goal":"not (not (m)) => (m)","type_family":"Int -> Int","context":["Assert","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_450","id":88449,"instance":450,"goal":"all v:Nat -> Nat. all g:Nat -> Nat. all u:Nat -> Nat. append(v, append(g, u)) = append(append(v, g), u)","type_family":"Nat -> Nat","context":["SplitOn","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_451","id":88450,"instance":451,"goal":"all e:Int. all w:Int. e + w = w + e","type_family":"Int","context":["RepUR","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_452","id":88451,"instance":452,"goal":"all x:Nat. all f:Nat. all h:Nat. (x+f)+h = x+(f+h)","type_family":"Nat","context":["BackThru","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_453","id":88452,"instance":453,"goal":"all phi:Real. phi + 0 = phi","type_family":"Real","context":["D 1","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_454","id":88453,"instance":454,"goal":"all a:Atom. all h:Atom. a * h = h * a","type_family":"Atom","context":["Thin","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_455","id":88454,"instance":455,"goal":"all v:Bool. v * 1 = v","type_family":"Bool","context":["Thin","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_456","id":88455,"instance":456,"goal":"all p:List(Int). all n:List(Int). all h:List(Int). p*(n+h) = p*n + p*h","type_family":"List(Int)","context":["RepUR","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_457","id":88456,"instance":457,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["Auto","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_458","id":88457,"instance":458,"goal":"all epsilon:List(Atom). all kappa:List(Atom). all phi:List(Atom). (epsilon <= kappa) => (kappa <= phi) => (epsilon <= phi)","type_family":"List(Atom)","context":["Thin","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_459","id":88458,"instance":459,"goal":"all h:Int -> Int. h = h","type_family":"Int -> Int","context":["D 0","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_460","id":88459,"instance":460,"goal":"all g:Nat -> Nat. all x:Nat -> Nat. (g = x) => (x = g)","type_family":"Nat -> Nat","context":["Reduce","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_461","id":88460,"instance":461,"goal":"(g) => (b) => (g)","type_family":"Int","context":["D -1","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_462","id":88461,"instance":462,"goal":"((w) & (e)) iff ((e) & (w))","type_family":"Nat","context":["Thin","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_463","id":88462,"instance":463,"goal":"((z) | (p)) iff ((p) | (z))","type_family":"Real","context":["D -1","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_464","id":88463,"instance":464,"goal":"not (not (m)) => (m)","type_family":"Atom","context":["Assert","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_465","id":88464,"instance":465,"goal":"all v:Bool. all f:Bool. all a:Bool. append(v, append(f, a)) = append(append(v, f), a)","type_family":"Bool","context":["Complete Auto","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_466","id":88465,"instance":466,"goal":"all b:List(Int). all n:List(Int). b + n = n + b","type_family":"List(Int)","context":["BackThru","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_467","id":88466,"instance":467,"goal":"all e:List(Nat). all b:List(Nat). all d:List(Nat). (e+b)+d = e+(b+d)","type_family":"List(Nat)","context":["Lemma","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_468","id":88467,"instance":468,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["DoSubst","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_469","id":88468,"instance":469,"goal":"all omega:Int -> Int. all gamma:Int -> Int. omega * gamma = gamma * omega","type_family":"Int -> Int","context":["D -1","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_470","id":88469,"instance":470,"goal":"all a:Nat -> Nat. a * 1 = a","type_family":"Nat -> Nat","context":["DoSubst","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_471","id":88470,"instance":471,"goal":"all m:Int. all v:Int. all y:Int. m*(v+y) = m*v + m*y","type_family":"Int","context":["D -1","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_472","id":88471,"instance":472,"goal":"all h:Nat. h <= h","type_family":"Nat","context":["Complete Auto","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_473","id":88472,"instance":473,"goal":"all d:Real. all g:Real. all h:Real. (d <= g) => (g <= h) => (d <= h)","type_family":"Real","context":["InstLemma","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_474","id":88473,"instance":474,"goal":"all h:Atom. h = h","type_family":"Atom","context":["D 1","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_475","id":88474,"instance":475,"goal":"all m:Bool. all h:Bool. (m = h) => (h = m)","type_family":"Bool","context":["InstLemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_476","id":88475,"instance":476,"goal":"(z) => (b) => (z)","type_family":"List(Int)","context":["BackThru","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_477","id":88476,"instance":477,"goal":"((x) & (d)) iff ((d) & (x))","type_family":"List(Nat)","context":["Complete Auto","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_478","id":88477,"instance":478,"goal":"((g) | (h)) iff ((h) | (g))","type_family":"List(Atom)","context":["Lemma","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_479","id":88478,"instance":479,"goal":"not (not (b)) => (b)","type_family":"Int -> Int","context":["SplitOn","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_480","id":88479,"instance":480,"goal":"all w:Nat -> Nat. all q:Nat -> Nat. all g:Nat -> Nat. append(w, append(q, g)) = append(append(w, q), g)","type_family":"Nat -> Nat","context":["Thin","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_481","id":88480,"instance":481,"goal":"all f:Int. all w:Int. f + w = w + f","type_family":"Int","context":["Reduce","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_482","id":88481,"instance":482,"goal":"all m:Nat. all e:Nat. all z:Nat. (m+e)+z = m+(e+z)","type_family":"Nat","context":["Auto","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_483","id":88482,"instance":483,"goal":"all v:Real. v + 0 = v","type_family":"Real","context":["D 0","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_484","id":88483,"instance":484,"goal":"all z:Atom. all b:Atom. z * b = b * z","type_family":"Atom","context":["RepUR","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_485","id":88484,"instance":485,"goal":"all b:Bool. b * 1 = b","type_family":"Bool","context":["InstLemma","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_486","id":88485,"instance":486,"goal":"all x:List(Int). all y:List(Int). all m:List(Int). x*(y+m) = x*y + x*m","type_family":"List(Int)","context":["Assert","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_487","id":88486,"instance":487,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["Complete Auto","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_488","id":88487,"instance":488,"goal":"all f:List(Atom). all w:List(Atom). all m:List(Atom). (f <= w) => (w <= m) => (f <= m)","type_family":"List(Atom)","context":["D 0","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_489","id":88488,"instance":489,"goal":"all v:Int -> Int. v = v","type_family":"Int -> Int","context":["DoSubst","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_490","id":88489,"instance":490,"goal":"all b:Nat -> Nat. all v:Nat -> Nat. (b = v) => (v = b)","type_family":"Nat -> Nat","context":["Complete Auto","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_491","id":88490,"instance":491,"goal":"(c) => (z) => (c)","type_family":"Int","context":["Assert","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_492","id":88491,"instance":492,"goal":"((d) & (h)) iff ((h) & (d))","type_family":"Nat","context":["Auto","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_493","id":88492,"instance":493,"goal":"((m) | (b)) iff ((b) | (m))","type_family":"Real","context":["Reduce","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_494","id":88493,"instance":494,"goal":"not (not (c)) => (c)","type_family":"Atom","context":["Complete Auto","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_495","id":88494,"instance":495,"goal":"all n:Bool. all m:Bool. all u:Bool. append(n, append(m, u)) = append(append(n, m), u)","type_family":"Bool","context":["Assert","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_496","id":88495,"instance":496,"goal":"all z:List(Int). all w:List(Int). z + w = w + z","type_family":"List(Int)","context":["Complete Auto","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_497","id":88496,"instance":497,"goal":"all y:List(Nat). all h:List(Nat). all e:List(Nat). (y+h)+e = y+(h+e)","type_family":"List(Nat)","context":["InstLemma","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_498","id":88497,"instance":498,"goal":"all delta:List(Atom). delta + 0 = delta","type_family":"List(Atom)","context":["Thin","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_499","id":88498,"instance":499,"goal":"all w:Int -> Int. all m:Int -> Int. w * m = m * w","type_family":"Int -> Int","context":["Auto","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_500","id":88499,"instance":500,"goal":"all c:Nat -> Nat. c * 1 = c","type_family":"Nat -> Nat","context":["Lemma","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_501","id":88500,"instance":501,"goal":"all x:Int. all e:Int. all z:Int. x*(e+z) = x*e + x*z","type_family":"Int","context":["D -1","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_502","id":88501,"instance":502,"goal":"all b:Nat. b <= b","type_family":"Nat","context":["D 0","D 1","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_503","id":88502,"instance":503,"goal":"all c:Real. all e:Real. all a:Real. (c <= e) => (e <= a) => (c <= a)","type_family":"Real","context":["D 0","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_504","id":88503,"instance":504,"goal":"all v:Atom. v = v","type_family":"Atom","context":["Thin","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_505","id":88504,"instance":505,"goal":"all z:Bool. all m:Bool. (z = m) => (m = z)","type_family":"Bool","context":["Auto","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_506","id":88505,"instance":506,"goal":"(n) => (f) => (n)","type_family":"List(Int)","context":["BackThru","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_507","id":88506,"instance":507,"goal":"((y) & (d)) iff ((d) & (y))","type_family":"List(Nat)","context":["D -1","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_508","id":88507,"instance":508,"goal":"((a) | (n)) iff ((n) | (a))","type_family":"List(Atom)","context":["Reduce","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_509","id":88508,"instance":509,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["Lemma","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_510","id":88509,"instance":510,"goal":"all e:Nat -> Nat. all d:Nat -> Nat. all b:Nat -> Nat. append(e, append(d, b)) = append(append(e, d), b)","type_family":"Nat -> Nat","context":["Lemma","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_511","id":88510,"instance":511,"goal":"all phi:Int. all gamma:Int. phi + gamma = gamma + phi","type_family":"Int","context":["Complete Auto","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_512","id":88511,"instance":512,"goal":"all b:Nat. all x:Nat. all m:Nat. (b+x)+m = b+(x+m)","type_family":"Nat","context":["RepUR","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_513","id":88512,"instance":513,"goal":"all h:Real. h + 0 = h","type_family":"Real","context":["Thin","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_514","id":88513,"instance":514,"goal":"all b:Atom. all m:Atom. b * m = m * b","type_family":"Atom","context":["InstLemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_515","id":88514,"instance":515,"goal":"all d:Bool. d * 1 = d","type_family":"Bool","context":["Reduce","D -1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_516","id":88515,"instance":516,"goal":"all p:List(Int). all a:List(Int). all q:List(Int). p*(a+q) = p*a + p*q","type_family":"List(Int)","context":["RepUR","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_517","id":88516,"instance":517,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["SplitOn","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_518","id":88517,"instance":518,"goal":"all x:List(Atom). all v:List(Atom). all n:List(Atom). (x <= v) => (v <= n) => (x <= n)","type_family":"List(Atom)","context":["Assert","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_519","id":88518,"instance":519,"goal":"all v:Int -> Int. v = v","type_family":"Int -> Int","context":["Thin","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_520","id":88519,"instance":520,"goal":"all d:Nat -> Nat. all h:Nat -> Nat. (d = h) => (h = d)","type_family":"Nat -> Nat","context":["RepUR","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_521","id":88520,"instance":521,"goal":"(f) => (z) => (f)","type_family":"Int","context":["Assert","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_522","id":88521,"instance":522,"goal":"((p) & (f)) iff ((f) & (p))","type_family":"Nat","context":["Assert","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_523","id":88522,"instance":523,"goal":"((x) | (h)) iff ((h) | (x))","type_family":"Real","context":["D 0","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_524","id":88523,"instance":524,"goal":"not (not (n)) => (n)","type_family":"Atom","context":["InstLemma","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_525","id":88524,"instance":525,"goal":"all m:Bool. all g:Bool. all d:Bool. append(m, append(g, d)) = append(append(m, g), d)","type_family":"Bool","context":["D 1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_526","id":88525,"instance":526,"goal":"all b:List(Int). all a:List(Int). b + a = a + b","type_family":"List(Int)","context":["Assert","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_527","id":88526,"instance":527,"goal":"all p:List(Nat). all d:List(Nat). all v:List(Nat). (p+d)+v = p+(d+v)","type_family":"List(Nat)","context":["BackThru","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_528","id":88527,"instance":528,"goal":"all d:List(Atom). d + 0 = d","type_family":"List(Atom)","context":["D 1","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_529","id":88528,"instance":529,"goal":"all alpha:Int -> Int. all delta:Int -> Int. alpha * delta = delta * alpha","type_family":"Int -> Int","context":["D 1","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_530","id":88529,"instance":530,"goal":"all u:Nat -> Nat. u * 1 = u","type_family":"Nat -> Nat","context":["D 0","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_531","id":88530,"instance":531,"goal":"all c:Int. all u:Int. all v:Int. c*(u+v) = c*u + c*v","type_family":"Int","context":["Lemma","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_532","id":88531,"instance":532,"goal":"all gamma:Nat. gamma <= gamma","type_family":"Nat","context":["D -1","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_533","id":88532,"instance":533,"goal":"all p:Real. all z:Real. all u:Real. (p <= z) => (z <= u) => (p <= u)","type_family":"Real","context":["Reduce","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_534","id":88533,"instance":534,"goal":"all e:Atom. e = e","type_family":"Atom","context":["InstLemma","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_535","id":88534,"instance":535,"goal":"all d:Bool. all x:Bool. (d = x) => (x = d)","type_family":"Bool","context":["D 0","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_536","id":88535,"instance":536,"goal":"(x) => (y) => (x)","type_family":"List(Int)","context":["Auto","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_537","id":88536,"instance":537,"goal":"((p) & (e)) iff ((e) & (p))","type_family":"List(Nat)","context":["BackThru","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_538","id":88537,"instance":538,"goal":"((q) | (n)) iff ((n) | (q))","type_family":"List(Atom)","context":["Auto","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_539","id":88538,"instance":539,"goal":"not (not (h)) => (h)","type_family":"Int -> Int","context":["Reduce","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_540","id":88539,"instance":540,"goal":"all c:Nat -> Nat. all y:Nat -> Nat. all a:Nat -> Nat. append(c, append(y, a)) = append(append(c, y), a)","type_family":"Nat -> Nat","context":["RepUR","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_541","id":88540,"instance":541,"goal":"all z:Int. all a:Int. z + a = a + z","type_family":"Int","context":["Lemma","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_542","id":88541,"instance":542,"goal":"all b:Nat. all z:Nat. all q:Nat. (b+z)+q = b+(z+q)","type_family":"Nat","context":["Auto","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_543","id":88542,"instance":543,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["Lemma","InstLemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_544","id":88543,"instance":544,"goal":"all v:Atom. all e:Atom. v * e = e * v","type_family":"Atom","context":["D -1","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_545","id":88544,"instance":545,"goal":"all y:Bool. y * 1 = y","type_family":"Bool","context":["InstLemma","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_546","id":88545,"instance":546,"goal":"all c:List(Int). all n:List(Int). all w:List(Int). c*(n+w) = c*n + c*w","type_family":"List(Int)","context":["D 1","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_547","id":88546,"instance":547,"goal":"all beta:List(Nat). beta <= beta","type_family":"List(Nat)","context":["Auto","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_548","id":88547,"instance":548,"goal":"all f:List(Atom). all n:List(Atom). all p:List(Atom). (f <= n) => (n <= p) => (f <= p)","type_family":"List(Atom)","context":["D -1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_549","id":88548,"instance":549,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["Reduce","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_550","id":88549,"instance":550,"goal":"all v:Nat -> Nat. all n:Nat -> Nat. (v = n) => (n = v)","type_family":"Nat -> Nat","context":["Lemma","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_551","id":88550,"instance":551,"goal":"(q) => (g) => (q)","type_family":"Int","context":["Auto","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_552","id":88551,"instance":552,"goal":"((z) & (u)) iff ((u) & (z))","type_family":"Nat","context":["InstLemma","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_553","id":88552,"instance":553,"goal":"((c) | (a)) iff ((a) | (c))","type_family":"Real","context":["SplitOn","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_554","id":88553,"instance":554,"goal":"not (not (n)) => (n)","type_family":"Atom","context":["D 1","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_555","id":88554,"instance":555,"goal":"all f:Bool. all c:Bool. all g:Bool. append(f, append(c, g)) = append(append(f, c), g)","type_family":"Bool","context":["Reduce","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_556","id":88555,"instance":556,"goal":"all e:List(Int). all w:List(Int). e + w = w + e","type_family":"List(Int)","context":["Complete Auto","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_557","id":88556,"instance":557,"goal":"all n:List(Nat). all m:List(Nat). all a:List(Nat). (n+m)+a = n+(m+a)","type_family":"List(Nat)","context":["Auto","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_558","id":88557,"instance":558,"goal":"all b:List(Atom). b + 0 = b","type_family":"List(Atom)","context":["SplitOn","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_559","id":88558,"instance":559,"goal":"all a:Int -> Int. all c:Int -> Int. a * c = c * a","type_family":"Int -> Int","context":["RepUR","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_560","id":88559,"instance":560,"goal":"all chi:Nat -> Nat. chi * 1 = chi","type_family":"Nat -> Nat","context":["D 0","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_561","id":88560,"instance":561,"goal":"all b:Int. all n:Int. all p:Int. b*(n+p) = b*n + b*p","type_family":"Int","context":["BackThru","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_562","id":88561,"instance":562,"goal":"all a:Nat. a <= a","type_family":"Nat","context":["D 0","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_563","id":88562,"instance":563,"goal":"all m:Real. all p:Real. all f:Real. (m <= p) => (p <= f) => (m <= f)","type_family":"Real","context":["Lemma","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_564","id":88563,"instance":564,"goal":"all v:Atom. v = v","type_family":"Atom","context":["BackThru","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_565","id":88564,"instance":565,"goal":"all g:Bool. all v:Bool. (g = v) => (v = g)","type_family":"Bool","context":["Auto","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_566","id":88565,"instance":566,"goal":"(q) => (g) => (q)","type_family":"List(Int)","context":["BackThru","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_567","id":88566,"instance":567,"goal":"((f) & (w)) iff ((w) & (f))","type_family":"List(Nat)","context":["DoSubst","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_568","id":88567,"instance":568,"goal":"((z) | (u)) iff ((u) | (z))","type_family":"List(Atom)","context":["Reduce","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_569","id":88568,"instance":569,"goal":"not (not (omega)) => (omega)","type_family":"Int -> Int","context":["D 1","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_570","id":88569,"instance":570,"goal":"all a:Nat -> Nat. all h:Nat -> Nat. all z:Nat -> Nat. append(a, append(h, z)) = append(append(a, h), z)","type_family":"Nat -> Nat","context":["D 0","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_571","id":88570,"instance":571,"goal":"all n:Int. all g:Int. n + g = g + n","type_family":"Int","context":["D -1","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_572","id":88571,"instance":572,"goal":"all v:Nat. all n:Nat. all m:Nat. (v+n)+m = v+(n+m)","type_family":"Nat","context":["SplitOn","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_573","id":88572,"instance":573,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["RepUR","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_574","id":88573,"instance":574,"goal":"all g:Atom. all u:Atom. g * u = u * g","type_family":"Atom","context":["Reduce","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_575","id":88574,"instance":575,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["InstLemma","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_576","id":88575,"instance":576,"goal":"all w:List(Int). all a:List(Int). all g:List(Int). w*(a+g) = w*a + w*g","type_family":"List(Int)","context":["BackThru","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_577","id":88576,"instance":577,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["DoSubst","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_578","id":88577,"instance":578,"goal":"all e:List(Atom). all h:List(Atom). all v:List(Atom). (e <= h) => (h <= v) => (e <= v)","type_family":"List(Atom)","context":["DoSubst","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_579","id":88578,"instance":579,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["SplitOn","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_580","id":88579,"instance":580,"goal":"all q:Nat -> Nat. all b:Nat -> Nat. (q = b) => (b = q)","type_family":"Nat -> Nat","context":["Complete Auto","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_581","id":88580,"instance":581,"goal":"(c) => (q) => (c)","type_family":"Int","context":["D 1","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_582","id":88581,"instance":582,"goal":"((q) & (w)) iff ((w) & (q))","type_family":"Nat","context":["InstLemma","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_583","id":88582,"instance":583,"goal":"((p) | (g)) iff ((g) | (p))","type_family":"Real","context":["Assert","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_584","id":88583,"instance":584,"goal":"not (not (g)) => (g)","type_family":"Atom","context":["Lemma","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_585","id":88584,"instance":585,"goal":"all h:Bool. all d:Bool. all a:Bool. append(h, append(d, a)) = append(append(h, d), a)","type_family":"Bool","context":["InstLemma","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_586","id":88585,"instance":586,"goal":"all x:List(Int). all n:List(Int). x + n = n + x","type_family":"List(Int)","context":["D 0","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_587","id":88586,"instance":587,"goal":"all a:List(Nat). all c:List(Nat). all g:List(Nat). (a+c)+g = a+(c+g)","type_family":"List(Nat)","context":["Assert","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_588","id":88587,"instance":588,"goal":"all h:List(Atom). h + 0 = h","type_family":"List(Atom)","context":["Thin","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_589","id":88588,"instance":589,"goal":"all g:Int -> Int. all f:Int -> Int. g * f = f * g","type_family":"Int -> Int","context":["DoSubst","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_590","id":88589,"instance":590,"goal":"all m:Nat -> Nat. m * 1 = m","type_family":"Nat -> Nat","context":["SplitOn","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_591","id":88590,"instance":591,"goal":"all phi:Int. all beta:Int. all kappa:Int. phi*(beta+kappa) = phi*beta + phi*kappa","type_family":"Int","context":["BackThru","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_592","id":88591,"instance":592,"goal":"all kappa:Nat. kappa <= kappa","type_family":"Nat","context":["DoSubst","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_593","id":88592,"instance":593,"goal":"all z:Real. all v:Real. all h:Real. (z <= v) => (v <= h) => (z <= h)","type_family":"Real","context":["Assert","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_594","id":88593,"instance":594,"goal":"all u:Atom. u = u","type_family":"Atom","context":["Assert","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_595","id":88594,"instance":595,"goal":"all g:Bool. all n:Bool. (g = n) => (n = g)","type_family":"Bool","context":["Complete Auto","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_596","id":88595,"instance":596,"goal":"(a) => (v) => (a)","type_family":"List(Int)","context":["Complete Auto","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_597","id":88596,"instance":597,"goal":"((m) & (p)) iff ((p) & (m))","type_family":"List(Nat)","context":["DoSubst","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_598","id":88597,"instance":598,"goal":"((z) | (m)) iff ((m) | (z))","type_family":"List(Atom)","context":["Complete Auto","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_599","id":88598,"instance":599,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["InstLemma","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_600","id":88599,"instance":600,"goal":"all g:Nat -> Nat. all z:Nat -> Nat. all w:Nat -> Nat. append(g, append(z, w)) = append(append(g, z), w)","type_family":"Nat -> Nat","context":["Lemma","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_601","id":88600,"instance":601,"goal":"all w:Int. all y:Int. w + y = y + w","type_family":"Int","context":["Auto","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_602","id":88601,"instance":602,"goal":"all h:Nat. all n:Nat. all f:Nat. (h+n)+f = h+(n+f)","type_family":"Nat","context":["Assert","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_603","id":88602,"instance":603,"goal":"all e:Real. e + 0 = e","type_family":"Real","context":["InstLemma","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_604","id":88603,"instance":604,"goal":"all g:Atom. all e:Atom. g * e = e * g","type_family":"Atom","context":["DoSubst","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_605","id":88604,"instance":605,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["InstLemma","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_606","id":88605,"instance":606,"goal":"all h:List(Int). all q:List(Int). all m:List(Int). h*(q+m) = h*q + h*m","type_family":"List(Int)","context":["D 0","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_607","id":88606,"instance":607,"goal":"all n:List(Nat). n <= n","type_family":"List(Nat)","context":["SplitOn","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_608","id":88607,"instance":608,"goal":"all z:List(Atom). all w:List(Atom). all p:List(Atom). (z <= w) => (w <= p) => (z <= p)","type_family":"List(Atom)","context":["Complete Auto","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_609","id":88608,"instance":609,"goal":"all b:Int -> Int. b = b","type_family":"Int -> Int","context":["SplitOn","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_610","id":88609,"instance":610,"goal":"all d:Nat -> Nat. all q:Nat -> Nat. (d = q) => (q = d)","type_family":"Nat -> Nat","context":["Auto","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_611","id":88610,"instance":611,"goal":"(f) => (w) => (f)","type_family":"Int","context":["Assert","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_612","id":88611,"instance":612,"goal":"((beta) & (gamma)) iff ((gamma) & (beta))","type_family":"Nat","context":["Assert","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_613","id":88612,"instance":613,"goal":"((d) | (w)) iff ((w) | (d))","type_family":"Real","context":["InstLemma","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_614","id":88613,"instance":614,"goal":"not (not (omega)) => (omega)","type_family":"Atom","context":["SplitOn","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_615","id":88614,"instance":615,"goal":"all q:Bool. all p:Bool. all m:Bool. append(q, append(p, m)) = append(append(q, p), m)","type_family":"Bool","context":["SplitOn","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_616","id":88615,"instance":616,"goal":"all y:List(Int). all n:List(Int). y + n = n + y","type_family":"List(Int)","context":["RepUR","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_617","id":88616,"instance":617,"goal":"all d:List(Nat). all q:List(Nat). all w:List(Nat). (d+q)+w = d+(q+w)","type_family":"List(Nat)","context":["D -1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_618","id":88617,"instance":618,"goal":"all a:List(Atom). a + 0 = a","type_family":"List(Atom)","context":["Complete Auto","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_619","id":88618,"instance":619,"goal":"all b:Int -> Int. all q:Int -> Int. b * q = q * b","type_family":"Int -> Int","context":["D 1","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_620","id":88619,"instance":620,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["D 0","DoSubst","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_621","id":88620,"instance":621,"goal":"all beta:Int. all omega:Int. all chi:Int. beta*(omega+chi) = beta*omega + beta*chi","type_family":"Int","context":["D 0","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_622","id":88621,"instance":622,"goal":"all gamma:Nat. gamma <= gamma","type_family":"Nat","context":["SplitOn","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_623","id":88622,"instance":623,"goal":"all g:Real. all c:Real. all b:Real. (g <= c) => (c <= b) => (g <= b)","type_family":"Real","context":["BackThru","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_624","id":88623,"instance":624,"goal":"all f:Atom. f = f","type_family":"Atom","context":["BackThru","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_625","id":88624,"instance":625,"goal":"all n:Bool. all b:Bool. (n = b) => (b = n)","type_family":"Bool","context":["DoSubst","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_626","id":88625,"instance":626,"goal":"(d) => (y) => (d)","type_family":"List(Int)","context":["BackThru","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_627","id":88626,"instance":627,"goal":"((c) & (w)) iff ((w) & (c))","type_family":"List(Nat)","context":["Thin","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_628","id":88627,"instance":628,"goal":"((d) | (f)) iff ((f) | (d))","type_family":"List(Atom)","context":["D 1","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_629","id":88628,"instance":629,"goal":"not (not (y)) => (y)","type_family":"Int -> Int","context":["RepUR","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_630","id":88629,"instance":630,"goal":"all e:Nat -> Nat. all x:Nat -> Nat. all c:Nat -> Nat. append(e, append(x, c)) = append(append(e, x), c)","type_family":"Nat -> Nat","context":["Lemma","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_631","id":88630,"instance":631,"goal":"all g:Int. all q:Int. g + q = q + g","type_family":"Int","context":["RepUR","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_632","id":88631,"instance":632,"goal":"all c:Nat. all u:Nat. all b:Nat. (c+u)+b = c+(u+b)","type_family":"Nat","context":["Lemma","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_633","id":88632,"instance":633,"goal":"all a:Real. a + 0 = a","type_family":"Real","context":["Lemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_634","id":88633,"instance":634,"goal":"all b:Atom. all p:Atom. b * p = p * b","type_family":"Atom","context":["RepUR","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_635","id":88634,"instance":635,"goal":"all delta:Bool. delta * 1 = delta","type_family":"Bool","context":["DoSubst","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_636","id":88635,"instance":636,"goal":"all v:List(Int). all p:List(Int). all c:List(Int). v*(p+c) = v*p + v*c","type_family":"List(Int)","context":["DoSubst","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_637","id":88636,"instance":637,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["Complete Auto","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_638","id":88637,"instance":638,"goal":"all y:List(Atom). all f:List(Atom). all n:List(Atom). (y <= f) => (f <= n) => (y <= n)","type_family":"List(Atom)","context":["InstLemma","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_639","id":88638,"instance":639,"goal":"all m:Int -> Int. m = m","type_family":"Int -> Int","context":["Auto","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_640","id":88639,"instance":640,"goal":"all x:Nat -> Nat. all g:Nat -> Nat. (x = g) => (g = x)","type_family":"Nat -> Nat","context":["Assert","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_641","id":88640,"instance":641,"goal":"(omega) => (alpha) => (omega)","type_family":"Int","context":["Auto","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_642","id":88641,"instance":642,"goal":"((delta) & (phi)) iff ((phi) & (delta))","type_family":"Nat","context":["DoSubst","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_643","id":88642,"instance":643,"goal":"((w) | (u)) iff ((u) | (w))","type_family":"Real","context":["DoSubst","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_644","id":88643,"instance":644,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["Lemma","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_645","id":88644,"instance":645,"goal":"all q:Bool. all d:Bool. all h:Bool. append(q, append(d, h)) = append(append(q, d), h)","type_family":"Bool","context":["RepUR","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_646","id":88645,"instance":646,"goal":"all q:List(Int). all u:List(Int). q + u = u + q","type_family":"List(Int)","context":["BackThru","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_647","id":88646,"instance":647,"goal":"all p:List(Nat). all y:List(Nat). all h:List(Nat). (p+y)+h = p+(y+h)","type_family":"List(Nat)","context":["SplitOn","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_648","id":88647,"instance":648,"goal":"all phi:List(Atom). phi + 0 = phi","type_family":"List(Atom)","context":["Reduce","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_649","id":88648,"instance":649,"goal":"all w:Int -> Int. all u:Int -> Int. w * u = u * w","type_family":"Int -> Int","context":["RepUR","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_650","id":88649,"instance":650,"goal":"all z:Nat -> Nat. z * 1 = z","type_family":"Nat -> Nat","context":["Thin","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_651","id":88650,"instance":651,"goal":"all a:Int. all m:Int. all y:Int. a*(m+y) = a*m + a*y","type_family":"Int","context":["RepUR","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_652","id":88651,"instance":652,"goal":"all e:Nat. e <= e","type_family":"Nat","context":["BackThru","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_653","id":88652,"instance":653,"goal":"all c:Real. all x:Real. all m:Real. (c <= x) => (x <= m) => (c <= m)","type_family":"Real","context":["D -1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_654","id":88653,"instance":654,"goal":"all a:Atom. a = a","type_family":"Atom","context":["Complete Auto","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_655","id":88654,"instance":655,"goal":"all p:Bool. all x:Bool. (p = x) => (x = p)","type_family":"Bool","context":["RepUR","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_656","id":88655,"instance":656,"goal":"(c) => (u) => (c)","type_family":"List(Int)","context":["Lemma","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_657","id":88656,"instance":657,"goal":"((z) & (a)) iff ((a) & (z))","type_family":"List(Nat)","context":["Lemma","InstLemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_658","id":88657,"instance":658,"goal":"((v) | (y)) iff ((y) | (v))","type_family":"List(Atom)","context":["Reduce","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_659","id":88658,"instance":659,"goal":"not (not (w)) => (w)","type_family":"Int -> Int","context":["D 0","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_660","id":88659,"instance":660,"goal":"all z:Nat -> Nat. all g:Nat -> Nat. all y:Nat -> Nat. append(z, append(g, y)) = append(append(z, g), y)","type_family":"Nat -> Nat","context":["Lemma","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_661","id":88660,"instance":661,"goal":"all f:Int. all p:Int. f + p = p + f","type_family":"Int","context":["Lemma","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_662","id":88661,"instance":662,"goal":"all u:Nat. all y:Nat. all e:Nat. (u+y)+e = u+(y+e)","type_family":"Nat","context":["DoSubst","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_663","id":88662,"instance":663,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["Auto","InstLemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_664","id":88663,"instance":664,"goal":"all m:Atom. all y:Atom. m * y = y * m","type_family":"Atom","context":["Assert","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_665","id":88664,"instance":665,"goal":"all delta:Bool. delta * 1 = delta","type_family":"Bool","context":["D -1","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_666","id":88665,"instance":666,"goal":"all m:List(Int). all c:List(Int). all w:List(Int). m*(c+w) = m*c + m*w","type_family":"List(Int)","context":["Complete Auto","Reduce","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_667","id":88666,"instance":667,"goal":"all x:List(Nat). x <= x","type_family":"List(Nat)","context":["SplitOn","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_668","id":88667,"instance":668,"goal":"all w:List(Atom). all d:List(Atom). all e:List(Atom). (w <= d) => (d <= e) => (w <= e)","type_family":"List(Atom)","context":["RepUR","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_669","id":88668,"instance":669,"goal":"all z:Int -> Int. z = z","type_family":"Int -> Int","context":["D 1","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_670","id":88669,"instance":670,"goal":"all h:Nat -> Nat. all c:Nat -> Nat. (h = c) => (c = h)","type_family":"Nat -> Nat","context":["BackThru","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_671","id":88670,"instance":671,"goal":"(z) => (n) => (z)","type_family":"Int","context":["RepUR","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_672","id":88671,"instance":672,"goal":"((f) & (e)) iff ((e) & (f))","type_family":"Nat","context":["Lemma","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_673","id":88672,"instance":673,"goal":"((w) | (x)) iff ((x) | (w))","type_family":"Real","context":["D 1","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_674","id":88673,"instance":674,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["D 1","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_675","id":88674,"instance":675,"goal":"all w:Bool. all f:Bool. all u:Bool. append(w, append(f, u)) = append(append(w, f), u)","type_family":"Bool","context":["SplitOn","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_676","id":88675,"instance":676,"goal":"all kappa:List(Int). all alpha:List(Int). kappa + alpha = alpha + kappa","type_family":"List(Int)","context":["SplitOn","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_677","id":88676,"instance":677,"goal":"all n:List(Nat). all z:List(Nat). all y:List(Nat). (n+z)+y = n+(z+y)","type_family":"List(Nat)","context":["Thin","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_678","id":88677,"instance":678,"goal":"all x:List(Atom). x + 0 = x","type_family":"List(Atom)","context":["D 0","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_679","id":88678,"instance":679,"goal":"all p:Int -> Int. all f:Int -> Int. p * f = f * p","type_family":"Int -> Int","context":["Reduce","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_680","id":88679,"instance":680,"goal":"all h:Nat -> Nat. h * 1 = h","type_family":"Nat -> Nat","context":["Reduce","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_681","id":88680,"instance":681,"goal":"all a:Int. all y:Int. all g:Int. a*(y+g) = a*y + a*g","type_family":"Int","context":["DoSubst","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_682","id":88681,"instance":682,"goal":"all h:Nat. h <= h","type_family":"Nat","context":["Assert","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_683","id":88682,"instance":683,"goal":"all y:Real. all e:Real. all q:Real. (y <= e) => (e <= q) => (y <= q)","type_family":"Real","context":["D -1","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_684","id":88683,"instance":684,"goal":"all u:Atom. u = u","type_family":"Atom","context":["Lemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_685","id":88684,"instance":685,"goal":"all m:Bool. all c:Bool. (m = c) => (c = m)","type_family":"Bool","context":["D 0","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_686","id":88685,"instance":686,"goal":"(g) => (e) => (g)","type_family":"List(Int)","context":["Assert","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_687","id":88686,"instance":687,"goal":"((n) & (y)) iff ((y) & (n))","type_family":"List(Nat)","context":["InstLemma","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_688","id":88687,"instance":688,"goal":"((w) | (c)) iff ((c) | (w))","type_family":"List(Atom)","context":["InstLemma","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_689","id":88688,"instance":689,"goal":"not (not (x)) => (x)","type_family":"Int -> Int","context":["InstLemma","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_690","id":88689,"instance":690,"goal":"all f:Nat -> Nat. all g:Nat -> Nat. all w:Nat -> Nat. append(f, append(g, w)) = append(append(f, g), w)","type_family":"Nat -> Nat","context":["Thin","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_691","id":88690,"instance":691,"goal":"all z:Int. all b:Int. z + b = b + z","type_family":"Int","context":["BackThru","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_692","id":88691,"instance":692,"goal":"all n:Nat. all a:Nat. all w:Nat. (n+a)+w = n+(a+w)","type_family":"Nat","context":["Complete Auto","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_693","id":88692,"instance":693,"goal":"all m:Real. m + 0 = m","type_family":"Real","context":["Reduce","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_694","id":88693,"instance":694,"goal":"all q:Atom. all h:Atom. q * h = h * q","type_family":"Atom","context":["RepUR","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_695","id":88694,"instance":695,"goal":"all f:Bool. f * 1 = f","type_family":"Bool","context":["BackThru","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_696","id":88695,"instance":696,"goal":"all c:List(Int). all p:List(Int). all h:List(Int). c*(p+h) = c*p + c*h","type_family":"List(Int)","context":["Reduce","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_697","id":88696,"instance":697,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["Assert","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_698","id":88697,"instance":698,"goal":"all psi:List(Atom). all beta:List(Atom). all chi:List(Atom). (psi <= beta) => (beta <= chi) => (psi <= chi)","type_family":"List(Atom)","context":["SplitOn","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_699","id":88698,"instance":699,"goal":"all v:Int -> Int. v = v","type_family":"Int -> Int","context":["Complete Auto","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_700","id":88699,"instance":700,"goal":"all b:Nat -> Nat. all a:Nat -> Nat. (b = a) => (a = b)","type_family":"Nat -> Nat","context":["Auto","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_701","id":88700,"instance":701,"goal":"(n) => (g) => (n)","type_family":"Int","context":["SplitOn","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_702","id":88701,"instance":702,"goal":"((n) & (m)) iff ((m) & (n))","type_family":"Nat","context":["DoSubst","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_703","id":88702,"instance":703,"goal":"((e) | (z)) iff ((z) | (e))","type_family":"Real","context":["BackThru","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_704","id":88703,"instance":704,"goal":"not (not (z)) => (z)","type_family":"Atom","context":["SplitOn","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_705","id":88704,"instance":705,"goal":"all w:Bool. all n:Bool. all f:Bool. append(w, append(n, f)) = append(append(w, n), f)","type_family":"Bool","context":["D 0","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_706","id":88705,"instance":706,"goal":"all f:List(Int). all e:List(Int). f + e = e + f","type_family":"List(Int)","context":["Reduce","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_707","id":88706,"instance":707,"goal":"all m:List(Nat). all q:List(Nat). all a:List(Nat). (m+q)+a = m+(q+a)","type_family":"List(Nat)","context":["Assert","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_708","id":88707,"instance":708,"goal":"all a:List(Atom). a + 0 = a","type_family":"List(Atom)","context":["D -1","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_709","id":88708,"instance":709,"goal":"all a:Int -> Int. all m:Int -> Int. a * m = m * a","type_family":"Int -> Int","context":["Thin","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_710","id":88709,"instance":710,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["Auto","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_711","id":88710,"instance":711,"goal":"all g:Int. all h:Int. all m:Int. g*(h+m) = g*h + g*m","type_family":"Int","context":["Complete Auto","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_712","id":88711,"instance":712,"goal":"all omega:Nat. omega <= omega","type_family":"Nat","context":["Lemma","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_713","id":88712,"instance":713,"goal":"all x:Real. all d:Real. all m:Real. (x <= d) => (d <= m) => (x <= m)","type_family":"Real","context":["D 1","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_714","id":88713,"instance":714,"goal":"all g:Atom. g = g","type_family":"Atom","context":["SplitOn","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_715","id":88714,"instance":715,"goal":"all x:Bool. all e:Bool. (x = e) => (e = x)","type_family":"Bool","context":["SplitOn","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_716","id":88715,"instance":716,"goal":"(kappa) => (beta) => (kappa)","type_family":"List(Int)","context":["Reduce","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_717","id":88716,"instance":717,"goal":"((a) & (e)) iff ((e) & (a))","type_family":"List(Nat)","context":["SplitOn","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_718","id":88717,"instance":718,"goal":"((d) | (h)) iff ((h) | (d))","type_family":"List(Atom)","context":["Auto","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_719","id":88718,"instance":719,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["D 1","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_720","id":88719,"instance":720,"goal":"all d:Nat -> Nat. all h:Nat -> Nat. all w:Nat -> Nat. append(d, append(h, w)) = append(append(d, h), w)","type_family":"Nat -> Nat","context":["Thin","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_721","id":88720,"instance":721,"goal":"all v:Int. all q:Int. v + q = q + v","type_family":"Int","context":["Reduce","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_722","id":88721,"instance":722,"goal":"all v:Nat. all b:Nat. all d:Nat. (v+b)+d = v+(b+d)","type_family":"Nat","context":["Lemma","D -1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_723","id":88722,"instance":723,"goal":"all m:Real. m + 0 = m","type_family":"Real","context":["Assert","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_724","id":88723,"instance":724,"goal":"all x:Atom. all d:Atom. x * d = d * x","type_family":"Atom","context":["DoSubst","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_725","id":88724,"instance":725,"goal":"all d:Bool. d * 1 = d","type_family":"Bool","context":["SplitOn","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_726","id":88725,"instance":726,"goal":"all y:List(Int). all f:List(Int). all z:List(Int). y*(f+z) = y*f + y*z","type_family":"List(Int)","context":["D -1","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_727","id":88726,"instance":727,"goal":"all y:List(Nat). y <= y","type_family":"List(Nat)","context":["DoSubst","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_728","id":88727,"instance":728,"goal":"all y:List(Atom). all x:List(Atom). all e:List(Atom). (y <= x) => (x <= e) => (y <= e)","type_family":"List(Atom)","context":["D -1","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_729","id":88728,"instance":729,"goal":"all z:Int -> Int. z = z","type_family":"Int -> Int","context":["Assert","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_730","id":88729,"instance":730,"goal":"all h:Nat -> Nat. all w:Nat -> Nat. (h = w) => (w = h)","type_family":"Nat -> Nat","context":["Auto","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_731","id":88730,"instance":731,"goal":"(d) => (c) => (d)","type_family":"Int","context":["D -1","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_732","id":88731,"instance":732,"goal":"((x) & (v)) iff ((v) & (x))","type_family":"Nat","context":["BackThru","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_733","id":88732,"instance":733,"goal":"((m) | (n)) iff ((n) | (m))","type_family":"Real","context":["D -1","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_734","id":88733,"instance":734,"goal":"not (not (b)) => (b)","type_family":"Atom","context":["D 0","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_735","id":88734,"instance":735,"goal":"all z:Bool. all w:Bool. all y:Bool. append(z, append(w, y)) = append(append(z, w), y)","type_family":"Bool","context":["Assert","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_736","id":88735,"instance":736,"goal":"all d:List(Int). all e:List(Int). d + e = e + d","type_family":"List(Int)","context":["Reduce","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_737","id":88736,"instance":737,"goal":"all b:List(Nat). all y:List(Nat). all n:List(Nat). (b+y)+n = b+(y+n)","type_family":"List(Nat)","context":["D -1","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_738","id":88737,"instance":738,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["Auto","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_739","id":88738,"instance":739,"goal":"all v:Int -> Int. all y:Int -> Int. v * y = y * v","type_family":"Int -> Int","context":["D 1","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_740","id":88739,"instance":740,"goal":"all w:Nat -> Nat. w * 1 = w","type_family":"Nat -> Nat","context":["Assert","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_741","id":88740,"instance":741,"goal":"all x:Int. all d:Int. all p:Int. x*(d+p) = x*d + x*p","type_family":"Int","context":["Thin","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_742","id":88741,"instance":742,"goal":"all c:Nat. c <= c","type_family":"Nat","context":["BackThru","D 1","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_743","id":88742,"instance":743,"goal":"all chi:Real. all kappa:Real. all delta:Real. (chi <= kappa) => (kappa <= delta) => (chi <= delta)","type_family":"Real","context":["SplitOn","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_744","id":88743,"instance":744,"goal":"all p:Atom. p = p","type_family":"Atom","context":["D 0","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_745","id":88744,"instance":745,"goal":"all g:Bool. all c:Bool. (g = c) => (c = g)","type_family":"Bool","context":["Reduce","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_746","id":88745,"instance":746,"goal":"(d) => (e) => (d)","type_family":"List(Int)","context":["InstLemma","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_747","id":88746,"instance":747,"goal":"((x) & (d)) iff ((d) & (x))","type_family":"List(Nat)","context":["Auto","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_748","id":88747,"instance":748,"goal":"((phi) | (omega)) iff ((omega) | (phi))","type_family":"List(Atom)","context":["Reduce","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_749","id":88748,"instance":749,"goal":"not (not (g)) => (g)","type_family":"Int -> Int","context":["SplitOn","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_750","id":88749,"instance":750,"goal":"all u:Nat -> Nat. all v:Nat -> Nat. all h:Nat -> Nat. append(u, append(v, h)) = append(append(u, v), h)","type_family":"Nat -> Nat","context":["Lemma","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_751","id":88750,"instance":751,"goal":"all b:Int. all n:Int. b + n = n + b","type_family":"Int","context":["BackThru","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_752","id":88751,"instance":752,"goal":"all alpha:Nat. all delta:Nat. all phi:Nat. (alpha+delta)+phi = alpha+(delta+phi)","type_family":"Nat","context":["Lemma","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_753","id":88752,"instance":753,"goal":"all m:Real. m + 0 = m","type_family":"Real","context":["BackThru","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_754","id":88753,"instance":754,"goal":"all f:Atom. all n:Atom. f * n = n * f","type_family":"Atom","context":["D -1","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_755","id":88754,"instance":755,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["SplitOn","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_756","id":88755,"instance":756,"goal":"all w:List(Int). all b:List(Int). all e:List(Int). w*(b+e) = w*b + w*e","type_family":"List(Int)","context":["RepUR","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_757","id":88756,"instance":757,"goal":"all e:List(Nat). e <= e","type_family":"List(Nat)","context":["DoSubst","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_758","id":88757,"instance":758,"goal":"all n:List(Atom). all w:List(Atom). all y:List(Atom). (n <= w) => (w <= y) => (n <= y)","type_family":"List(Atom)","context":["Complete Auto","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_759","id":88758,"instance":759,"goal":"all h:Int -> Int. h = h","type_family":"Int -> Int","context":["SplitOn","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_760","id":88759,"instance":760,"goal":"all n:Nat -> Nat. all e:Nat -> Nat. (n = e) => (e = n)","type_family":"Nat -> Nat","context":["D 1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_761","id":88760,"instance":761,"goal":"(u) => (w) => (u)","type_family":"Int","context":["D 1","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_762","id":88761,"instance":762,"goal":"((h) & (b)) iff ((b) & (h))","type_family":"Nat","context":["D -1","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_763","id":88762,"instance":763,"goal":"((z) | (y)) iff ((y) | (z))","type_family":"Real","context":["InstLemma","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_764","id":88763,"instance":764,"goal":"not (not (v)) => (v)","type_family":"Atom","context":["D -1","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_765","id":88764,"instance":765,"goal":"all g:Bool. all f:Bool. all q:Bool. append(g, append(f, q)) = append(append(g, f), q)","type_family":"Bool","context":["Reduce","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_766","id":88765,"instance":766,"goal":"all y:List(Int). all g:List(Int). y + g = g + y","type_family":"List(Int)","context":["D 0","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_767","id":88766,"instance":767,"goal":"all e:List(Nat). all n:List(Nat). all m:List(Nat). (e+n)+m = e+(n+m)","type_family":"List(Nat)","context":["Complete Auto","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_768","id":88767,"instance":768,"goal":"all x:List(Atom). x + 0 = x","type_family":"List(Atom)","context":["D 0","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_769","id":88768,"instance":769,"goal":"all e:Int -> Int. all x:Int -> Int. e * x = x * e","type_family":"Int -> Int","context":["SplitOn","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_770","id":88769,"instance":770,"goal":"all d:Nat -> Nat. d * 1 = d","type_family":"Nat -> Nat","context":["RepUR","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_771","id":88770,"instance":771,"goal":"all n:Int. all q:Int. all e:Int. n*(q+e) = n*q + n*e","type_family":"Int","context":["D 1","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_772","id":88771,"instance":772,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["D 1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_773","id":88772,"instance":773,"goal":"all h:Real. all v:Real. all z:Real. (h <= v) => (v <= z) => (h <= z)","type_family":"Real","context":["Auto","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_774","id":88773,"instance":774,"goal":"all z:Atom. z = z","type_family":"Atom","context":["BackThru","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_775","id":88774,"instance":775,"goal":"all x:Bool. all a:Bool. (x = a) => (a = x)","type_family":"Bool","context":["Reduce","D -1","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_776","id":88775,"instance":776,"goal":"(q) => (v) => (q)","type_family":"List(Int)","context":["D -1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_777","id":88776,"instance":777,"goal":"((f) & (d)) iff ((d) & (f))","type_family":"List(Nat)","context":["DoSubst","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_778","id":88777,"instance":778,"goal":"((x) | (n)) iff ((n) | (x))","type_family":"List(Atom)","context":["Complete Auto","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_779","id":88778,"instance":779,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["D -1","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_780","id":88779,"instance":780,"goal":"all p:Nat -> Nat. all w:Nat -> Nat. all m:Nat -> Nat. append(p, append(w, m)) = append(append(p, w), m)","type_family":"Nat -> Nat","context":["D -1","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_781","id":88780,"instance":781,"goal":"all d:Int. all x:Int. d + x = x + d","type_family":"Int","context":["D 1","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_782","id":88781,"instance":782,"goal":"all a:Nat. all h:Nat. all f:Nat. (a+h)+f = a+(h+f)","type_family":"Nat","context":["Thin","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_783","id":88782,"instance":783,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["D -1","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_784","id":88783,"instance":784,"goal":"all m:Atom. all z:Atom. m * z = z * m","type_family":"Atom","context":["SplitOn","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_785","id":88784,"instance":785,"goal":"all q:Bool. q * 1 = q","type_family":"Bool","context":["Reduce","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_786","id":88785,"instance":786,"goal":"all a:List(Int). all z:List(Int). all b:List(Int). a*(z+b) = a*z + a*b","type_family":"List(Int)","context":["D -1","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_787","id":88786,"instance":787,"goal":"all alpha:List(Nat). alpha <= alpha","type_family":"List(Nat)","context":["DoSubst","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_788","id":88787,"instance":788,"goal":"all v:List(Atom). all d:List(Atom). all n:List(Atom). (v <= d) => (d <= n) => (v <= n)","type_family":"List(Atom)","context":["Complete Auto","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_789","id":88788,"instance":789,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["DoSubst","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_790","id":88789,"instance":790,"goal":"all m:Nat -> Nat. all a:Nat -> Nat. (m = a) => (a = m)","type_family":"Nat -> Nat","context":["Auto","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_791","id":88790,"instance":791,"goal":"(p) => (c) => (p)","type_family":"Int","context":["Lemma","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_792","id":88791,"instance":792,"goal":"((p) & (f)) iff ((f) & (p))","type_family":"Nat","context":["D 1","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_793","id":88792,"instance":793,"goal":"((h) | (y)) iff ((y) | (h))","type_family":"Real","context":["InstLemma","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_794","id":88793,"instance":794,"goal":"not (not (a)) => (a)","type_family":"Atom","context":["D -1","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_795","id":88794,"instance":795,"goal":"all y:Bool. all n:Bool. all g:Bool. append(y, append(n, g)) = append(append(y, n), g)","type_family":"Bool","context":["RepUR","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_796","id":88795,"instance":796,"goal":"all n:List(Int). all m:List(Int). n + m = m + n","type_family":"List(Int)","context":["Assert","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_797","id":88796,"instance":797,"goal":"all w:List(Nat). all e:List(Nat). all c:List(Nat). (w+e)+c = w+(e+c)","type_family":"List(Nat)","context":["Complete Auto","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_798","id":88797,"instance":798,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["D 0","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_799","id":88798,"instance":799,"goal":"all b:Int -> Int. all z:Int -> Int. b * z = z * b","type_family":"Int -> Int","context":["InstLemma","InstLemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_800","id":88799,"instance":800,"goal":"all b:Nat -> Nat. b * 1 = b","type_family":"Nat -> Nat","context":["Assert","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_801","id":88800,"instance":801,"goal":"all f:Int. all p:Int. all a:Int. f*(p+a) = f*p + f*a","type_family":"Int","context":["Lemma","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_802","id":88801,"instance":802,"goal":"all e:Nat. e <= e","type_family":"Nat","context":["D 1","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_803","id":88802,"instance":803,"goal":"all u:Real. all p:Real. all b:Real. (u <= p) => (p <= b) => (u <= b)","type_family":"Real","context":["Complete Auto","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_804","id":88803,"instance":804,"goal":"all m:Atom. m = m","type_family":"Atom","context":["Thin","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_805","id":88804,"instance":805,"goal":"all x:Bool. all d:Bool. (x = d) => (d = x)","type_family":"Bool","context":["Reduce","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_806","id":88805,"instance":806,"goal":"(alpha) => (chi) => (alpha)","type_family":"List(Int)","context":["Reduce","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_807","id":88806,"instance":807,"goal":"((x) & (g)) iff ((g) & (x))","type_family":"List(Nat)","context":["SplitOn","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_808","id":88807,"instance":808,"goal":"((e) | (y)) iff ((y) | (e))","type_family":"List(Atom)","context":["D 1","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_809","id":88808,"instance":809,"goal":"not (not (m)) => (m)","type_family":"Int -> Int","context":["InstLemma","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_810","id":88809,"instance":810,"goal":"all psi:Nat -> Nat. all epsilon:Nat -> Nat. all beta:Nat -> Nat. append(psi, append(epsilon, beta)) = append(append(psi, epsilon), beta)","type_family":"Nat -> Nat","context":["D 0","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_811","id":88810,"instance":811,"goal":"all w:Int. all u:Int. w + u = u + w","type_family":"Int","context":["Auto","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_812","id":88811,"instance":812,"goal":"all phi:Nat. all alpha:Nat. all psi:Nat. (phi+alpha)+psi = phi+(alpha+psi)","type_family":"Nat","context":["BackThru","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_813","id":88812,"instance":813,"goal":"all q:Real. q + 0 = q","type_family":"Real","context":["D 1","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_814","id":88813,"instance":814,"goal":"all m:Atom. all g:Atom. m * g = g * m","type_family":"Atom","context":["D 0","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_815","id":88814,"instance":815,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["RepUR","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_816","id":88815,"instance":816,"goal":"all u:List(Int). all p:List(Int). all e:List(Int). u*(p+e) = u*p + u*e","type_family":"List(Int)","context":["D -1","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_817","id":88816,"instance":817,"goal":"all c:List(Nat). c <= c","type_family":"List(Nat)","context":["D -1","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_818","id":88817,"instance":818,"goal":"all f:List(Atom). all e:List(Atom). all z:List(Atom). (f <= e) => (e <= z) => (f <= z)","type_family":"List(Atom)","context":["D 0","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_819","id":88818,"instance":819,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["Auto","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_820","id":88819,"instance":820,"goal":"all b:Nat -> Nat. all x:Nat -> Nat. (b = x) => (x = b)","type_family":"Nat -> Nat","context":["Reduce","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_821","id":88820,"instance":821,"goal":"(m) => (h) => (m)","type_family":"Int","context":["Auto","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_822","id":88821,"instance":822,"goal":"((y) & (z)) iff ((z) & (y))","type_family":"Nat","context":["RepUR","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_823","id":88822,"instance":823,"goal":"((p) | (e)) iff ((e) | (p))","type_family":"Real","context":["Assert","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_824","id":88823,"instance":824,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["D 1","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_825","id":88824,"instance":825,"goal":"all n:Bool. all c:Bool. all g:Bool. append(n, append(c, g)) = append(append(n, c), g)","type_family":"Bool","context":["InstLemma","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_826","id":88825,"instance":826,"goal":"all a:List(Int). all z:List(Int). a + z = z + a","type_family":"List(Int)","context":["InstLemma","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_827","id":88826,"instance":827,"goal":"all a:List(Nat). all u:List(Nat). all d:List(Nat). (a+u)+d = a+(u+d)","type_family":"List(Nat)","context":["D 0","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_828","id":88827,"instance":828,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["D 1","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_829","id":88828,"instance":829,"goal":"all g:Int -> Int. all a:Int -> Int. g * a = a * g","type_family":"Int -> Int","context":["DoSubst","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_830","id":88829,"instance":830,"goal":"all d:Nat -> Nat. d * 1 = d","type_family":"Nat -> Nat","context":["Assert","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_831","id":88830,"instance":831,"goal":"all n:Int. all p:Int. all e:Int. n*(p+e) = n*p + n*e","type_family":"Int","context":["Assert","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_832","id":88831,"instance":832,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["D 1","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_833","id":88832,"instance":833,"goal":"all q:Real. all y:Real. all m:Real. (q <= y) => (y <= m) => (q <= m)","type_family":"Real","context":["Auto","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_834","id":88833,"instance":834,"goal":"all x:Atom. x = x","type_family":"Atom","context":["BackThru","SplitOn","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_835","id":88834,"instance":835,"goal":"all p:Bool. all f:Bool. (p = f) => (f = p)","type_family":"Bool","context":["SplitOn","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_836","id":88835,"instance":836,"goal":"(n) => (e) => (n)","type_family":"List(Int)","context":["SplitOn","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_837","id":88836,"instance":837,"goal":"((f) & (z)) iff ((z) & (f))","type_family":"List(Nat)","context":["SplitOn","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_838","id":88837,"instance":838,"goal":"((b) | (q)) iff ((q) | (b))","type_family":"List(Atom)","context":["Auto","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_839","id":88838,"instance":839,"goal":"not (not (alpha)) => (alpha)","type_family":"Int -> Int","context":["Lemma","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_840","id":88839,"instance":840,"goal":"all f:Nat -> Nat. all q:Nat -> Nat. all b:Nat -> Nat. append(f, append(q, b)) = append(append(f, q), b)","type_family":"Nat -> Nat","context":["Auto","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_841","id":88840,"instance":841,"goal":"all m:Int. all u:Int. m + u = u + m","type_family":"Int","context":["Reduce","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_842","id":88841,"instance":842,"goal":"all z:Nat. all b:Nat. all e:Nat. (z+b)+e = z+(b+e)","type_family":"Nat","context":["D 1","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_843","id":88842,"instance":843,"goal":"all w:Real. w + 0 = w","type_family":"Real","context":["Auto","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_844","id":88843,"instance":844,"goal":"all a:Atom. all b:Atom. a * b = b * a","type_family":"Atom","context":["D -1","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_845","id":88844,"instance":845,"goal":"all y:Bool. y * 1 = y","type_family":"Bool","context":["Assert","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_846","id":88845,"instance":846,"goal":"all u:List(Int). all w:List(Int). all y:List(Int). u*(w+y) = u*w + u*y","type_family":"List(Int)","context":["InstLemma","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_847","id":88846,"instance":847,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["D -1","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_848","id":88847,"instance":848,"goal":"all d:List(Atom). all f:List(Atom). all q:List(Atom). (d <= f) => (f <= q) => (d <= q)","type_family":"List(Atom)","context":["Auto","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_849","id":88848,"instance":849,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["D 1","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_850","id":88849,"instance":850,"goal":"all m:Nat -> Nat. all e:Nat -> Nat. (m = e) => (e = m)","type_family":"Nat -> Nat","context":["Assert","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_851","id":88850,"instance":851,"goal":"(z) => (y) => (z)","type_family":"Int","context":["D 1","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_852","id":88851,"instance":852,"goal":"((kappa) & (phi)) iff ((phi) & (kappa))","type_family":"Nat","context":["InstLemma","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_853","id":88852,"instance":853,"goal":"((d) | (g)) iff ((g) | (d))","type_family":"Real","context":["DoSubst","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_854","id":88853,"instance":854,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Auto","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_855","id":88854,"instance":855,"goal":"all d:Bool. all w:Bool. all y:Bool. append(d, append(w, y)) = append(append(d, w), y)","type_family":"Bool","context":["D 0","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_856","id":88855,"instance":856,"goal":"all c:List(Int). all f:List(Int). c + f = f + c","type_family":"List(Int)","context":["D 0","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_857","id":88856,"instance":857,"goal":"all b:List(Nat). all w:List(Nat). all h:List(Nat). (b+w)+h = b+(w+h)","type_family":"List(Nat)","context":["D -1","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_858","id":88857,"instance":858,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["D 1","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_859","id":88858,"instance":859,"goal":"all h:Int -> Int. all x:Int -> Int. h * x = x * h","type_family":"Int -> Int","context":["D -1","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_860","id":88859,"instance":860,"goal":"all kappa:Nat -> Nat. kappa * 1 = kappa","type_family":"Nat -> Nat","context":["Complete Auto","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_861","id":88860,"instance":861,"goal":"all f:Int. all h:Int. all x:Int. f*(h+x) = f*h + f*x","type_family":"Int","context":["Complete Auto","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_862","id":88861,"instance":862,"goal":"all d:Nat. d <= d","type_family":"Nat","context":["D -1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_863","id":88862,"instance":863,"goal":"all a:Real. all w:Real. all v:Real. (a <= w) => (w <= v) => (a <= v)","type_family":"Real","context":["BackThru","D -1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_864","id":88863,"instance":864,"goal":"all q:Atom. q = q","type_family":"Atom","context":["Thin","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_865","id":88864,"instance":865,"goal":"all c:Bool. all a:Bool. (c = a) => (a = c)","type_family":"Bool","context":["Thin","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_866","id":88865,"instance":866,"goal":"(p) => (w) => (p)","type_family":"List(Int)","context":["Lemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_867","id":88866,"instance":867,"goal":"((a) & (f)) iff ((f) & (a))","type_family":"List(Nat)","context":["Lemma","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_868","id":88867,"instance":868,"goal":"((v) | (z)) iff ((z) | (v))","type_family":"List(Atom)","context":["D 0","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_869","id":88868,"instance":869,"goal":"not (not (y)) => (y)","type_family":"Int -> Int","context":["D 1","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_870","id":88869,"instance":870,"goal":"all h:Nat -> Nat. all n:Nat -> Nat. all e:Nat -> Nat. append(h, append(n, e)) = append(append(h, n), e)","type_family":"Nat -> Nat","context":["RepUR","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_871","id":88870,"instance":871,"goal":"all y:Int. all b:Int. y + b = b + y","type_family":"Int","context":["D 0","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_872","id":88871,"instance":872,"goal":"all z:Nat. all a:Nat. all x:Nat. (z+a)+x = z+(a+x)","type_family":"Nat","context":["Lemma","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_873","id":88872,"instance":873,"goal":"all h:Real. h + 0 = h","type_family":"Real","context":["Thin","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_874","id":88873,"instance":874,"goal":"all m:Atom. all v:Atom. m * v = v * m","type_family":"Atom","context":["Thin","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_875","id":88874,"instance":875,"goal":"all d:Bool. d * 1 = d","type_family":"Bool","context":["DoSubst","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_876","id":88875,"instance":876,"goal":"all f:List(Int). all x:List(Int). all c:List(Int). f*(x+c) = f*x + f*c","type_family":"List(Int)","context":["D -1","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_877","id":88876,"instance":877,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["D 1","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_878","id":88877,"instance":878,"goal":"all p:List(Atom). all g:List(Atom). all b:List(Atom). (p <= g) => (g <= b) => (p <= b)","type_family":"List(Atom)","context":["Thin","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_879","id":88878,"instance":879,"goal":"all q:Int -> Int. q = q","type_family":"Int -> Int","context":["Complete Auto","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_880","id":88879,"instance":880,"goal":"all e:Nat -> Nat. all u:Nat -> Nat. (e = u) => (u = e)","type_family":"Nat -> Nat","context":["D -1","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_881","id":88880,"instance":881,"goal":"(w) => (p) => (w)","type_family":"Int","context":["D -1","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_882","id":88881,"instance":882,"goal":"((w) & (z)) iff ((z) & (w))","type_family":"Nat","context":["D 0","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_883","id":88882,"instance":883,"goal":"((g) | (n)) iff ((n) | (g))","type_family":"Real","context":["Reduce","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_884","id":88883,"instance":884,"goal":"not (not (v)) => (v)","type_family":"Atom","context":["BackThru","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_885","id":88884,"instance":885,"goal":"all w:Bool. all h:Bool. all g:Bool. append(w, append(h, g)) = append(append(w, h), g)","type_family":"Bool","context":["D 0","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_886","id":88885,"instance":886,"goal":"all psi:List(Int). all delta:List(Int). psi + delta = delta + psi","type_family":"List(Int)","context":["Lemma","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_887","id":88886,"instance":887,"goal":"all h:List(Nat). all f:List(Nat). all e:List(Nat). (h+f)+e = h+(f+e)","type_family":"List(Nat)","context":["Reduce","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_888","id":88887,"instance":888,"goal":"all omega:List(Atom). omega + 0 = omega","type_family":"List(Atom)","context":["DoSubst","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_889","id":88888,"instance":889,"goal":"all alpha:Int -> Int. all omega:Int -> Int. alpha * omega = omega * alpha","type_family":"Int -> Int","context":["D -1","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_890","id":88889,"instance":890,"goal":"all phi:Nat -> Nat. phi * 1 = phi","type_family":"Nat -> Nat","context":["InstLemma","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_891","id":88890,"instance":891,"goal":"all f:Int. all c:Int. all u:Int. f*(c+u) = f*c + f*u","type_family":"Int","context":["Assert","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_892","id":88891,"instance":892,"goal":"all v:Nat. v <= v","type_family":"Nat","context":["Reduce","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_893","id":88892,"instance":893,"goal":"all psi:Real. all beta:Real. all delta:Real. (psi <= beta) => (beta <= delta) => (psi <= delta)","type_family":"Real","context":["Complete Auto","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_894","id":88893,"instance":894,"goal":"all h:Atom. h = h","type_family":"Atom","context":["SplitOn","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_895","id":88894,"instance":895,"goal":"all w:Bool. all g:Bool. (w = g) => (g = w)","type_family":"Bool","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_896","id":88895,"instance":896,"goal":"(z) => (f) => (z)","type_family":"List(Int)","context":["D 0","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_897","id":88896,"instance":897,"goal":"((epsilon) & (alpha)) iff ((alpha) & (epsilon))","type_family":"List(Nat)","context":["DoSubst","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_898","id":88897,"instance":898,"goal":"((u) | (w)) iff ((w) | (u))","type_family":"List(Atom)","context":["RepUR","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_899","id":88898,"instance":899,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["InstLemma","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_900","id":88899,"instance":900,"goal":"all h:Nat -> Nat. all f:Nat -> Nat. all u:Nat -> Nat. append(h, append(f, u)) = append(append(h, f), u)","type_family":"Nat -> Nat","context":["Auto","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_901","id":88900,"instance":901,"goal":"all b:Int. all w:Int. b + w = w + b","type_family":"Int","context":["Auto","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_902","id":88901,"instance":902,"goal":"all m:Nat. all p:Nat. all a:Nat. (m+p)+a = m+(p+a)","type_family":"Nat","context":["DoSubst","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_903","id":88902,"instance":903,"goal":"all a:Real. a + 0 = a","type_family":"Real","context":["Thin","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_904","id":88903,"instance":904,"goal":"all h:Atom. all y:Atom. h * y = y * h","type_family":"Atom","context":["DoSubst","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_905","id":88904,"instance":905,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["D 0","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_906","id":88905,"instance":906,"goal":"all alpha:List(Int). all phi:List(Int). all gamma:List(Int). alpha*(phi+gamma) = alpha*phi + alpha*gamma","type_family":"List(Int)","context":["Assert","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_907","id":88906,"instance":907,"goal":"all y:List(Nat). y <= y","type_family":"List(Nat)","context":["D 1","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_908","id":88907,"instance":908,"goal":"all p:List(Atom). all m:List(Atom). all e:List(Atom). (p <= m) => (m <= e) => (p <= e)","type_family":"List(Atom)","context":["BackThru","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_909","id":88908,"instance":909,"goal":"all z:Int -> Int. z = z","type_family":"Int -> Int","context":["Reduce","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_910","id":88909,"instance":910,"goal":"all beta:Nat -> Nat. all delta:Nat -> Nat. (beta = delta) => (delta = beta)","type_family":"Nat -> Nat","context":["InstLemma","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_911","id":88910,"instance":911,"goal":"(m) => (p) => (m)","type_family":"Int","context":["DoSubst","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_912","id":88911,"instance":912,"goal":"((omega) & (phi)) iff ((phi) & (omega))","type_family":"Nat","context":["D 1","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_913","id":88912,"instance":913,"goal":"((w) | (m)) iff ((m) | (w))","type_family":"Real","context":["Assert","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_914","id":88913,"instance":914,"goal":"not (not (n)) => (n)","type_family":"Atom","context":["Complete Auto","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_915","id":88914,"instance":915,"goal":"all w:Bool. all a:Bool. all e:Bool. append(w, append(a, e)) = append(append(w, a), e)","type_family":"Bool","context":["DoSubst","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_916","id":88915,"instance":916,"goal":"all g:List(Int). all u:List(Int). g + u = u + g","type_family":"List(Int)","context":["DoSubst","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_917","id":88916,"instance":917,"goal":"all chi:List(Nat). all phi:List(Nat). all omega:List(Nat). (chi+phi)+omega = chi+(phi+omega)","type_family":"List(Nat)","context":["D 0","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_918","id":88917,"instance":918,"goal":"all w:List(Atom). w + 0 = w","type_family":"List(Atom)","context":["RepUR","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_919","id":88918,"instance":919,"goal":"all alpha:Int -> Int. all beta:Int -> Int. alpha * beta = beta * alpha","type_family":"Int -> Int","context":["DoSubst","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_920","id":88919,"instance":920,"goal":"all n:Nat -> Nat. n * 1 = n","type_family":"Nat -> Nat","context":["SplitOn","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_921","id":88920,"instance":921,"goal":"all b:Int. all z:Int. all v:Int. b*(z+v) = b*z + b*v","type_family":"Int","context":["BackThru","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_922","id":88921,"instance":922,"goal":"all q:Nat. q <= q","type_family":"Nat","context":["InstLemma","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_923","id":88922,"instance":923,"goal":"all h:Real. all q:Real. all f:Real. (h <= q) => (q <= f) => (h <= f)","type_family":"Real","context":["Reduce","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_924","id":88923,"instance":924,"goal":"all b:Atom. b = b","type_family":"Atom","context":["Auto","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_925","id":88924,"instance":925,"goal":"all b:Bool. all h:Bool. (b = h) => (h = b)","type_family":"Bool","context":["Reduce","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_926","id":88925,"instance":926,"goal":"(z) => (p) => (z)","type_family":"List(Int)","context":["SplitOn","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_927","id":88926,"instance":927,"goal":"((h) & (u)) iff ((u) & (h))","type_family":"List(Nat)","context":["Thin","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_928","id":88927,"instance":928,"goal":"((e) | (q)) iff ((q) | (e))","type_family":"List(Atom)","context":["RepUR","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_929","id":88928,"instance":929,"goal":"not (not (gamma)) => (gamma)","type_family":"Int -> Int","context":["RepUR","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_930","id":88929,"instance":930,"goal":"all x:Nat -> Nat. all p:Nat -> Nat. all u:Nat -> Nat. append(x, append(p, u)) = append(append(x, p), u)","type_family":"Nat -> Nat","context":["RepUR","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_931","id":88930,"instance":931,"goal":"all h:Int. all p:Int. h + p = p + h","type_family":"Int","context":["D 1","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_932","id":88931,"instance":932,"goal":"all a:Nat. all p:Nat. all x:Nat. (a+p)+x = a+(p+x)","type_family":"Nat","context":["D 1","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_933","id":88932,"instance":933,"goal":"all x:Real. x + 0 = x","type_family":"Real","context":["InstLemma","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_934","id":88933,"instance":934,"goal":"all epsilon:Atom. all alpha:Atom. epsilon * alpha = alpha * epsilon","type_family":"Atom","context":["Thin","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_935","id":88934,"instance":935,"goal":"all v:Bool. v * 1 = v","type_family":"Bool","context":["D -1","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_936","id":88935,"instance":936,"goal":"all z:List(Int). all w:List(Int). all m:List(Int). z*(w+m) = z*w + z*m","type_family":"List(Int)","context":["D -1","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_937","id":88936,"instance":937,"goal":"all g:List(Nat). g <= g","type_family":"List(Nat)","context":["D 1","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_938","id":88937,"instance":938,"goal":"all v:List(Atom). all h:List(Atom). all n:List(Atom). (v <= h) => (h <= n) => (v <= n)","type_family":"List(Atom)","context":["DoSubst","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_939","id":88938,"instance":939,"goal":"all v:Int -> Int. v = v","type_family":"Int -> Int","context":["D -1","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_940","id":88939,"instance":940,"goal":"all f:Nat -> Nat. all m:Nat -> Nat. (f = m) => (m = f)","type_family":"Nat -> Nat","context":["Complete Auto","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_941","id":88940,"instance":941,"goal":"(c) => (n) => (c)","type_family":"Int","context":["Complete Auto","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_942","id":88941,"instance":942,"goal":"((z) & (f)) iff ((f) & (z))","type_family":"Nat","context":["Thin","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_943","id":88942,"instance":943,"goal":"((u) | (c)) iff ((c) | (u))","type_family":"Real","context":["D 0","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_944","id":88943,"instance":944,"goal":"not (not (gamma)) => (gamma)","type_family":"Atom","context":["D 0","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_945","id":88944,"instance":945,"goal":"all x:Bool. all z:Bool. all h:Bool. append(x, append(z, h)) = append(append(x, z), h)","type_family":"Bool","context":["D 0","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_946","id":88945,"instance":946,"goal":"all d:List(Int). all n:List(Int). d + n = n + d","type_family":"List(Int)","context":["SplitOn","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_947","id":88946,"instance":947,"goal":"all g:List(Nat). all e:List(Nat). all u:List(Nat). (g+e)+u = g+(e+u)","type_family":"List(Nat)","context":["D -1","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_948","id":88947,"instance":948,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["Assert","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_949","id":88948,"instance":949,"goal":"all n:Int -> Int. all g:Int -> Int. n * g = g * n","type_family":"Int -> Int","context":["Auto","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_950","id":88949,"instance":950,"goal":"all y:Nat -> Nat. y * 1 = y","type_family":"Nat -> Nat","context":["SplitOn","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_951","id":88950,"instance":951,"goal":"all b:Int. all u:Int. all w:Int. b*(u+w) = b*u + b*w","type_family":"Int","context":["D 1","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_952","id":88951,"instance":952,"goal":"all h:Nat. h <= h","type_family":"Nat","context":["RepUR","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_953","id":88952,"instance":953,"goal":"all q:Real. all p:Real. all f:Real. (q <= p) => (p <= f) => (q <= f)","type_family":"Real","context":["Auto","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_954","id":88953,"instance":954,"goal":"all v:Atom. v = v","type_family":"Atom","context":["Assert","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_955","id":88954,"instance":955,"goal":"all x:Bool. all d:Bool. (x = d) => (d = x)","type_family":"Bool","context":["D 1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_956","id":88955,"instance":956,"goal":"(b) => (u) => (b)","type_family":"List(Int)","context":["Thin","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_957","id":88956,"instance":957,"goal":"((q) & (g)) iff ((g) & (q))","type_family":"List(Nat)","context":["Assert","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_958","id":88957,"instance":958,"goal":"((epsilon) | (beta)) iff ((beta) | (epsilon))","type_family":"List(Atom)","context":["Lemma","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_959","id":88958,"instance":959,"goal":"not (not (d)) => (d)","type_family":"Int -> Int","context":["Auto","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_960","id":88959,"instance":960,"goal":"all chi:Nat -> Nat. all omega:Nat -> Nat. all psi:Nat -> Nat. append(chi, append(omega, psi)) = append(append(chi, omega), psi)","type_family":"Nat -> Nat","context":["Reduce","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_961","id":88960,"instance":961,"goal":"all c:Int. all b:Int. c + b = b + c","type_family":"Int","context":["D 1","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_962","id":88961,"instance":962,"goal":"all h:Nat. all v:Nat. all n:Nat. (h+v)+n = h+(v+n)","type_family":"Nat","context":["D 1","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_963","id":88962,"instance":963,"goal":"all q:Real. q + 0 = q","type_family":"Real","context":["InstLemma","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_964","id":88963,"instance":964,"goal":"all n:Atom. all u:Atom. n * u = u * n","type_family":"Atom","context":["BackThru","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_965","id":88964,"instance":965,"goal":"all alpha:Bool. alpha * 1 = alpha","type_family":"Bool","context":["RepUR","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_966","id":88965,"instance":966,"goal":"all m:List(Int). all y:List(Int). all v:List(Int). m*(y+v) = m*y + m*v","type_family":"List(Int)","context":["Lemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_967","id":88966,"instance":967,"goal":"all x:List(Nat). x <= x","type_family":"List(Nat)","context":["BackThru","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_968","id":88967,"instance":968,"goal":"all z:List(Atom). all n:List(Atom). all e:List(Atom). (z <= n) => (n <= e) => (z <= e)","type_family":"List(Atom)","context":["Assert","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_969","id":88968,"instance":969,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["InstLemma","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_970","id":88969,"instance":970,"goal":"all y:Nat -> Nat. all d:Nat -> Nat. (y = d) => (d = y)","type_family":"Nat -> Nat","context":["BackThru","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_971","id":88970,"instance":971,"goal":"(omega) => (beta) => (omega)","type_family":"Int","context":["DoSubst","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_972","id":88971,"instance":972,"goal":"((v) & (q)) iff ((q) & (v))","type_family":"Nat","context":["D -1","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_973","id":88972,"instance":973,"goal":"((w) | (d)) iff ((d) | (w))","type_family":"Real","context":["Complete Auto","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_974","id":88973,"instance":974,"goal":"not (not (z)) => (z)","type_family":"Atom","context":["Thin","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_975","id":88974,"instance":975,"goal":"all a:Bool. all p:Bool. all n:Bool. append(a, append(p, n)) = append(append(a, p), n)","type_family":"Bool","context":["Thin","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_976","id":88975,"instance":976,"goal":"all z:List(Int). all m:List(Int). z + m = m + z","type_family":"List(Int)","context":["Assert","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_977","id":88976,"instance":977,"goal":"all u:List(Nat). all z:List(Nat). all q:List(Nat). (u+z)+q = u+(z+q)","type_family":"List(Nat)","context":["Lemma","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_978","id":88977,"instance":978,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["D 0","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_979","id":88978,"instance":979,"goal":"all h:Int -> Int. all d:Int -> Int. h * d = d * h","type_family":"Int -> Int","context":["RepUR","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_980","id":88979,"instance":980,"goal":"all h:Nat -> Nat. h * 1 = h","type_family":"Nat -> Nat","context":["D -1","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_981","id":88980,"instance":981,"goal":"all m:Int. all y:Int. all g:Int. m*(y+g) = m*y + m*g","type_family":"Int","context":["D -1","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_982","id":88981,"instance":982,"goal":"all w:Nat. w <= w","type_family":"Nat","context":["D -1","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_983","id":88982,"instance":983,"goal":"all n:Real. all h:Real. all a:Real. (n <= h) => (h <= a) => (n <= a)","type_family":"Real","context":["BackThru","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_984","id":88983,"instance":984,"goal":"all x:Atom. x = x","type_family":"Atom","context":["D 0","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_985","id":88984,"instance":985,"goal":"all z:Bool. all h:Bool. (z = h) => (h = z)","type_family":"Bool","context":["BackThru","InstLemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_986","id":88985,"instance":986,"goal":"(v) => (p) => (v)","type_family":"List(Int)","context":["D 0","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_987","id":88986,"instance":987,"goal":"((a) & (e)) iff ((e) & (a))","type_family":"List(Nat)","context":["Reduce","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_988","id":88987,"instance":988,"goal":"((c) | (m)) iff ((m) | (c))","type_family":"List(Atom)","context":["D -1","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_989","id":88988,"instance":989,"goal":"not (not (f)) => (f)","type_family":"Int -> Int","context":["D 0","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_990","id":88989,"instance":990,"goal":"all e:Nat -> Nat. all n:Nat -> Nat. all a:Nat -> Nat. append(e, append(n, a)) = append(append(e, n), a)","type_family":"Nat -> Nat","context":["D -1","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_991","id":88990,"instance":991,"goal":"all c:Int. all p:Int. c + p = p + c","type_family":"Int","context":["Reduce","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_992","id":88991,"instance":992,"goal":"all e:Nat. all b:Nat. all n:Nat. (e+b)+n = e+(b+n)","type_family":"Nat","context":["SplitOn","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_993","id":88992,"instance":993,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["D -1","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_994","id":88993,"instance":994,"goal":"all d:Atom. all n:Atom. d * n = n * d","type_family":"Atom","context":["Auto","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_995","id":88994,"instance":995,"goal":"all a:Bool. a * 1 = a","type_family":"Bool","context":["Lemma","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_996","id":88995,"instance":996,"goal":"all a:List(Int). all b:List(Int). all q:List(Int). a*(b+q) = a*b + a*q","type_family":"List(Int)","context":["D 0","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_997","id":88996,"instance":997,"goal":"all b:List(Nat). b <= b","type_family":"List(Nat)","context":["DoSubst","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_998","id":88997,"instance":998,"goal":"all g:List(Atom). all z:List(Atom). all a:List(Atom). (g <= z) => (z <= a) => (g <= a)","type_family":"List(Atom)","context":["Thin","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_999","id":88998,"instance":999,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["Assert","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1000","id":88999,"instance":1000,"goal":"all n:Nat -> Nat. all h:Nat -> Nat. (n = h) => (h = n)","type_family":"Nat -> Nat","context":["Auto","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1001","id":89000,"instance":1001,"goal":"(alpha) => (kappa) => (alpha)","type_family":"Int","context":["Assert","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1002","id":89001,"instance":1002,"goal":"((v) & (q)) iff ((q) & (v))","type_family":"Nat","context":["Complete Auto","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1003","id":89002,"instance":1003,"goal":"((b) | (c)) iff ((c) | (b))","type_family":"Real","context":["Complete Auto","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1004","id":89003,"instance":1004,"goal":"not (not (w)) => (w)","type_family":"Atom","context":["D 1","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1005","id":89004,"instance":1005,"goal":"all a:Bool. all b:Bool. all g:Bool. append(a, append(b, g)) = append(append(a, b), g)","type_family":"Bool","context":["Lemma","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1006","id":89005,"instance":1006,"goal":"all n:List(Int). all v:List(Int). n + v = v + n","type_family":"List(Int)","context":["Complete Auto","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1007","id":89006,"instance":1007,"goal":"all x:List(Nat). all d:List(Nat). all a:List(Nat). (x+d)+a = x+(d+a)","type_family":"List(Nat)","context":["BackThru","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1008","id":89007,"instance":1008,"goal":"all chi:List(Atom). chi + 0 = chi","type_family":"List(Atom)","context":["D 0","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1009","id":89008,"instance":1009,"goal":"all y:Int -> Int. all q:Int -> Int. y * q = q * y","type_family":"Int -> Int","context":["RepUR","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1010","id":89009,"instance":1010,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["D 0","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1011","id":89010,"instance":1011,"goal":"all h:Int. all f:Int. all b:Int. h*(f+b) = h*f + h*b","type_family":"Int","context":["SplitOn","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1012","id":89011,"instance":1012,"goal":"all d:Nat. d <= d","type_family":"Nat","context":["InstLemma","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1013","id":89012,"instance":1013,"goal":"all x:Real. all m:Real. all u:Real. (x <= m) => (m <= u) => (x <= u)","type_family":"Real","context":["DoSubst","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1014","id":89013,"instance":1014,"goal":"all psi:Atom. psi = psi","type_family":"Atom","context":["BackThru","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1015","id":89014,"instance":1015,"goal":"all u:Bool. all m:Bool. (u = m) => (m = u)","type_family":"Bool","context":["SplitOn","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1016","id":89015,"instance":1016,"goal":"(m) => (a) => (m)","type_family":"List(Int)","context":["Lemma","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1017","id":89016,"instance":1017,"goal":"((p) & (b)) iff ((b) & (p))","type_family":"List(Nat)","context":["Reduce","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1018","id":89017,"instance":1018,"goal":"((m) | (x)) iff ((x) | (m))","type_family":"List(Atom)","context":["Lemma","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1019","id":89018,"instance":1019,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["DoSubst","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1020","id":89019,"instance":1020,"goal":"all z:Nat -> Nat. all y:Nat -> Nat. all m:Nat -> Nat. append(z, append(y, m)) = append(append(z, y), m)","type_family":"Nat -> Nat","context":["SplitOn","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1021","id":89020,"instance":1021,"goal":"all g:Int. all u:Int. g + u = u + g","type_family":"Int","context":["Complete Auto","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1022","id":89021,"instance":1022,"goal":"all u:Nat. all g:Nat. all h:Nat. (u+g)+h = u+(g+h)","type_family":"Nat","context":["Complete Auto","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1023","id":89022,"instance":1023,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["BackThru","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1024","id":89023,"instance":1024,"goal":"all m:Atom. all e:Atom. m * e = e * m","type_family":"Atom","context":["Lemma","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1025","id":89024,"instance":1025,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["RepUR","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1026","id":89025,"instance":1026,"goal":"all f:List(Int). all v:List(Int). all w:List(Int). f*(v+w) = f*v + f*w","type_family":"List(Int)","context":["Complete Auto","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1027","id":89026,"instance":1027,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["SplitOn","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1028","id":89027,"instance":1028,"goal":"all g:List(Atom). all h:List(Atom). all y:List(Atom). (g <= h) => (h <= y) => (g <= y)","type_family":"List(Atom)","context":["DoSubst","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1029","id":89028,"instance":1029,"goal":"all chi:Int -> Int. chi = chi","type_family":"Int -> Int","context":["Reduce","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1030","id":89029,"instance":1030,"goal":"all e:Nat -> Nat. all q:Nat -> Nat. (e = q) => (q = e)","type_family":"Nat -> Nat","context":["DoSubst","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1031","id":89030,"instance":1031,"goal":"(f) => (b) => (f)","type_family":"Int","context":["Lemma","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1032","id":89031,"instance":1032,"goal":"((n) & (c)) iff ((c) & (n))","type_family":"Nat","context":["Assert","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1033","id":89032,"instance":1033,"goal":"((q) | (n)) iff ((n) | (q))","type_family":"Real","context":["Complete Auto","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1034","id":89033,"instance":1034,"goal":"not (not (psi)) => (psi)","type_family":"Atom","context":["Lemma","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1035","id":89034,"instance":1035,"goal":"all y:Bool. all u:Bool. all m:Bool. append(y, append(u, m)) = append(append(y, u), m)","type_family":"Bool","context":["InstLemma","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1036","id":89035,"instance":1036,"goal":"all a:List(Int). all g:List(Int). a + g = g + a","type_family":"List(Int)","context":["SplitOn","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1037","id":89036,"instance":1037,"goal":"all m:List(Nat). all q:List(Nat). all x:List(Nat). (m+q)+x = m+(q+x)","type_family":"List(Nat)","context":["InstLemma","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1038","id":89037,"instance":1038,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["Assert","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1039","id":89038,"instance":1039,"goal":"all g:Int -> Int. all n:Int -> Int. g * n = n * g","type_family":"Int -> Int","context":["Complete Auto","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1040","id":89039,"instance":1040,"goal":"all a:Nat -> Nat. a * 1 = a","type_family":"Nat -> Nat","context":["RepUR","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1041","id":89040,"instance":1041,"goal":"all m:Int. all y:Int. all b:Int. m*(y+b) = m*y + m*b","type_family":"Int","context":["SplitOn","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1042","id":89041,"instance":1042,"goal":"all h:Nat. h <= h","type_family":"Nat","context":["SplitOn","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1043","id":89042,"instance":1043,"goal":"all y:Real. all f:Real. all v:Real. (y <= f) => (f <= v) => (y <= v)","type_family":"Real","context":["Complete Auto","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1044","id":89043,"instance":1044,"goal":"all h:Atom. h = h","type_family":"Atom","context":["D 1","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1045","id":89044,"instance":1045,"goal":"all d:Bool. all v:Bool. (d = v) => (v = d)","type_family":"Bool","context":["Reduce","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1046","id":89045,"instance":1046,"goal":"(p) => (h) => (p)","type_family":"List(Int)","context":["D 1","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1047","id":89046,"instance":1047,"goal":"((p) & (f)) iff ((f) & (p))","type_family":"List(Nat)","context":["BackThru","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1048","id":89047,"instance":1048,"goal":"((x) | (d)) iff ((d) | (x))","type_family":"List(Atom)","context":["BackThru","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1049","id":89048,"instance":1049,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["Auto","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1050","id":89049,"instance":1050,"goal":"all a:Nat -> Nat. all c:Nat -> Nat. all f:Nat -> Nat. append(a, append(c, f)) = append(append(a, c), f)","type_family":"Nat -> Nat","context":["D -1","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1051","id":89050,"instance":1051,"goal":"all f:Int. all g:Int. f + g = g + f","type_family":"Int","context":["D -1","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1052","id":89051,"instance":1052,"goal":"all a:Nat. all e:Nat. all g:Nat. (a+e)+g = a+(e+g)","type_family":"Nat","context":["SplitOn","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1053","id":89052,"instance":1053,"goal":"all delta:Real. delta + 0 = delta","type_family":"Real","context":["InstLemma","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1054","id":89053,"instance":1054,"goal":"all q:Atom. all n:Atom. q * n = n * q","type_family":"Atom","context":["Reduce","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1055","id":89054,"instance":1055,"goal":"all p:Bool. p * 1 = p","type_family":"Bool","context":["D 0","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1056","id":89055,"instance":1056,"goal":"all omega:List(Int). all epsilon:List(Int). all phi:List(Int). omega*(epsilon+phi) = omega*epsilon + omega*phi","type_family":"List(Int)","context":["BackThru","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1057","id":89056,"instance":1057,"goal":"all kappa:List(Nat). kappa <= kappa","type_family":"List(Nat)","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1058","id":89057,"instance":1058,"goal":"all w:List(Atom). all m:List(Atom). all x:List(Atom). (w <= m) => (m <= x) => (w <= x)","type_family":"List(Atom)","context":["D -1","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1059","id":89058,"instance":1059,"goal":"all delta:Int -> Int. delta = delta","type_family":"Int -> Int","context":["Complete Auto","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1060","id":89059,"instance":1060,"goal":"all a:Nat -> Nat. all w:Nat -> Nat. (a = w) => (w = a)","type_family":"Nat -> Nat","context":["DoSubst","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1061","id":89060,"instance":1061,"goal":"(g) => (x) => (g)","type_family":"Int","context":["Complete Auto","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1062","id":89061,"instance":1062,"goal":"((omega) & (alpha)) iff ((alpha) & (omega))","type_family":"Nat","context":["InstLemma","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1063","id":89062,"instance":1063,"goal":"((y) | (x)) iff ((x) | (y))","type_family":"Real","context":["Lemma","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1064","id":89063,"instance":1064,"goal":"not (not (y)) => (y)","type_family":"Atom","context":["Thin","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1065","id":89064,"instance":1065,"goal":"all y:Bool. all q:Bool. all c:Bool. append(y, append(q, c)) = append(append(y, q), c)","type_family":"Bool","context":["Assert","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1066","id":89065,"instance":1066,"goal":"all h:List(Int). all c:List(Int). h + c = c + h","type_family":"List(Int)","context":["Lemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1067","id":89066,"instance":1067,"goal":"all q:List(Nat). all y:List(Nat). all e:List(Nat). (q+y)+e = q+(y+e)","type_family":"List(Nat)","context":["BackThru","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1068","id":89067,"instance":1068,"goal":"all a:List(Atom). a + 0 = a","type_family":"List(Atom)","context":["Assert","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1069","id":89068,"instance":1069,"goal":"all v:Int -> Int. all b:Int -> Int. v * b = b * v","type_family":"Int -> Int","context":["RepUR","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1070","id":89069,"instance":1070,"goal":"all v:Nat -> Nat. v * 1 = v","type_family":"Nat -> Nat","context":["Complete Auto","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1071","id":89070,"instance":1071,"goal":"all e:Int. all z:Int. all v:Int. e*(z+v) = e*z + e*v","type_family":"Int","context":["BackThru","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1072","id":89071,"instance":1072,"goal":"all e:Nat. e <= e","type_family":"Nat","context":["SplitOn","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1073","id":89072,"instance":1073,"goal":"all f:Real. all h:Real. all u:Real. (f <= h) => (h <= u) => (f <= u)","type_family":"Real","context":["RepUR","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1074","id":89073,"instance":1074,"goal":"all z:Atom. z = z","type_family":"Atom","context":["InstLemma","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1075","id":89074,"instance":1075,"goal":"all y:Bool. all h:Bool. (y = h) => (h = y)","type_family":"Bool","context":["Assert","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1076","id":89075,"instance":1076,"goal":"(b) => (n) => (b)","type_family":"List(Int)","context":["D 1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1077","id":89076,"instance":1077,"goal":"((d) & (e)) iff ((e) & (d))","type_family":"List(Nat)","context":["DoSubst","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1078","id":89077,"instance":1078,"goal":"((d) | (n)) iff ((n) | (d))","type_family":"List(Atom)","context":["D 1","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1079","id":89078,"instance":1079,"goal":"not (not (d)) => (d)","type_family":"Int -> Int","context":["Thin","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1080","id":89079,"instance":1080,"goal":"all e:Nat -> Nat. all a:Nat -> Nat. all p:Nat -> Nat. append(e, append(a, p)) = append(append(e, a), p)","type_family":"Nat -> Nat","context":["Complete Auto","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1081","id":89080,"instance":1081,"goal":"all v:Int. all z:Int. v + z = z + v","type_family":"Int","context":["SplitOn","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1082","id":89081,"instance":1082,"goal":"all p:Nat. all c:Nat. all a:Nat. (p+c)+a = p+(c+a)","type_family":"Nat","context":["Complete Auto","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1083","id":89082,"instance":1083,"goal":"all y:Real. y + 0 = y","type_family":"Real","context":["Lemma","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1084","id":89083,"instance":1084,"goal":"all p:Atom. all g:Atom. p * g = g * p","type_family":"Atom","context":["BackThru","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1085","id":89084,"instance":1085,"goal":"all u:Bool. u * 1 = u","type_family":"Bool","context":["InstLemma","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1086","id":89085,"instance":1086,"goal":"all c:List(Int). all d:List(Int). all v:List(Int). c*(d+v) = c*d + c*v","type_family":"List(Int)","context":["SplitOn","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1087","id":89086,"instance":1087,"goal":"all m:List(Nat). m <= m","type_family":"List(Nat)","context":["BackThru","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1088","id":89087,"instance":1088,"goal":"all epsilon:List(Atom). all kappa:List(Atom). all alpha:List(Atom). (epsilon <= kappa) => (kappa <= alpha) => (epsilon <= alpha)","type_family":"List(Atom)","context":["RepUR","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1089","id":89088,"instance":1089,"goal":"all v:Int -> Int. v = v","type_family":"Int -> Int","context":["D 1","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1090","id":89089,"instance":1090,"goal":"all q:Nat -> Nat. all w:Nat -> Nat. (q = w) => (w = q)","type_family":"Nat -> Nat","context":["BackThru","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1091","id":89090,"instance":1091,"goal":"(h) => (m) => (h)","type_family":"Int","context":["BackThru","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1092","id":89091,"instance":1092,"goal":"((h) & (q)) iff ((q) & (h))","type_family":"Nat","context":["Lemma","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1093","id":89092,"instance":1093,"goal":"((c) | (q)) iff ((q) | (c))","type_family":"Real","context":["RepUR","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1094","id":89093,"instance":1094,"goal":"not (not (u)) => (u)","type_family":"Atom","context":["BackThru","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1095","id":89094,"instance":1095,"goal":"all e:Bool. all p:Bool. all z:Bool. append(e, append(p, z)) = append(append(e, p), z)","type_family":"Bool","context":["D 1","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1096","id":89095,"instance":1096,"goal":"all n:List(Int). all v:List(Int). n + v = v + n","type_family":"List(Int)","context":["D 1","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1097","id":89096,"instance":1097,"goal":"all f:List(Nat). all q:List(Nat). all m:List(Nat). (f+q)+m = f+(q+m)","type_family":"List(Nat)","context":["D -1","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1098","id":89097,"instance":1098,"goal":"all x:List(Atom). x + 0 = x","type_family":"List(Atom)","context":["D 1","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1099","id":89098,"instance":1099,"goal":"all f:Int -> Int. all m:Int -> Int. f * m = m * f","type_family":"Int -> Int","context":["BackThru","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1100","id":89099,"instance":1100,"goal":"all h:Nat -> Nat. h * 1 = h","type_family":"Nat -> Nat","context":["D -1","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1101","id":89100,"instance":1101,"goal":"all q:Int. all e:Int. all n:Int. q*(e+n) = q*e + q*n","type_family":"Int","context":["D -1","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1102","id":89101,"instance":1102,"goal":"all g:Nat. g <= g","type_family":"Nat","context":["Auto","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1103","id":89102,"instance":1103,"goal":"all n:Real. all p:Real. all z:Real. (n <= p) => (p <= z) => (n <= z)","type_family":"Real","context":["InstLemma","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1104","id":89103,"instance":1104,"goal":"all w:Atom. w = w","type_family":"Atom","context":["Reduce","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1105","id":89104,"instance":1105,"goal":"all x:Bool. all g:Bool. (x = g) => (g = x)","type_family":"Bool","context":["InstLemma","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1106","id":89105,"instance":1106,"goal":"(d) => (z) => (d)","type_family":"List(Int)","context":["D -1","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1107","id":89106,"instance":1107,"goal":"((n) & (c)) iff ((c) & (n))","type_family":"List(Nat)","context":["BackThru","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1108","id":89107,"instance":1108,"goal":"((chi) | (epsilon)) iff ((epsilon) | (chi))","type_family":"List(Atom)","context":["Reduce","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1109","id":89108,"instance":1109,"goal":"not (not (x)) => (x)","type_family":"Int -> Int","context":["Complete Auto","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1110","id":89109,"instance":1110,"goal":"all omega:Nat -> Nat. all alpha:Nat -> Nat. all beta:Nat -> Nat. append(omega, append(alpha, beta)) = append(append(omega, alpha), beta)","type_family":"Nat -> Nat","context":["RepUR","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1111","id":89110,"instance":1111,"goal":"all w:Int. all x:Int. w + x = x + w","type_family":"Int","context":["D 1","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1112","id":89111,"instance":1112,"goal":"all u:Nat. all f:Nat. all p:Nat. (u+f)+p = u+(f+p)","type_family":"Nat","context":["BackThru","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1113","id":89112,"instance":1113,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["D 0","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1114","id":89113,"instance":1114,"goal":"all z:Atom. all e:Atom. z * e = e * z","type_family":"Atom","context":["D 1","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1115","id":89114,"instance":1115,"goal":"all chi:Bool. chi * 1 = chi","type_family":"Bool","context":["Lemma","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1116","id":89115,"instance":1116,"goal":"all f:List(Int). all x:List(Int). all v:List(Int). f*(x+v) = f*x + f*v","type_family":"List(Int)","context":["D 1","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1117","id":89116,"instance":1117,"goal":"all omega:List(Nat). omega <= omega","type_family":"List(Nat)","context":["DoSubst","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1118","id":89117,"instance":1118,"goal":"all f:List(Atom). all n:List(Atom). all q:List(Atom). (f <= n) => (n <= q) => (f <= q)","type_family":"List(Atom)","context":["BackThru","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1119","id":89118,"instance":1119,"goal":"all a:Int -> Int. a = a","type_family":"Int -> Int","context":["Assert","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1120","id":89119,"instance":1120,"goal":"all z:Nat -> Nat. all u:Nat -> Nat. (z = u) => (u = z)","type_family":"Nat -> Nat","context":["InstLemma","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1121","id":89120,"instance":1121,"goal":"(y) => (x) => (y)","type_family":"Int","context":["D 1","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1122","id":89121,"instance":1122,"goal":"((a) & (p)) iff ((p) & (a))","type_family":"Nat","context":["Assert","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1123","id":89122,"instance":1123,"goal":"((chi) | (psi)) iff ((psi) | (chi))","type_family":"Real","context":["Assert","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1124","id":89123,"instance":1124,"goal":"not (not (a)) => (a)","type_family":"Atom","context":["DoSubst","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1125","id":89124,"instance":1125,"goal":"all c:Bool. all w:Bool. all v:Bool. append(c, append(w, v)) = append(append(c, w), v)","type_family":"Bool","context":["DoSubst","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1126","id":89125,"instance":1126,"goal":"all d:List(Int). all w:List(Int). d + w = w + d","type_family":"List(Int)","context":["D 1","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1127","id":89126,"instance":1127,"goal":"all u:List(Nat). all m:List(Nat). all v:List(Nat). (u+m)+v = u+(m+v)","type_family":"List(Nat)","context":["SplitOn","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1128","id":89127,"instance":1128,"goal":"all d:List(Atom). d + 0 = d","type_family":"List(Atom)","context":["Lemma","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1129","id":89128,"instance":1129,"goal":"all psi:Int -> Int. all omega:Int -> Int. psi * omega = omega * psi","type_family":"Int -> Int","context":["D 1","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1130","id":89129,"instance":1130,"goal":"all chi:Nat -> Nat. chi * 1 = chi","type_family":"Nat -> Nat","context":["D -1","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1131","id":89130,"instance":1131,"goal":"all h:Int. all q:Int. all w:Int. h*(q+w) = h*q + h*w","type_family":"Int","context":["Reduce","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1132","id":89131,"instance":1132,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["SplitOn","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1133","id":89132,"instance":1133,"goal":"all c:Real. all w:Real. all a:Real. (c <= w) => (w <= a) => (c <= a)","type_family":"Real","context":["Assert","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1134","id":89133,"instance":1134,"goal":"all d:Atom. d = d","type_family":"Atom","context":["SplitOn","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1135","id":89134,"instance":1135,"goal":"all x:Bool. all n:Bool. (x = n) => (n = x)","type_family":"Bool","context":["Reduce","Reduce","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1136","id":89135,"instance":1136,"goal":"(f) => (z) => (f)","type_family":"List(Int)","context":["InstLemma","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1137","id":89136,"instance":1137,"goal":"((f) & (z)) iff ((z) & (f))","type_family":"List(Nat)","context":["Assert","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1138","id":89137,"instance":1138,"goal":"((v) | (e)) iff ((e) | (v))","type_family":"List(Atom)","context":["Complete Auto","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1139","id":89138,"instance":1139,"goal":"not (not (c)) => (c)","type_family":"Int -> Int","context":["Complete Auto","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1140","id":89139,"instance":1140,"goal":"all d:Nat -> Nat. all w:Nat -> Nat. all b:Nat -> Nat. append(d, append(w, b)) = append(append(d, w), b)","type_family":"Nat -> Nat","context":["Lemma","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1141","id":89140,"instance":1141,"goal":"all phi:Int. all psi:Int. phi + psi = psi + phi","type_family":"Int","context":["InstLemma","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1142","id":89141,"instance":1142,"goal":"all v:Nat. all m:Nat. all b:Nat. (v+m)+b = v+(m+b)","type_family":"Nat","context":["InstLemma","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1143","id":89142,"instance":1143,"goal":"all g:Real. g + 0 = g","type_family":"Real","context":["SplitOn","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1144","id":89143,"instance":1144,"goal":"all z:Atom. all d:Atom. z * d = d * z","type_family":"Atom","context":["Auto","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1145","id":89144,"instance":1145,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["D 1","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1146","id":89145,"instance":1146,"goal":"all x:List(Int). all v:List(Int). all b:List(Int). x*(v+b) = x*v + x*b","type_family":"List(Int)","context":["Reduce","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1147","id":89146,"instance":1147,"goal":"all f:List(Nat). f <= f","type_family":"List(Nat)","context":["Complete Auto","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1148","id":89147,"instance":1148,"goal":"all y:List(Atom). all d:List(Atom). all q:List(Atom). (y <= d) => (d <= q) => (y <= q)","type_family":"List(Atom)","context":["Thin","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1149","id":89148,"instance":1149,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["Reduce","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1150","id":89149,"instance":1150,"goal":"all a:Nat -> Nat. all q:Nat -> Nat. (a = q) => (q = a)","type_family":"Nat -> Nat","context":["D -1","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1151","id":89150,"instance":1151,"goal":"(y) => (n) => (y)","type_family":"Int","context":["Thin","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1152","id":89151,"instance":1152,"goal":"((y) & (h)) iff ((h) & (y))","type_family":"Nat","context":["BackThru","DoSubst","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1153","id":89152,"instance":1153,"goal":"((z) | (x)) iff ((x) | (z))","type_family":"Real","context":["InstLemma","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1154","id":89153,"instance":1154,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Lemma","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1155","id":89154,"instance":1155,"goal":"all q:Bool. all a:Bool. all z:Bool. append(q, append(a, z)) = append(append(q, a), z)","type_family":"Bool","context":["Thin","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1156","id":89155,"instance":1156,"goal":"all c:List(Int). all g:List(Int). c + g = g + c","type_family":"List(Int)","context":["Lemma","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1157","id":89156,"instance":1157,"goal":"all z:List(Nat). all y:List(Nat). all a:List(Nat). (z+y)+a = z+(y+a)","type_family":"List(Nat)","context":["Reduce","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1158","id":89157,"instance":1158,"goal":"all y:List(Atom). y + 0 = y","type_family":"List(Atom)","context":["D 1","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1159","id":89158,"instance":1159,"goal":"all f:Int -> Int. all x:Int -> Int. f * x = x * f","type_family":"Int -> Int","context":["D 0","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1160","id":89159,"instance":1160,"goal":"all w:Nat -> Nat. w * 1 = w","type_family":"Nat -> Nat","context":["InstLemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1161","id":89160,"instance":1161,"goal":"all q:Int. all e:Int. all n:Int. q*(e+n) = q*e + q*n","type_family":"Int","context":["D 1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1162","id":89161,"instance":1162,"goal":"all f:Nat. f <= f","type_family":"Nat","context":["Complete Auto","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1163","id":89162,"instance":1163,"goal":"all b:Real. all u:Real. all p:Real. (b <= u) => (u <= p) => (b <= p)","type_family":"Real","context":["InstLemma","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1164","id":89163,"instance":1164,"goal":"all f:Atom. f = f","type_family":"Atom","context":["Reduce","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1165","id":89164,"instance":1165,"goal":"all kappa:Bool. all omega:Bool. (kappa = omega) => (omega = kappa)","type_family":"Bool","context":["Thin","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1166","id":89165,"instance":1166,"goal":"(a) => (v) => (a)","type_family":"List(Int)","context":["D 0","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1167","id":89166,"instance":1167,"goal":"((m) & (h)) iff ((h) & (m))","type_family":"List(Nat)","context":["Lemma","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1168","id":89167,"instance":1168,"goal":"((u) | (c)) iff ((c) | (u))","type_family":"List(Atom)","context":["Assert","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1169","id":89168,"instance":1169,"goal":"not (not (z)) => (z)","type_family":"Int -> Int","context":["Assert","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1170","id":89169,"instance":1170,"goal":"all u:Nat -> Nat. all z:Nat -> Nat. all p:Nat -> Nat. append(u, append(z, p)) = append(append(u, z), p)","type_family":"Nat -> Nat","context":["D 0","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1171","id":89170,"instance":1171,"goal":"all y:Int. all p:Int. y + p = p + y","type_family":"Int","context":["Auto","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1172","id":89171,"instance":1172,"goal":"all v:Nat. all n:Nat. all h:Nat. (v+n)+h = v+(n+h)","type_family":"Nat","context":["Lemma","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1173","id":89172,"instance":1173,"goal":"all u:Real. u + 0 = u","type_family":"Real","context":["Lemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1174","id":89173,"instance":1174,"goal":"all n:Atom. all b:Atom. n * b = b * n","type_family":"Atom","context":["InstLemma","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1175","id":89174,"instance":1175,"goal":"all beta:Bool. beta * 1 = beta","type_family":"Bool","context":["D 1","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1176","id":89175,"instance":1176,"goal":"all x:List(Int). all a:List(Int). all v:List(Int). x*(a+v) = x*a + x*v","type_family":"List(Int)","context":["Lemma","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1177","id":89176,"instance":1177,"goal":"all alpha:List(Nat). alpha <= alpha","type_family":"List(Nat)","context":["D 0","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1178","id":89177,"instance":1178,"goal":"all p:List(Atom). all v:List(Atom). all w:List(Atom). (p <= v) => (v <= w) => (p <= w)","type_family":"List(Atom)","context":["InstLemma","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1179","id":89178,"instance":1179,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["Auto","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1180","id":89179,"instance":1180,"goal":"all a:Nat -> Nat. all c:Nat -> Nat. (a = c) => (c = a)","type_family":"Nat -> Nat","context":["DoSubst","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1181","id":89180,"instance":1181,"goal":"(u) => (e) => (u)","type_family":"Int","context":["D 1","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1182","id":89181,"instance":1182,"goal":"((f) & (x)) iff ((x) & (f))","type_family":"Nat","context":["D 0","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1183","id":89182,"instance":1183,"goal":"((n) | (x)) iff ((x) | (n))","type_family":"Real","context":["DoSubst","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1184","id":89183,"instance":1184,"goal":"not (not (a)) => (a)","type_family":"Atom","context":["D 0","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1185","id":89184,"instance":1185,"goal":"all n:Bool. all b:Bool. all v:Bool. append(n, append(b, v)) = append(append(n, b), v)","type_family":"Bool","context":["SplitOn","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1186","id":89185,"instance":1186,"goal":"all a:List(Int). all c:List(Int). a + c = c + a","type_family":"List(Int)","context":["DoSubst","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1187","id":89186,"instance":1187,"goal":"all a:List(Nat). all m:List(Nat). all p:List(Nat). (a+m)+p = a+(m+p)","type_family":"List(Nat)","context":["SplitOn","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1188","id":89187,"instance":1188,"goal":"all omega:List(Atom). omega + 0 = omega","type_family":"List(Atom)","context":["Reduce","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1189","id":89188,"instance":1189,"goal":"all f:Int -> Int. all v:Int -> Int. f * v = v * f","type_family":"Int -> Int","context":["D 1","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1190","id":89189,"instance":1190,"goal":"all kappa:Nat -> Nat. kappa * 1 = kappa","type_family":"Nat -> Nat","context":["RepUR","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1191","id":89190,"instance":1191,"goal":"all y:Int. all z:Int. all v:Int. y*(z+v) = y*z + y*v","type_family":"Int","context":["Reduce","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1192","id":89191,"instance":1192,"goal":"all kappa:Nat. kappa <= kappa","type_family":"Nat","context":["InstLemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1193","id":89192,"instance":1193,"goal":"all d:Real. all v:Real. all u:Real. (d <= v) => (v <= u) => (d <= u)","type_family":"Real","context":["D -1","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1194","id":89193,"instance":1194,"goal":"all a:Atom. a = a","type_family":"Atom","context":["D 1","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1195","id":89194,"instance":1195,"goal":"all u:Bool. all q:Bool. (u = q) => (q = u)","type_family":"Bool","context":["DoSubst","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1196","id":89195,"instance":1196,"goal":"(p) => (e) => (p)","type_family":"List(Int)","context":["D 0","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1197","id":89196,"instance":1197,"goal":"((h) & (c)) iff ((c) & (h))","type_family":"List(Nat)","context":["D -1","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1198","id":89197,"instance":1198,"goal":"((c) | (b)) iff ((b) | (c))","type_family":"List(Atom)","context":["SplitOn","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1199","id":89198,"instance":1199,"goal":"not (not (m)) => (m)","type_family":"Int -> Int","context":["Assert","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1200","id":89199,"instance":1200,"goal":"all delta:Nat -> Nat. all alpha:Nat -> Nat. all beta:Nat -> Nat. append(delta, append(alpha, beta)) = append(append(delta, alpha), beta)","type_family":"Nat -> Nat","context":["Auto","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1201","id":89200,"instance":1201,"goal":"all e:Int. all p:Int. e + p = p + e","type_family":"Int","context":["DoSubst","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1202","id":89201,"instance":1202,"goal":"all chi:Nat. all phi:Nat. all gamma:Nat. (chi+phi)+gamma = chi+(phi+gamma)","type_family":"Nat","context":["Complete Auto","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1203","id":89202,"instance":1203,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["D 0","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1204","id":89203,"instance":1204,"goal":"all z:Atom. all x:Atom. z * x = x * z","type_family":"Atom","context":["Reduce","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1205","id":89204,"instance":1205,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["RepUR","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1206","id":89205,"instance":1206,"goal":"all d:List(Int). all z:List(Int). all f:List(Int). d*(z+f) = d*z + d*f","type_family":"List(Int)","context":["Thin","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1207","id":89206,"instance":1207,"goal":"all q:List(Nat). q <= q","type_family":"List(Nat)","context":["D 0","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1208","id":89207,"instance":1208,"goal":"all h:List(Atom). all w:List(Atom). all y:List(Atom). (h <= w) => (w <= y) => (h <= y)","type_family":"List(Atom)","context":["D 1","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1209","id":89208,"instance":1209,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["D 0","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1210","id":89209,"instance":1210,"goal":"all g:Nat -> Nat. all n:Nat -> Nat. (g = n) => (n = g)","type_family":"Nat -> Nat","context":["Auto","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1211","id":89210,"instance":1211,"goal":"(h) => (z) => (h)","type_family":"Int","context":["SplitOn","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1212","id":89211,"instance":1212,"goal":"((g) & (u)) iff ((u) & (g))","type_family":"Nat","context":["RepUR","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1213","id":89212,"instance":1213,"goal":"((a) | (e)) iff ((e) | (a))","type_family":"Real","context":["D -1","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1214","id":89213,"instance":1214,"goal":"not (not (alpha)) => (alpha)","type_family":"Atom","context":["RepUR","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1215","id":89214,"instance":1215,"goal":"all alpha:Bool. all beta:Bool. all delta:Bool. append(alpha, append(beta, delta)) = append(append(alpha, beta), delta)","type_family":"Bool","context":["D 0","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1216","id":89215,"instance":1216,"goal":"all c:List(Int). all f:List(Int). c + f = f + c","type_family":"List(Int)","context":["Auto","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1217","id":89216,"instance":1217,"goal":"all q:List(Nat). all e:List(Nat). all a:List(Nat). (q+e)+a = q+(e+a)","type_family":"List(Nat)","context":["Assert","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1218","id":89217,"instance":1218,"goal":"all y:List(Atom). y + 0 = y","type_family":"List(Atom)","context":["InstLemma","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1219","id":89218,"instance":1219,"goal":"all q:Int -> Int. all h:Int -> Int. q * h = h * q","type_family":"Int -> Int","context":["D -1","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1220","id":89219,"instance":1220,"goal":"all psi:Nat -> Nat. psi * 1 = psi","type_family":"Nat -> Nat","context":["Complete Auto","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1221","id":89220,"instance":1221,"goal":"all p:Int. all u:Int. all a:Int. p*(u+a) = p*u + p*a","type_family":"Int","context":["D 0","RepUR","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1222","id":89221,"instance":1222,"goal":"all q:Nat. q <= q","type_family":"Nat","context":["Complete Auto","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1223","id":89222,"instance":1223,"goal":"all u:Real. all z:Real. all e:Real. (u <= z) => (z <= e) => (u <= e)","type_family":"Real","context":["Auto","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1224","id":89223,"instance":1224,"goal":"all u:Atom. u = u","type_family":"Atom","context":["SplitOn","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1225","id":89224,"instance":1225,"goal":"all w:Bool. all b:Bool. (w = b) => (b = w)","type_family":"Bool","context":["Complete Auto","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1226","id":89225,"instance":1226,"goal":"(y) => (x) => (y)","type_family":"List(Int)","context":["D 0","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1227","id":89226,"instance":1227,"goal":"((m) & (n)) iff ((n) & (m))","type_family":"List(Nat)","context":["Thin","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1228","id":89227,"instance":1228,"goal":"((h) | (q)) iff ((q) | (h))","type_family":"List(Atom)","context":["Complete Auto","D -1","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1229","id":89228,"instance":1229,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["D 1","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1230","id":89229,"instance":1230,"goal":"all e:Nat -> Nat. all g:Nat -> Nat. all d:Nat -> Nat. append(e, append(g, d)) = append(append(e, g), d)","type_family":"Nat -> Nat","context":["InstLemma","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1231","id":89230,"instance":1231,"goal":"all p:Int. all f:Int. p + f = f + p","type_family":"Int","context":["Thin","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1232","id":89231,"instance":1232,"goal":"all g:Nat. all e:Nat. all q:Nat. (g+e)+q = g+(e+q)","type_family":"Nat","context":["Complete Auto","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1233","id":89232,"instance":1233,"goal":"all u:Real. u + 0 = u","type_family":"Real","context":["D -1","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1234","id":89233,"instance":1234,"goal":"all q:Atom. all e:Atom. q * e = e * q","type_family":"Atom","context":["BackThru","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1235","id":89234,"instance":1235,"goal":"all n:Bool. n * 1 = n","type_family":"Bool","context":["D 0","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1236","id":89235,"instance":1236,"goal":"all z:List(Int). all x:List(Int). all m:List(Int). z*(x+m) = z*x + z*m","type_family":"List(Int)","context":["D -1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1237","id":89236,"instance":1237,"goal":"all z:List(Nat). z <= z","type_family":"List(Nat)","context":["InstLemma","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1238","id":89237,"instance":1238,"goal":"all n:List(Atom). all m:List(Atom). all c:List(Atom). (n <= m) => (m <= c) => (n <= c)","type_family":"List(Atom)","context":["D 1","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1239","id":89238,"instance":1239,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["Complete Auto","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1240","id":89239,"instance":1240,"goal":"all p:Nat -> Nat. all n:Nat -> Nat. (p = n) => (n = p)","type_family":"Nat -> Nat","context":["D 0","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1241","id":89240,"instance":1241,"goal":"(delta) => (chi) => (delta)","type_family":"Int","context":["Complete Auto","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1242","id":89241,"instance":1242,"goal":"((z) & (e)) iff ((e) & (z))","type_family":"Nat","context":["Auto","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1243","id":89242,"instance":1243,"goal":"((x) | (m)) iff ((m) | (x))","type_family":"Real","context":["InstLemma","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1244","id":89243,"instance":1244,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Reduce","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1245","id":89244,"instance":1245,"goal":"all p:Bool. all u:Bool. all b:Bool. append(p, append(u, b)) = append(append(p, u), b)","type_family":"Bool","context":["Auto","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1246","id":89245,"instance":1246,"goal":"all n:List(Int). all x:List(Int). n + x = x + n","type_family":"List(Int)","context":["Lemma","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1247","id":89246,"instance":1247,"goal":"all omega:List(Nat). all epsilon:List(Nat). all phi:List(Nat). (omega+epsilon)+phi = omega+(epsilon+phi)","type_family":"List(Nat)","context":["Assert","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1248","id":89247,"instance":1248,"goal":"all beta:List(Atom). beta + 0 = beta","type_family":"List(Atom)","context":["Lemma","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1249","id":89248,"instance":1249,"goal":"all p:Int -> Int. all q:Int -> Int. p * q = q * p","type_family":"Int -> Int","context":["D -1","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1250","id":89249,"instance":1250,"goal":"all u:Nat -> Nat. u * 1 = u","type_family":"Nat -> Nat","context":["D -1","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1251","id":89250,"instance":1251,"goal":"all z:Int. all m:Int. all d:Int. z*(m+d) = z*m + z*d","type_family":"Int","context":["RepUR","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1252","id":89251,"instance":1252,"goal":"all z:Nat. z <= z","type_family":"Nat","context":["BackThru","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1253","id":89252,"instance":1253,"goal":"all z:Real. all f:Real. all p:Real. (z <= f) => (f <= p) => (z <= p)","type_family":"Real","context":["Reduce","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1254","id":89253,"instance":1254,"goal":"all m:Atom. m = m","type_family":"Atom","context":["RepUR","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1255","id":89254,"instance":1255,"goal":"all y:Bool. all f:Bool. (y = f) => (f = y)","type_family":"Bool","context":["D 1","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1256","id":89255,"instance":1256,"goal":"(g) => (n) => (g)","type_family":"List(Int)","context":["Reduce","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1257","id":89256,"instance":1257,"goal":"((phi) & (epsilon)) iff ((epsilon) & (phi))","type_family":"List(Nat)","context":["SplitOn","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1258","id":89257,"instance":1258,"goal":"((b) | (n)) iff ((n) | (b))","type_family":"List(Atom)","context":["Auto","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1259","id":89258,"instance":1259,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["D 0","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1260","id":89259,"instance":1260,"goal":"all v:Nat -> Nat. all u:Nat -> Nat. all b:Nat -> Nat. append(v, append(u, b)) = append(append(v, u), b)","type_family":"Nat -> Nat","context":["Auto","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1261","id":89260,"instance":1261,"goal":"all a:Int. all v:Int. a + v = v + a","type_family":"Int","context":["D 1","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1262","id":89261,"instance":1262,"goal":"all gamma:Nat. all kappa:Nat. all beta:Nat. (gamma+kappa)+beta = gamma+(kappa+beta)","type_family":"Nat","context":["Auto","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1263","id":89262,"instance":1263,"goal":"all q:Real. q + 0 = q","type_family":"Real","context":["D -1","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1264","id":89263,"instance":1264,"goal":"all p:Atom. all c:Atom. p * c = c * p","type_family":"Atom","context":["BackThru","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1265","id":89264,"instance":1265,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["D 0","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1266","id":89265,"instance":1266,"goal":"all m:List(Int). all b:List(Int). all z:List(Int). m*(b+z) = m*b + m*z","type_family":"List(Int)","context":["D -1","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1267","id":89266,"instance":1267,"goal":"all m:List(Nat). m <= m","type_family":"List(Nat)","context":["D 1","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1268","id":89267,"instance":1268,"goal":"all h:List(Atom). all d:List(Atom). all a:List(Atom). (h <= d) => (d <= a) => (h <= a)","type_family":"List(Atom)","context":["Reduce","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1269","id":89268,"instance":1269,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["D -1","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1270","id":89269,"instance":1270,"goal":"all d:Nat -> Nat. all m:Nat -> Nat. (d = m) => (m = d)","type_family":"Nat -> Nat","context":["BackThru","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1271","id":89270,"instance":1271,"goal":"(d) => (u) => (d)","type_family":"Int","context":["Thin","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1272","id":89271,"instance":1272,"goal":"((z) & (c)) iff ((c) & (z))","type_family":"Nat","context":["RepUR","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1273","id":89272,"instance":1273,"goal":"((n) | (a)) iff ((a) | (n))","type_family":"Real","context":["Lemma","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1274","id":89273,"instance":1274,"goal":"not (not (epsilon)) => (epsilon)","type_family":"Atom","context":["D 1","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1275","id":89274,"instance":1275,"goal":"all p:Bool. all d:Bool. all q:Bool. append(p, append(d, q)) = append(append(p, d), q)","type_family":"Bool","context":["Lemma","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1276","id":89275,"instance":1276,"goal":"all u:List(Int). all m:List(Int). u + m = m + u","type_family":"List(Int)","context":["Reduce","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1277","id":89276,"instance":1277,"goal":"all e:List(Nat). all b:List(Nat). all y:List(Nat). (e+b)+y = e+(b+y)","type_family":"List(Nat)","context":["BackThru","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1278","id":89277,"instance":1278,"goal":"all g:List(Atom). g + 0 = g","type_family":"List(Atom)","context":["BackThru","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1279","id":89278,"instance":1279,"goal":"all psi:Int -> Int. all chi:Int -> Int. psi * chi = chi * psi","type_family":"Int -> Int","context":["InstLemma","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1280","id":89279,"instance":1280,"goal":"all h:Nat -> Nat. h * 1 = h","type_family":"Nat -> Nat","context":["SplitOn","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1281","id":89280,"instance":1281,"goal":"all f:Int. all c:Int. all u:Int. f*(c+u) = f*c + f*u","type_family":"Int","context":["Assert","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1282","id":89281,"instance":1282,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["Complete Auto","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1283","id":89282,"instance":1283,"goal":"all m:Real. all g:Real. all u:Real. (m <= g) => (g <= u) => (m <= u)","type_family":"Real","context":["D 0","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1284","id":89283,"instance":1284,"goal":"all b:Atom. b = b","type_family":"Atom","context":["Assert","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1285","id":89284,"instance":1285,"goal":"all c:Bool. all x:Bool. (c = x) => (x = c)","type_family":"Bool","context":["BackThru","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1286","id":89285,"instance":1286,"goal":"(z) => (w) => (z)","type_family":"List(Int)","context":["Assert","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1287","id":89286,"instance":1287,"goal":"((x) & (g)) iff ((g) & (x))","type_family":"List(Nat)","context":["InstLemma","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1288","id":89287,"instance":1288,"goal":"((p) | (f)) iff ((f) | (p))","type_family":"List(Atom)","context":["D 0","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1289","id":89288,"instance":1289,"goal":"not (not (h)) => (h)","type_family":"Int -> Int","context":["Reduce","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1290","id":89289,"instance":1290,"goal":"all m:Nat -> Nat. all w:Nat -> Nat. all e:Nat -> Nat. append(m, append(w, e)) = append(append(m, w), e)","type_family":"Nat -> Nat","context":["D 1","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1291","id":89290,"instance":1291,"goal":"all f:Int. all u:Int. f + u = u + f","type_family":"Int","context":["Auto","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1292","id":89291,"instance":1292,"goal":"all n:Nat. all p:Nat. all m:Nat. (n+p)+m = n+(p+m)","type_family":"Nat","context":["Complete Auto","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1293","id":89292,"instance":1293,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["InstLemma","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1294","id":89293,"instance":1294,"goal":"all g:Atom. all n:Atom. g * n = n * g","type_family":"Atom","context":["InstLemma","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1295","id":89294,"instance":1295,"goal":"all n:Bool. n * 1 = n","type_family":"Bool","context":["Thin","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1296","id":89295,"instance":1296,"goal":"all g:List(Int). all x:List(Int). all e:List(Int). g*(x+e) = g*x + g*e","type_family":"List(Int)","context":["BackThru","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1297","id":89296,"instance":1297,"goal":"all v:List(Nat). v <= v","type_family":"List(Nat)","context":["Complete Auto","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1298","id":89297,"instance":1298,"goal":"all f:List(Atom). all p:List(Atom). all h:List(Atom). (f <= p) => (p <= h) => (f <= h)","type_family":"List(Atom)","context":["Thin","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1299","id":89298,"instance":1299,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["SplitOn","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1300","id":89299,"instance":1300,"goal":"all a:Nat -> Nat. all w:Nat -> Nat. (a = w) => (w = a)","type_family":"Nat -> Nat","context":["SplitOn","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1301","id":89300,"instance":1301,"goal":"(w) => (c) => (w)","type_family":"Int","context":["BackThru","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1302","id":89301,"instance":1302,"goal":"((f) & (a)) iff ((a) & (f))","type_family":"Nat","context":["Auto","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1303","id":89302,"instance":1303,"goal":"((e) | (x)) iff ((x) | (e))","type_family":"Real","context":["Reduce","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1304","id":89303,"instance":1304,"goal":"not (not (beta)) => (beta)","type_family":"Atom","context":["D 1","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1305","id":89304,"instance":1305,"goal":"all n:Bool. all d:Bool. all v:Bool. append(n, append(d, v)) = append(append(n, d), v)","type_family":"Bool","context":["InstLemma","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1306","id":89305,"instance":1306,"goal":"all p:List(Int). all x:List(Int). p + x = x + p","type_family":"List(Int)","context":["Complete Auto","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1307","id":89306,"instance":1307,"goal":"all d:List(Nat). all c:List(Nat). all b:List(Nat). (d+c)+b = d+(c+b)","type_family":"List(Nat)","context":["Assert","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1308","id":89307,"instance":1308,"goal":"all omega:List(Atom). omega + 0 = omega","type_family":"List(Atom)","context":["SplitOn","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1309","id":89308,"instance":1309,"goal":"all w:Int -> Int. all v:Int -> Int. w * v = v * w","type_family":"Int -> Int","context":["BackThru","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1310","id":89309,"instance":1310,"goal":"all m:Nat -> Nat. m * 1 = m","type_family":"Nat -> Nat","context":["Assert","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1311","id":89310,"instance":1311,"goal":"all z:Int. all f:Int. all h:Int. z*(f+h) = z*f + z*h","type_family":"Int","context":["Lemma","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1312","id":89311,"instance":1312,"goal":"all n:Nat. n <= n","type_family":"Nat","context":["BackThru","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1313","id":89312,"instance":1313,"goal":"all e:Real. all c:Real. all g:Real. (e <= c) => (c <= g) => (e <= g)","type_family":"Real","context":["Thin","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1314","id":89313,"instance":1314,"goal":"all v:Atom. v = v","type_family":"Atom","context":["RepUR","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1315","id":89314,"instance":1315,"goal":"all b:Bool. all m:Bool. (b = m) => (m = b)","type_family":"Bool","context":["SplitOn","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1316","id":89315,"instance":1316,"goal":"(a) => (v) => (a)","type_family":"List(Int)","context":["D 1","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1317","id":89316,"instance":1317,"goal":"((w) & (b)) iff ((b) & (w))","type_family":"List(Nat)","context":["InstLemma","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1318","id":89317,"instance":1318,"goal":"((delta) | (phi)) iff ((phi) | (delta))","type_family":"List(Atom)","context":["InstLemma","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1319","id":89318,"instance":1319,"goal":"not (not (c)) => (c)","type_family":"Int -> Int","context":["Lemma","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1320","id":89319,"instance":1320,"goal":"all g:Nat -> Nat. all v:Nat -> Nat. all c:Nat -> Nat. append(g, append(v, c)) = append(append(g, v), c)","type_family":"Nat -> Nat","context":["D 0","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1321","id":89320,"instance":1321,"goal":"all z:Int. all u:Int. z + u = u + z","type_family":"Int","context":["Complete Auto","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1322","id":89321,"instance":1322,"goal":"all g:Nat. all f:Nat. all w:Nat. (g+f)+w = g+(f+w)","type_family":"Nat","context":["Reduce","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1323","id":89322,"instance":1323,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["Lemma","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1324","id":89323,"instance":1324,"goal":"all n:Atom. all p:Atom. n * p = p * n","type_family":"Atom","context":["InstLemma","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1325","id":89324,"instance":1325,"goal":"all p:Bool. p * 1 = p","type_family":"Bool","context":["BackThru","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1326","id":89325,"instance":1326,"goal":"all p:List(Int). all u:List(Int). all n:List(Int). p*(u+n) = p*u + p*n","type_family":"List(Int)","context":["Reduce","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1327","id":89326,"instance":1327,"goal":"all m:List(Nat). m <= m","type_family":"List(Nat)","context":["Lemma","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1328","id":89327,"instance":1328,"goal":"all w:List(Atom). all c:List(Atom). all v:List(Atom). (w <= c) => (c <= v) => (w <= v)","type_family":"List(Atom)","context":["Lemma","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1329","id":89328,"instance":1329,"goal":"all q:Int -> Int. q = q","type_family":"Int -> Int","context":["Complete Auto","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1330","id":89329,"instance":1330,"goal":"all c:Nat -> Nat. all m:Nat -> Nat. (c = m) => (m = c)","type_family":"Nat -> Nat","context":["D 0","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1331","id":89330,"instance":1331,"goal":"(w) => (u) => (w)","type_family":"Int","context":["Lemma","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1332","id":89331,"instance":1332,"goal":"((p) & (m)) iff ((m) & (p))","type_family":"Nat","context":["Reduce","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1333","id":89332,"instance":1333,"goal":"((a) | (x)) iff ((x) | (a))","type_family":"Real","context":["Reduce","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1334","id":89333,"instance":1334,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["DoSubst","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1335","id":89334,"instance":1335,"goal":"all q:Bool. all b:Bool. all f:Bool. append(q, append(b, f)) = append(append(q, b), f)","type_family":"Bool","context":["SplitOn","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1336","id":89335,"instance":1336,"goal":"all b:List(Int). all e:List(Int). b + e = e + b","type_family":"List(Int)","context":["Assert","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1337","id":89336,"instance":1337,"goal":"all d:List(Nat). all a:List(Nat). all b:List(Nat). (d+a)+b = d+(a+b)","type_family":"List(Nat)","context":["Reduce","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1338","id":89337,"instance":1338,"goal":"all w:List(Atom). w + 0 = w","type_family":"List(Atom)","context":["D 0","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1339","id":89338,"instance":1339,"goal":"all u:Int -> Int. all m:Int -> Int. u * m = m * u","type_family":"Int -> Int","context":["D -1","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 1; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1340","id":89339,"instance":1340,"goal":"all f:Nat -> Nat. f * 1 = f","type_family":"Nat -> Nat","context":["RepUR","D 1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1341","id":89340,"instance":1341,"goal":"all a:Int. all u:Int. all d:Int. a*(u+d) = a*u + a*d","type_family":"Int","context":["D 1","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1342","id":89341,"instance":1342,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["Thin","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1343","id":89342,"instance":1343,"goal":"all f:Real. all q:Real. all h:Real. (f <= q) => (q <= h) => (f <= h)","type_family":"Real","context":["BackThru","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1344","id":89343,"instance":1344,"goal":"all f:Atom. f = f","type_family":"Atom","context":["SplitOn","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1345","id":89344,"instance":1345,"goal":"all beta:Bool. all delta:Bool. (beta = delta) => (delta = beta)","type_family":"Bool","context":["RepUR","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1346","id":89345,"instance":1346,"goal":"(u) => (x) => (u)","type_family":"List(Int)","context":["InstLemma","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1347","id":89346,"instance":1347,"goal":"((w) & (n)) iff ((n) & (w))","type_family":"List(Nat)","context":["DoSubst","InstLemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1348","id":89347,"instance":1348,"goal":"((y) | (m)) iff ((m) | (y))","type_family":"List(Atom)","context":["Assert","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1349","id":89348,"instance":1349,"goal":"not (not (d)) => (d)","type_family":"Int -> Int","context":["DoSubst","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1350","id":89349,"instance":1350,"goal":"all alpha:Nat -> Nat. all psi:Nat -> Nat. all delta:Nat -> Nat. append(alpha, append(psi, delta)) = append(append(alpha, psi), delta)","type_family":"Nat -> Nat","context":["Complete Auto","RepUR","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1351","id":89350,"instance":1351,"goal":"all w:Int. all a:Int. w + a = a + w","type_family":"Int","context":["Complete Auto","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1352","id":89351,"instance":1352,"goal":"all q:Nat. all n:Nat. all y:Nat. (q+n)+y = q+(n+y)","type_family":"Nat","context":["DoSubst","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1353","id":89352,"instance":1353,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["Lemma","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1354","id":89353,"instance":1354,"goal":"all z:Atom. all e:Atom. z * e = e * z","type_family":"Atom","context":["SplitOn","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1355","id":89354,"instance":1355,"goal":"all g:Bool. g * 1 = g","type_family":"Bool","context":["Thin","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1356","id":89355,"instance":1356,"goal":"all f:List(Int). all h:List(Int). all g:List(Int). f*(h+g) = f*h + f*g","type_family":"List(Int)","context":["RepUR","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1357","id":89356,"instance":1357,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["Thin","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1358","id":89357,"instance":1358,"goal":"all x:List(Atom). all p:List(Atom). all v:List(Atom). (x <= p) => (p <= v) => (x <= v)","type_family":"List(Atom)","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1359","id":89358,"instance":1359,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["Lemma","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1360","id":89359,"instance":1360,"goal":"all e:Nat -> Nat. all m:Nat -> Nat. (e = m) => (m = e)","type_family":"Nat -> Nat","context":["Auto","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1361","id":89360,"instance":1361,"goal":"(z) => (w) => (z)","type_family":"Int","context":["Auto","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1362","id":89361,"instance":1362,"goal":"((e) & (v)) iff ((v) & (e))","type_family":"Nat","context":["Reduce","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1363","id":89362,"instance":1363,"goal":"((a) | (g)) iff ((g) | (a))","type_family":"Real","context":["SplitOn","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1364","id":89363,"instance":1364,"goal":"not (not (m)) => (m)","type_family":"Atom","context":["DoSubst","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1365","id":89364,"instance":1365,"goal":"all f:Bool. all z:Bool. all n:Bool. append(f, append(z, n)) = append(append(f, z), n)","type_family":"Bool","context":["Complete Auto","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1366","id":89365,"instance":1366,"goal":"all v:List(Int). all e:List(Int). v + e = e + v","type_family":"List(Int)","context":["InstLemma","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1367","id":89366,"instance":1367,"goal":"all f:List(Nat). all d:List(Nat). all x:List(Nat). (f+d)+x = f+(d+x)","type_family":"List(Nat)","context":["RepUR","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1368","id":89367,"instance":1368,"goal":"all d:List(Atom). d + 0 = d","type_family":"List(Atom)","context":["Lemma","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1369","id":89368,"instance":1369,"goal":"all z:Int -> Int. all v:Int -> Int. z * v = v * z","type_family":"Int -> Int","context":["D -1","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1370","id":89369,"instance":1370,"goal":"all c:Nat -> Nat. c * 1 = c","type_family":"Nat -> Nat","context":["Auto","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1371","id":89370,"instance":1371,"goal":"all x:Int. all v:Int. all a:Int. x*(v+a) = x*v + x*a","type_family":"Int","context":["Reduce","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1372","id":89371,"instance":1372,"goal":"all c:Nat. c <= c","type_family":"Nat","context":["D 1","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1373","id":89372,"instance":1373,"goal":"all n:Real. all d:Real. all x:Real. (n <= d) => (d <= x) => (n <= x)","type_family":"Real","context":["Complete Auto","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1374","id":89373,"instance":1374,"goal":"all phi:Atom. phi = phi","type_family":"Atom","context":["RepUR","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1375","id":89374,"instance":1375,"goal":"all epsilon:Bool. all chi:Bool. (epsilon = chi) => (chi = epsilon)","type_family":"Bool","context":["InstLemma","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1376","id":89375,"instance":1376,"goal":"(f) => (z) => (f)","type_family":"List(Int)","context":["InstLemma","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; SplitOn; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1377","id":89376,"instance":1377,"goal":"((chi) & (omega)) iff ((omega) & (chi))","type_family":"List(Nat)","context":["D 0","SplitOn","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1378","id":89377,"instance":1378,"goal":"((chi) | (gamma)) iff ((gamma) | (chi))","type_family":"List(Atom)","context":["Auto","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1379","id":89378,"instance":1379,"goal":"not (not (c)) => (c)","type_family":"Int -> Int","context":["Complete Auto","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1380","id":89379,"instance":1380,"goal":"all u:Nat -> Nat. all w:Nat -> Nat. all a:Nat -> Nat. append(u, append(w, a)) = append(append(u, w), a)","type_family":"Nat -> Nat","context":["DoSubst","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1381","id":89380,"instance":1381,"goal":"all f:Int. all e:Int. f + e = e + f","type_family":"Int","context":["Thin","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1382","id":89381,"instance":1382,"goal":"all u:Nat. all e:Nat. all w:Nat. (u+e)+w = u+(e+w)","type_family":"Nat","context":["Complete Auto","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1383","id":89382,"instance":1383,"goal":"all v:Real. v + 0 = v","type_family":"Real","context":["Complete Auto","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1384","id":89383,"instance":1384,"goal":"all v:Atom. all p:Atom. v * p = p * v","type_family":"Atom","context":["Thin","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1385","id":89384,"instance":1385,"goal":"all g:Bool. g * 1 = g","type_family":"Bool","context":["Reduce","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1386","id":89385,"instance":1386,"goal":"all w:List(Int). all y:List(Int). all n:List(Int). w*(y+n) = w*y + w*n","type_family":"List(Int)","context":["D -1","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1387","id":89386,"instance":1387,"goal":"all y:List(Nat). y <= y","type_family":"List(Nat)","context":["D 0","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1388","id":89387,"instance":1388,"goal":"all v:List(Atom). all z:List(Atom). all q:List(Atom). (v <= z) => (z <= q) => (v <= q)","type_family":"List(Atom)","context":["D -1","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1389","id":89388,"instance":1389,"goal":"all b:Int -> Int. b = b","type_family":"Int -> Int","context":["InstLemma","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1390","id":89389,"instance":1390,"goal":"all c:Nat -> Nat. all u:Nat -> Nat. (c = u) => (u = c)","type_family":"Nat -> Nat","context":["D 1","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1391","id":89390,"instance":1391,"goal":"(epsilon) => (kappa) => (epsilon)","type_family":"Int","context":["SplitOn","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1392","id":89391,"instance":1392,"goal":"((c) & (h)) iff ((h) & (c))","type_family":"Nat","context":["BackThru","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1393","id":89392,"instance":1393,"goal":"((m) | (x)) iff ((x) | (m))","type_family":"Real","context":["RepUR","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1394","id":89393,"instance":1394,"goal":"not (not (c)) => (c)","type_family":"Atom","context":["SplitOn","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1395","id":89394,"instance":1395,"goal":"all q:Bool. all f:Bool. all y:Bool. append(q, append(f, y)) = append(append(q, f), y)","type_family":"Bool","context":["RepUR","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1396","id":89395,"instance":1396,"goal":"all v:List(Int). all c:List(Int). v + c = c + v","type_family":"List(Int)","context":["DoSubst","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1397","id":89396,"instance":1397,"goal":"all m:List(Nat). all n:List(Nat). all a:List(Nat). (m+n)+a = m+(n+a)","type_family":"List(Nat)","context":["D 1","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1398","id":89397,"instance":1398,"goal":"all w:List(Atom). w + 0 = w","type_family":"List(Atom)","context":["Reduce","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1399","id":89398,"instance":1399,"goal":"all a:Int -> Int. all c:Int -> Int. a * c = c * a","type_family":"Int -> Int","context":["InstLemma","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1400","id":89399,"instance":1400,"goal":"all p:Nat -> Nat. p * 1 = p","type_family":"Nat -> Nat","context":["DoSubst","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1401","id":89400,"instance":1401,"goal":"all psi:Int. all epsilon:Int. all delta:Int. psi*(epsilon+delta) = psi*epsilon + psi*delta","type_family":"Int","context":["D 0","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1402","id":89401,"instance":1402,"goal":"all w:Nat. w <= w","type_family":"Nat","context":["Auto","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1403","id":89402,"instance":1403,"goal":"all q:Real. all v:Real. all d:Real. (q <= v) => (v <= d) => (q <= d)","type_family":"Real","context":["DoSubst","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1404","id":89403,"instance":1404,"goal":"all b:Atom. b = b","type_family":"Atom","context":["DoSubst","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1405","id":89404,"instance":1405,"goal":"all d:Bool. all c:Bool. (d = c) => (c = d)","type_family":"Bool","context":["D 0","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1406","id":89405,"instance":1406,"goal":"(p) => (w) => (p)","type_family":"List(Int)","context":["D -1","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1407","id":89406,"instance":1407,"goal":"((d) & (p)) iff ((p) & (d))","type_family":"List(Nat)","context":["Auto","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1408","id":89407,"instance":1408,"goal":"((f) | (g)) iff ((g) | (f))","type_family":"List(Atom)","context":["Complete Auto","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1409","id":89408,"instance":1409,"goal":"not (not (y)) => (y)","type_family":"Int -> Int","context":["Auto","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1410","id":89409,"instance":1410,"goal":"all m:Nat -> Nat. all y:Nat -> Nat. all w:Nat -> Nat. append(m, append(y, w)) = append(append(m, y), w)","type_family":"Nat -> Nat","context":["Lemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1411","id":89410,"instance":1411,"goal":"all z:Int. all q:Int. z + q = q + z","type_family":"Int","context":["Auto","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1412","id":89411,"instance":1412,"goal":"all g:Nat. all y:Nat. all n:Nat. (g+y)+n = g+(y+n)","type_family":"Nat","context":["D 0","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1413","id":89412,"instance":1413,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["Lemma","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1414","id":89413,"instance":1414,"goal":"all y:Atom. all n:Atom. y * n = n * y","type_family":"Atom","context":["Reduce","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1415","id":89414,"instance":1415,"goal":"all delta:Bool. delta * 1 = delta","type_family":"Bool","context":["D 0","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1416","id":89415,"instance":1416,"goal":"all v:List(Int). all f:List(Int). all y:List(Int). v*(f+y) = v*f + v*y","type_family":"List(Int)","context":["Thin","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1417","id":89416,"instance":1417,"goal":"all p:List(Nat). p <= p","type_family":"List(Nat)","context":["Auto","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1418","id":89417,"instance":1418,"goal":"all c:List(Atom). all n:List(Atom). all a:List(Atom). (c <= n) => (n <= a) => (c <= a)","type_family":"List(Atom)","context":["D 1","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1419","id":89418,"instance":1419,"goal":"all q:Int -> Int. q = q","type_family":"Int -> Int","context":["BackThru","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1420","id":89419,"instance":1420,"goal":"all c:Nat -> Nat. all y:Nat -> Nat. (c = y) => (y = c)","type_family":"Nat -> Nat","context":["Thin","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1421","id":89420,"instance":1421,"goal":"(z) => (b) => (z)","type_family":"Int","context":["D -1","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1422","id":89421,"instance":1422,"goal":"((h) & (c)) iff ((c) & (h))","type_family":"Nat","context":["Auto","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1423","id":89422,"instance":1423,"goal":"((p) | (q)) iff ((q) | (p))","type_family":"Real","context":["SplitOn","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1424","id":89423,"instance":1424,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["Assert","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1425","id":89424,"instance":1425,"goal":"all phi:Bool. all beta:Bool. all epsilon:Bool. append(phi, append(beta, epsilon)) = append(append(phi, beta), epsilon)","type_family":"Bool","context":["D 1","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1426","id":89425,"instance":1426,"goal":"all v:List(Int). all w:List(Int). v + w = w + v","type_family":"List(Int)","context":["D 0","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1427","id":89426,"instance":1427,"goal":"all c:List(Nat). all z:List(Nat). all p:List(Nat). (c+z)+p = c+(z+p)","type_family":"List(Nat)","context":["Reduce","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1428","id":89427,"instance":1428,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["Lemma","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1429","id":89428,"instance":1429,"goal":"all q:Int -> Int. all u:Int -> Int. q * u = u * q","type_family":"Int -> Int","context":["D 0","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1430","id":89429,"instance":1430,"goal":"all n:Nat -> Nat. n * 1 = n","type_family":"Nat -> Nat","context":["DoSubst","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1431","id":89430,"instance":1431,"goal":"all p:Int. all g:Int. all w:Int. p*(g+w) = p*g + p*w","type_family":"Int","context":["D 0","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1432","id":89431,"instance":1432,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["RepUR","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1433","id":89432,"instance":1433,"goal":"all a:Real. all m:Real. all x:Real. (a <= m) => (m <= x) => (a <= x)","type_family":"Real","context":["BackThru","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1434","id":89433,"instance":1434,"goal":"all q:Atom. q = q","type_family":"Atom","context":["Complete Auto","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1435","id":89434,"instance":1435,"goal":"all n:Bool. all v:Bool. (n = v) => (v = n)","type_family":"Bool","context":["RepUR","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1436","id":89435,"instance":1436,"goal":"(b) => (y) => (b)","type_family":"List(Int)","context":["RepUR","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1437","id":89436,"instance":1437,"goal":"((e) & (n)) iff ((n) & (e))","type_family":"List(Nat)","context":["InstLemma","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1438","id":89437,"instance":1438,"goal":"((c) | (d)) iff ((d) | (c))","type_family":"List(Atom)","context":["InstLemma","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1439","id":89438,"instance":1439,"goal":"not (not (d)) => (d)","type_family":"Int -> Int","context":["Auto","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1440","id":89439,"instance":1440,"goal":"all x:Nat -> Nat. all d:Nat -> Nat. all a:Nat -> Nat. append(x, append(d, a)) = append(append(x, d), a)","type_family":"Nat -> Nat","context":["SplitOn","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1441","id":89440,"instance":1441,"goal":"all c:Int. all m:Int. c + m = m + c","type_family":"Int","context":["InstLemma","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1442","id":89441,"instance":1442,"goal":"all w:Nat. all p:Nat. all q:Nat. (w+p)+q = w+(p+q)","type_family":"Nat","context":["Lemma","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1443","id":89442,"instance":1443,"goal":"all a:Real. a + 0 = a","type_family":"Real","context":["D 0","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1444","id":89443,"instance":1444,"goal":"all b:Atom. all g:Atom. b * g = g * b","type_family":"Atom","context":["D -1","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1445","id":89444,"instance":1445,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["RepUR","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1446","id":89445,"instance":1446,"goal":"all c:List(Int). all x:List(Int). all b:List(Int). c*(x+b) = c*x + c*b","type_family":"List(Int)","context":["Reduce","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1447","id":89446,"instance":1447,"goal":"all a:List(Nat). a <= a","type_family":"List(Nat)","context":["Auto","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1448","id":89447,"instance":1448,"goal":"all p:List(Atom). all h:List(Atom). all f:List(Atom). (p <= h) => (h <= f) => (p <= f)","type_family":"List(Atom)","context":["Assert","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1449","id":89448,"instance":1449,"goal":"all f:Int -> Int. f = f","type_family":"Int -> Int","context":["InstLemma","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1450","id":89449,"instance":1450,"goal":"all f:Nat -> Nat. all q:Nat -> Nat. (f = q) => (q = f)","type_family":"Nat -> Nat","context":["D 1","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1451","id":89450,"instance":1451,"goal":"(b) => (x) => (b)","type_family":"Int","context":["RepUR","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1452","id":89451,"instance":1452,"goal":"((kappa) & (psi)) iff ((psi) & (kappa))","type_family":"Nat","context":["BackThru","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1453","id":89452,"instance":1453,"goal":"((phi) | (psi)) iff ((psi) | (phi))","type_family":"Real","context":["DoSubst","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1454","id":89453,"instance":1454,"goal":"not (not (c)) => (c)","type_family":"Atom","context":["Lemma","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1455","id":89454,"instance":1455,"goal":"all f:Bool. all y:Bool. all w:Bool. append(f, append(y, w)) = append(append(f, y), w)","type_family":"Bool","context":["RepUR","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1456","id":89455,"instance":1456,"goal":"all z:List(Int). all d:List(Int). z + d = d + z","type_family":"List(Int)","context":["D -1","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1457","id":89456,"instance":1457,"goal":"all v:List(Nat). all g:List(Nat). all c:List(Nat). (v+g)+c = v+(g+c)","type_family":"List(Nat)","context":["RepUR","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1458","id":89457,"instance":1458,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["BackThru","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1459","id":89458,"instance":1459,"goal":"all delta:Int -> Int. all omega:Int -> Int. delta * omega = omega * delta","type_family":"Int -> Int","context":["D 1","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1460","id":89459,"instance":1460,"goal":"all c:Nat -> Nat. c * 1 = c","type_family":"Nat -> Nat","context":["BackThru","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1461","id":89460,"instance":1461,"goal":"all d:Int. all n:Int. all y:Int. d*(n+y) = d*n + d*y","type_family":"Int","context":["Lemma","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1462","id":89461,"instance":1462,"goal":"all e:Nat. e <= e","type_family":"Nat","context":["D 1","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1463","id":89462,"instance":1463,"goal":"all b:Real. all h:Real. all z:Real. (b <= h) => (h <= z) => (b <= z)","type_family":"Real","context":["RepUR","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1464","id":89463,"instance":1464,"goal":"all e:Atom. e = e","type_family":"Atom","context":["Auto","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1465","id":89464,"instance":1465,"goal":"all beta:Bool. all delta:Bool. (beta = delta) => (delta = beta)","type_family":"Bool","context":["D -1","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1466","id":89465,"instance":1466,"goal":"(v) => (d) => (v)","type_family":"List(Int)","context":["Thin","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1467","id":89466,"instance":1467,"goal":"((kappa) & (phi)) iff ((phi) & (kappa))","type_family":"List(Nat)","context":["D -1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1468","id":89467,"instance":1468,"goal":"((m) | (d)) iff ((d) | (m))","type_family":"List(Atom)","context":["Complete Auto","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1469","id":89468,"instance":1469,"goal":"not (not (w)) => (w)","type_family":"Int -> Int","context":["Assert","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1470","id":89469,"instance":1470,"goal":"all u:Nat -> Nat. all v:Nat -> Nat. all b:Nat -> Nat. append(u, append(v, b)) = append(append(u, v), b)","type_family":"Nat -> Nat","context":["BackThru","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1471","id":89470,"instance":1471,"goal":"all chi:Int. all delta:Int. chi + delta = delta + chi","type_family":"Int","context":["RepUR","D -1","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1472","id":89471,"instance":1472,"goal":"all g:Nat. all u:Nat. all d:Nat. (g+u)+d = g+(u+d)","type_family":"Nat","context":["D 1","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1473","id":89472,"instance":1473,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["Complete Auto","D -1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1474","id":89473,"instance":1474,"goal":"all y:Atom. all q:Atom. y * q = q * y","type_family":"Atom","context":["Complete Auto","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1475","id":89474,"instance":1475,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["Thin","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1476","id":89475,"instance":1476,"goal":"all g:List(Int). all p:List(Int). all h:List(Int). g*(p+h) = g*p + g*h","type_family":"List(Int)","context":["SplitOn","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1477","id":89476,"instance":1477,"goal":"all c:List(Nat). c <= c","type_family":"List(Nat)","context":["D 0","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1478","id":89477,"instance":1478,"goal":"all q:List(Atom). all d:List(Atom). all x:List(Atom). (q <= d) => (d <= x) => (q <= x)","type_family":"List(Atom)","context":["InstLemma","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1479","id":89478,"instance":1479,"goal":"all z:Int -> Int. z = z","type_family":"Int -> Int","context":["D -1","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1480","id":89479,"instance":1480,"goal":"all m:Nat -> Nat. all y:Nat -> Nat. (m = y) => (y = m)","type_family":"Nat -> Nat","context":["BackThru","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1481","id":89480,"instance":1481,"goal":"(c) => (d) => (c)","type_family":"Int","context":["InstLemma","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1482","id":89481,"instance":1482,"goal":"((c) & (b)) iff ((b) & (c))","type_family":"Nat","context":["Lemma","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1483","id":89482,"instance":1483,"goal":"((v) | (y)) iff ((y) | (v))","type_family":"Real","context":["SplitOn","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1484","id":89483,"instance":1484,"goal":"not (not (b)) => (b)","type_family":"Atom","context":["Complete Auto","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1485","id":89484,"instance":1485,"goal":"all g:Bool. all v:Bool. all m:Bool. append(g, append(v, m)) = append(append(g, v), m)","type_family":"Bool","context":["Lemma","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1486","id":89485,"instance":1486,"goal":"all h:List(Int). all g:List(Int). h + g = g + h","type_family":"List(Int)","context":["Auto","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1487","id":89486,"instance":1487,"goal":"all x:List(Nat). all d:List(Nat). all n:List(Nat). (x+d)+n = x+(d+n)","type_family":"List(Nat)","context":["RepUR","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1488","id":89487,"instance":1488,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["SplitOn","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1489","id":89488,"instance":1489,"goal":"all q:Int -> Int. all m:Int -> Int. q * m = m * q","type_family":"Int -> Int","context":["BackThru","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1490","id":89489,"instance":1490,"goal":"all a:Nat -> Nat. a * 1 = a","type_family":"Nat -> Nat","context":["Auto","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1491","id":89490,"instance":1491,"goal":"all y:Int. all m:Int. all u:Int. y*(m+u) = y*m + y*u","type_family":"Int","context":["D -1","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1492","id":89491,"instance":1492,"goal":"all b:Nat. b <= b","type_family":"Nat","context":["Complete Auto","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1493","id":89492,"instance":1493,"goal":"all m:Real. all a:Real. all p:Real. (m <= a) => (a <= p) => (m <= p)","type_family":"Real","context":["Lemma","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1494","id":89493,"instance":1494,"goal":"all epsilon:Atom. epsilon = epsilon","type_family":"Atom","context":["BackThru","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1495","id":89494,"instance":1495,"goal":"all g:Bool. all e:Bool. (g = e) => (e = g)","type_family":"Bool","context":["Lemma","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1496","id":89495,"instance":1496,"goal":"(z) => (p) => (z)","type_family":"List(Int)","context":["Lemma","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1497","id":89496,"instance":1497,"goal":"((n) & (v)) iff ((v) & (n))","type_family":"List(Nat)","context":["D -1","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1498","id":89497,"instance":1498,"goal":"((f) | (h)) iff ((h) | (f))","type_family":"List(Atom)","context":["Assert","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1499","id":89498,"instance":1499,"goal":"not (not (z)) => (z)","type_family":"Int -> Int","context":["D 1","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1500","id":89499,"instance":1500,"goal":"all y:Nat -> Nat. all w:Nat -> Nat. all d:Nat -> Nat. append(y, append(w, d)) = append(append(y, w), d)","type_family":"Nat -> Nat","context":["BackThru","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1501","id":89500,"instance":1501,"goal":"all g:Int. all z:Int. g + z = z + g","type_family":"Int","context":["D -1","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1502","id":89501,"instance":1502,"goal":"all h:Nat. all n:Nat. all w:Nat. (h+n)+w = h+(n+w)","type_family":"Nat","context":["D 1","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1503","id":89502,"instance":1503,"goal":"all u:Real. u + 0 = u","type_family":"Real","context":["Complete Auto","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1504","id":89503,"instance":1504,"goal":"all p:Atom. all g:Atom. p * g = g * p","type_family":"Atom","context":["Complete Auto","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1505","id":89504,"instance":1505,"goal":"all u:Bool. u * 1 = u","type_family":"Bool","context":["SplitOn","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1506","id":89505,"instance":1506,"goal":"all n:List(Int). all v:List(Int). all x:List(Int). n*(v+x) = n*v + n*x","type_family":"List(Int)","context":["Lemma","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1507","id":89506,"instance":1507,"goal":"all b:List(Nat). b <= b","type_family":"List(Nat)","context":["SplitOn","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1508","id":89507,"instance":1508,"goal":"all h:List(Atom). all w:List(Atom). all c:List(Atom). (h <= w) => (w <= c) => (h <= c)","type_family":"List(Atom)","context":["D 0","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1509","id":89508,"instance":1509,"goal":"all x:Int -> Int. x = x","type_family":"Int -> Int","context":["DoSubst","D -1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1510","id":89509,"instance":1510,"goal":"all d:Nat -> Nat. all u:Nat -> Nat. (d = u) => (u = d)","type_family":"Nat -> Nat","context":["D 1","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1511","id":89510,"instance":1511,"goal":"(q) => (z) => (q)","type_family":"Int","context":["Complete Auto","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1512","id":89511,"instance":1512,"goal":"((p) & (x)) iff ((x) & (p))","type_family":"Nat","context":["Auto","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1513","id":89512,"instance":1513,"goal":"((g) | (a)) iff ((a) | (g))","type_family":"Real","context":["Complete Auto","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1514","id":89513,"instance":1514,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Complete Auto","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1515","id":89514,"instance":1515,"goal":"all w:Bool. all g:Bool. all y:Bool. append(w, append(g, y)) = append(append(w, g), y)","type_family":"Bool","context":["DoSubst","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1516","id":89515,"instance":1516,"goal":"all q:List(Int). all x:List(Int). q + x = x + q","type_family":"List(Int)","context":["Assert","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1517","id":89516,"instance":1517,"goal":"all b:List(Nat). all z:List(Nat). all y:List(Nat). (b+z)+y = b+(z+y)","type_family":"List(Nat)","context":["SplitOn","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1518","id":89517,"instance":1518,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["RepUR","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1519","id":89518,"instance":1519,"goal":"all beta:Int -> Int. all chi:Int -> Int. beta * chi = chi * beta","type_family":"Int -> Int","context":["InstLemma","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1520","id":89519,"instance":1520,"goal":"all v:Nat -> Nat. v * 1 = v","type_family":"Nat -> Nat","context":["D 1","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1521","id":89520,"instance":1521,"goal":"all u:Int. all g:Int. all x:Int. u*(g+x) = u*g + u*x","type_family":"Int","context":["InstLemma","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1522","id":89521,"instance":1522,"goal":"all c:Nat. c <= c","type_family":"Nat","context":["BackThru","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1523","id":89522,"instance":1523,"goal":"all y:Real. all n:Real. all d:Real. (y <= n) => (n <= d) => (y <= d)","type_family":"Real","context":["InstLemma","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1524","id":89523,"instance":1524,"goal":"all e:Atom. e = e","type_family":"Atom","context":["DoSubst","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1525","id":89524,"instance":1525,"goal":"all p:Bool. all n:Bool. (p = n) => (n = p)","type_family":"Bool","context":["DoSubst","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1526","id":89525,"instance":1526,"goal":"(e) => (a) => (e)","type_family":"List(Int)","context":["D -1","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1527","id":89526,"instance":1527,"goal":"((e) & (z)) iff ((z) & (e))","type_family":"List(Nat)","context":["SplitOn","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1528","id":89527,"instance":1528,"goal":"((u) | (c)) iff ((c) | (u))","type_family":"List(Atom)","context":["SplitOn","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1529","id":89528,"instance":1529,"goal":"not (not (omega)) => (omega)","type_family":"Int -> Int","context":["Thin","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1530","id":89529,"instance":1530,"goal":"all d:Nat -> Nat. all b:Nat -> Nat. all n:Nat -> Nat. append(d, append(b, n)) = append(append(d, b), n)","type_family":"Nat -> Nat","context":["Reduce","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1531","id":89530,"instance":1531,"goal":"all g:Int. all e:Int. g + e = e + g","type_family":"Int","context":["D 1","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1532","id":89531,"instance":1532,"goal":"all n:Nat. all a:Nat. all f:Nat. (n+a)+f = n+(a+f)","type_family":"Nat","context":["Reduce","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1533","id":89532,"instance":1533,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["SplitOn","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1534","id":89533,"instance":1534,"goal":"all p:Atom. all m:Atom. p * m = m * p","type_family":"Atom","context":["DoSubst","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1535","id":89534,"instance":1535,"goal":"all p:Bool. p * 1 = p","type_family":"Bool","context":["DoSubst","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1536","id":89535,"instance":1536,"goal":"all x:List(Int). all a:List(Int). all n:List(Int). x*(a+n) = x*a + x*n","type_family":"List(Int)","context":["Complete Auto","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1537","id":89536,"instance":1537,"goal":"all v:List(Nat). v <= v","type_family":"List(Nat)","context":["Thin","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1538","id":89537,"instance":1538,"goal":"all y:List(Atom). all g:List(Atom). all p:List(Atom). (y <= g) => (g <= p) => (y <= p)","type_family":"List(Atom)","context":["BackThru","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1539","id":89538,"instance":1539,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["Assert","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1540","id":89539,"instance":1540,"goal":"all beta:Nat -> Nat. all omega:Nat -> Nat. (beta = omega) => (omega = beta)","type_family":"Nat -> Nat","context":["Complete Auto","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1541","id":89540,"instance":1541,"goal":"(b) => (q) => (b)","type_family":"Int","context":["BackThru","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1542","id":89541,"instance":1542,"goal":"((h) & (a)) iff ((a) & (h))","type_family":"Nat","context":["Assert","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1543","id":89542,"instance":1543,"goal":"((m) | (c)) iff ((c) | (m))","type_family":"Real","context":["D 0","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1544","id":89543,"instance":1544,"goal":"not (not (g)) => (g)","type_family":"Atom","context":["D 0","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1545","id":89544,"instance":1545,"goal":"all d:Bool. all f:Bool. all m:Bool. append(d, append(f, m)) = append(append(d, f), m)","type_family":"Bool","context":["Thin","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1546","id":89545,"instance":1546,"goal":"all b:List(Int). all w:List(Int). b + w = w + b","type_family":"List(Int)","context":["Thin","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1547","id":89546,"instance":1547,"goal":"all c:List(Nat). all m:List(Nat). all w:List(Nat). (c+m)+w = c+(m+w)","type_family":"List(Nat)","context":["Thin","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1548","id":89547,"instance":1548,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["Reduce","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1549","id":89548,"instance":1549,"goal":"all e:Int -> Int. all b:Int -> Int. e * b = b * e","type_family":"Int -> Int","context":["DoSubst","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1550","id":89549,"instance":1550,"goal":"all m:Nat -> Nat. m * 1 = m","type_family":"Nat -> Nat","context":["BackThru","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1551","id":89550,"instance":1551,"goal":"all c:Int. all q:Int. all u:Int. c*(q+u) = c*q + c*u","type_family":"Int","context":["D -1","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1552","id":89551,"instance":1552,"goal":"all w:Nat. w <= w","type_family":"Nat","context":["D -1","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1553","id":89552,"instance":1553,"goal":"all w:Real. all a:Real. all n:Real. (w <= a) => (a <= n) => (w <= n)","type_family":"Real","context":["Thin","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1554","id":89553,"instance":1554,"goal":"all u:Atom. u = u","type_family":"Atom","context":["Lemma","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1555","id":89554,"instance":1555,"goal":"all q:Bool. all p:Bool. (q = p) => (p = q)","type_family":"Bool","context":["Complete Auto","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1556","id":89555,"instance":1556,"goal":"(q) => (x) => (q)","type_family":"List(Int)","context":["BackThru","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1557","id":89556,"instance":1557,"goal":"((v) & (x)) iff ((x) & (v))","type_family":"List(Nat)","context":["Assert","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1558","id":89557,"instance":1558,"goal":"((phi) | (delta)) iff ((delta) | (phi))","type_family":"List(Atom)","context":["Reduce","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1559","id":89558,"instance":1559,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["SplitOn","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1560","id":89559,"instance":1560,"goal":"all m:Nat -> Nat. all v:Nat -> Nat. all d:Nat -> Nat. append(m, append(v, d)) = append(append(m, v), d)","type_family":"Nat -> Nat","context":["RepUR","D 0","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Auto; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1561","id":89560,"instance":1561,"goal":"all v:Int. all m:Int. v + m = m + v","type_family":"Int","context":["Auto","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1562","id":89561,"instance":1562,"goal":"all p:Nat. all x:Nat. all q:Nat. (p+x)+q = p+(x+q)","type_family":"Nat","context":["D -1","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1563","id":89562,"instance":1563,"goal":"all v:Real. v + 0 = v","type_family":"Real","context":["Lemma","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1564","id":89563,"instance":1564,"goal":"all g:Atom. all b:Atom. g * b = b * g","type_family":"Atom","context":["InstLemma","BackThru","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1565","id":89564,"instance":1565,"goal":"all v:Bool. v * 1 = v","type_family":"Bool","context":["BackThru","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1566","id":89565,"instance":1566,"goal":"all q:List(Int). all n:List(Int). all f:List(Int). q*(n+f) = q*n + q*f","type_family":"List(Int)","context":["Reduce","SplitOn","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1567","id":89566,"instance":1567,"goal":"all b:List(Nat). b <= b","type_family":"List(Nat)","context":["Thin","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1568","id":89567,"instance":1568,"goal":"all u:List(Atom). all v:List(Atom). all h:List(Atom). (u <= v) => (v <= h) => (u <= h)","type_family":"List(Atom)","context":["D 0","D -1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1569","id":89568,"instance":1569,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["SplitOn","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1570","id":89569,"instance":1570,"goal":"all n:Nat -> Nat. all a:Nat -> Nat. (n = a) => (a = n)","type_family":"Nat -> Nat","context":["Auto","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1571","id":89570,"instance":1571,"goal":"(kappa) => (phi) => (kappa)","type_family":"Int","context":["D 0","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1572","id":89571,"instance":1572,"goal":"((q) & (m)) iff ((m) & (q))","type_family":"Nat","context":["BackThru","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1573","id":89572,"instance":1573,"goal":"((kappa) | (gamma)) iff ((gamma) | (kappa))","type_family":"Real","context":["Auto","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1574","id":89573,"instance":1574,"goal":"not (not (m)) => (m)","type_family":"Atom","context":["D 0","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1575","id":89574,"instance":1575,"goal":"all f:Bool. all g:Bool. all x:Bool. append(f, append(g, x)) = append(append(f, g), x)","type_family":"Bool","context":["D 1","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1576","id":89575,"instance":1576,"goal":"all p:List(Int). all m:List(Int). p + m = m + p","type_family":"List(Int)","context":["D -1","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1577","id":89576,"instance":1577,"goal":"all p:List(Nat). all u:List(Nat). all y:List(Nat). (p+u)+y = p+(u+y)","type_family":"List(Nat)","context":["BackThru","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1578","id":89577,"instance":1578,"goal":"all d:List(Atom). d + 0 = d","type_family":"List(Atom)","context":["D 1","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1579","id":89578,"instance":1579,"goal":"all d:Int -> Int. all f:Int -> Int. d * f = f * d","type_family":"Int -> Int","context":["RepUR","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1580","id":89579,"instance":1580,"goal":"all h:Nat -> Nat. h * 1 = h","type_family":"Nat -> Nat","context":["Assert","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1581","id":89580,"instance":1581,"goal":"all n:Int. all d:Int. all z:Int. n*(d+z) = n*d + n*z","type_family":"Int","context":["SplitOn","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1582","id":89581,"instance":1582,"goal":"all y:Nat. y <= y","type_family":"Nat","context":["D 0","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1583","id":89582,"instance":1583,"goal":"all p:Real. all w:Real. all c:Real. (p <= w) => (w <= c) => (p <= c)","type_family":"Real","context":["Reduce","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1584","id":89583,"instance":1584,"goal":"all p:Atom. p = p","type_family":"Atom","context":["Complete Auto","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1585","id":89584,"instance":1585,"goal":"all v:Bool. all z:Bool. (v = z) => (z = v)","type_family":"Bool","context":["DoSubst","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1586","id":89585,"instance":1586,"goal":"(chi) => (phi) => (chi)","type_family":"List(Int)","context":["D 0","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1587","id":89586,"instance":1587,"goal":"((x) & (u)) iff ((u) & (x))","type_family":"List(Nat)","context":["DoSubst","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1588","id":89587,"instance":1588,"goal":"((h) | (u)) iff ((u) | (h))","type_family":"List(Atom)","context":["Thin","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1589","id":89588,"instance":1589,"goal":"not (not (p)) => (p)","type_family":"Int -> Int","context":["Thin","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1590","id":89589,"instance":1590,"goal":"all h:Nat -> Nat. all m:Nat -> Nat. all z:Nat -> Nat. append(h, append(m, z)) = append(append(h, m), z)","type_family":"Nat -> Nat","context":["DoSubst","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1591","id":89590,"instance":1591,"goal":"all delta:Int. all omega:Int. delta + omega = omega + delta","type_family":"Int","context":["Reduce","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1592","id":89591,"instance":1592,"goal":"all d:Nat. all b:Nat. all u:Nat. (d+b)+u = d+(b+u)","type_family":"Nat","context":["Auto","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1593","id":89592,"instance":1593,"goal":"all z:Real. z + 0 = z","type_family":"Real","context":["Complete Auto","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1594","id":89593,"instance":1594,"goal":"all a:Atom. all m:Atom. a * m = m * a","type_family":"Atom","context":["SplitOn","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1595","id":89594,"instance":1595,"goal":"all chi:Bool. chi * 1 = chi","type_family":"Bool","context":["D 1","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1596","id":89595,"instance":1596,"goal":"all u:List(Int). all z:List(Int). all c:List(Int). u*(z+c) = u*z + u*c","type_family":"List(Int)","context":["Thin","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1597","id":89596,"instance":1597,"goal":"all w:List(Nat). w <= w","type_family":"List(Nat)","context":["Lemma","Complete Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1598","id":89597,"instance":1598,"goal":"all e:List(Atom). all f:List(Atom). all y:List(Atom). (e <= f) => (f <= y) => (e <= y)","type_family":"List(Atom)","context":["Complete Auto","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1599","id":89598,"instance":1599,"goal":"all z:Int -> Int. z = z","type_family":"Int -> Int","context":["BackThru","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1600","id":89599,"instance":1600,"goal":"all m:Nat -> Nat. all g:Nat -> Nat. (m = g) => (g = m)","type_family":"Nat -> Nat","context":["SplitOn","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1601","id":89600,"instance":1601,"goal":"(e) => (g) => (e)","type_family":"Int","context":["Reduce","SplitOn","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1602","id":89601,"instance":1602,"goal":"((chi) & (kappa)) iff ((kappa) & (chi))","type_family":"Nat","context":["Complete Auto","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1603","id":89602,"instance":1603,"goal":"((u) | (f)) iff ((f) | (u))","type_family":"Real","context":["D 0","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1604","id":89603,"instance":1604,"goal":"not (not (w)) => (w)","type_family":"Atom","context":["Auto","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1605","id":89604,"instance":1605,"goal":"all c:Bool. all n:Bool. all b:Bool. append(c, append(n, b)) = append(append(c, n), b)","type_family":"Bool","context":["D -1","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1606","id":89605,"instance":1606,"goal":"all epsilon:List(Int). all phi:List(Int). epsilon + phi = phi + epsilon","type_family":"List(Int)","context":["Lemma","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1607","id":89606,"instance":1607,"goal":"all n:List(Nat). all w:List(Nat). all e:List(Nat). (n+w)+e = n+(w+e)","type_family":"List(Nat)","context":["BackThru","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1608","id":89607,"instance":1608,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["DoSubst","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1609","id":89608,"instance":1609,"goal":"all phi:Int -> Int. all kappa:Int -> Int. phi * kappa = kappa * phi","type_family":"Int -> Int","context":["D -1","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1610","id":89609,"instance":1610,"goal":"all b:Nat -> Nat. b * 1 = b","type_family":"Nat -> Nat","context":["Thin","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1611","id":89610,"instance":1611,"goal":"all f:Int. all c:Int. all d:Int. f*(c+d) = f*c + f*d","type_family":"Int","context":["D 0","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1612","id":89611,"instance":1612,"goal":"all n:Nat. n <= n","type_family":"Nat","context":["Complete Auto","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1613","id":89612,"instance":1613,"goal":"all z:Real. all e:Real. all w:Real. (z <= e) => (e <= w) => (z <= w)","type_family":"Real","context":["Thin","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1614","id":89613,"instance":1614,"goal":"all n:Atom. n = n","type_family":"Atom","context":["Complete Auto","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1615","id":89614,"instance":1615,"goal":"all a:Bool. all g:Bool. (a = g) => (g = a)","type_family":"Bool","context":["Auto","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1616","id":89615,"instance":1616,"goal":"(y) => (h) => (y)","type_family":"List(Int)","context":["RepUR","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1617","id":89616,"instance":1617,"goal":"((u) & (z)) iff ((z) & (u))","type_family":"List(Nat)","context":["Assert","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1618","id":89617,"instance":1618,"goal":"((b) | (v)) iff ((v) | (b))","type_family":"List(Atom)","context":["Reduce","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1619","id":89618,"instance":1619,"goal":"not (not (n)) => (n)","type_family":"Int -> Int","context":["D -1","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1620","id":89619,"instance":1620,"goal":"all w:Nat -> Nat. all f:Nat -> Nat. all b:Nat -> Nat. append(w, append(f, b)) = append(append(w, f), b)","type_family":"Nat -> Nat","context":["DoSubst","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1621","id":89620,"instance":1621,"goal":"all w:Int. all u:Int. w + u = u + w","type_family":"Int","context":["Thin","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1622","id":89621,"instance":1622,"goal":"all d:Nat. all m:Nat. all z:Nat. (d+m)+z = d+(m+z)","type_family":"Nat","context":["BackThru","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1623","id":89622,"instance":1623,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["RepUR","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1624","id":89623,"instance":1624,"goal":"all p:Atom. all f:Atom. p * f = f * p","type_family":"Atom","context":["Reduce","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1625","id":89624,"instance":1625,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["Reduce","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1626","id":89625,"instance":1626,"goal":"all e:List(Int). all u:List(Int). all b:List(Int). e*(u+b) = e*u + e*b","type_family":"List(Int)","context":["DoSubst","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1627","id":89626,"instance":1627,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["Thin","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1628","id":89627,"instance":1628,"goal":"all z:List(Atom). all h:List(Atom). all p:List(Atom). (z <= h) => (h <= p) => (z <= p)","type_family":"List(Atom)","context":["DoSubst","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1629","id":89628,"instance":1629,"goal":"all alpha:Int -> Int. alpha = alpha","type_family":"Int -> Int","context":["Lemma","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1630","id":89629,"instance":1630,"goal":"all b:Nat -> Nat. all c:Nat -> Nat. (b = c) => (c = b)","type_family":"Nat -> Nat","context":["D 1","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1631","id":89630,"instance":1631,"goal":"(m) => (z) => (m)","type_family":"Int","context":["Complete Auto","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1632","id":89631,"instance":1632,"goal":"((beta) & (psi)) iff ((psi) & (beta))","type_family":"Nat","context":["D 1","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1633","id":89632,"instance":1633,"goal":"((g) | (m)) iff ((m) | (g))","type_family":"Real","context":["D -1","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1634","id":89633,"instance":1634,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["Lemma","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1635","id":89634,"instance":1635,"goal":"all h:Bool. all z:Bool. all f:Bool. append(h, append(z, f)) = append(append(h, z), f)","type_family":"Bool","context":["Thin","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1636","id":89635,"instance":1636,"goal":"all kappa:List(Int). all alpha:List(Int). kappa + alpha = alpha + kappa","type_family":"List(Int)","context":["BackThru","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1637","id":89636,"instance":1637,"goal":"all d:List(Nat). all q:List(Nat). all y:List(Nat). (d+q)+y = d+(q+y)","type_family":"List(Nat)","context":["InstLemma","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1638","id":89637,"instance":1638,"goal":"all epsilon:List(Atom). epsilon + 0 = epsilon","type_family":"List(Atom)","context":["SplitOn","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1639","id":89638,"instance":1639,"goal":"all n:Int -> Int. all y:Int -> Int. n * y = y * n","type_family":"Int -> Int","context":["D -1","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1640","id":89639,"instance":1640,"goal":"all z:Nat -> Nat. z * 1 = z","type_family":"Nat -> Nat","context":["Auto","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1641","id":89640,"instance":1641,"goal":"all n:Int. all h:Int. all c:Int. n*(h+c) = n*h + n*c","type_family":"Int","context":["D 0","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1642","id":89641,"instance":1642,"goal":"all q:Nat. q <= q","type_family":"Nat","context":["BackThru","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1643","id":89642,"instance":1643,"goal":"all f:Real. all w:Real. all g:Real. (f <= w) => (w <= g) => (f <= g)","type_family":"Real","context":["Auto","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1644","id":89643,"instance":1644,"goal":"all z:Atom. z = z","type_family":"Atom","context":["Complete Auto","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1645","id":89644,"instance":1645,"goal":"all p:Bool. all g:Bool. (p = g) => (g = p)","type_family":"Bool","context":["Auto","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1646","id":89645,"instance":1646,"goal":"(x) => (e) => (x)","type_family":"List(Int)","context":["Complete Auto","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1647","id":89646,"instance":1647,"goal":"((c) & (n)) iff ((n) & (c))","type_family":"List(Nat)","context":["D 0","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1648","id":89647,"instance":1648,"goal":"((v) | (b)) iff ((b) | (v))","type_family":"List(Atom)","context":["Reduce","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1649","id":89648,"instance":1649,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["Reduce","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1650","id":89649,"instance":1650,"goal":"all p:Nat -> Nat. all g:Nat -> Nat. all q:Nat -> Nat. append(p, append(g, q)) = append(append(p, g), q)","type_family":"Nat -> Nat","context":["Complete Auto","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1651","id":89650,"instance":1651,"goal":"all w:Int. all h:Int. w + h = h + w","type_family":"Int","context":["D -1","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1652","id":89651,"instance":1652,"goal":"all z:Nat. all m:Nat. all x:Nat. (z+m)+x = z+(m+x)","type_family":"Nat","context":["RepUR","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1653","id":89652,"instance":1653,"goal":"all v:Real. v + 0 = v","type_family":"Real","context":["D 1","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1654","id":89653,"instance":1654,"goal":"all e:Atom. all m:Atom. e * m = m * e","type_family":"Atom","context":["Complete Auto","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1655","id":89654,"instance":1655,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["Complete Auto","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1656","id":89655,"instance":1656,"goal":"all b:List(Int). all m:List(Int). all u:List(Int). b*(m+u) = b*m + b*u","type_family":"List(Int)","context":["Lemma","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1657","id":89656,"instance":1657,"goal":"all u:List(Nat). u <= u","type_family":"List(Nat)","context":["BackThru","D 1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1658","id":89657,"instance":1658,"goal":"all z:List(Atom). all w:List(Atom). all f:List(Atom). (z <= w) => (w <= f) => (z <= f)","type_family":"List(Atom)","context":["Reduce","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1659","id":89658,"instance":1659,"goal":"all y:Int -> Int. y = y","type_family":"Int -> Int","context":["D -1","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1660","id":89659,"instance":1660,"goal":"all kappa:Nat -> Nat. all beta:Nat -> Nat. (kappa = beta) => (beta = kappa)","type_family":"Nat -> Nat","context":["BackThru","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1661","id":89660,"instance":1661,"goal":"(alpha) => (phi) => (alpha)","type_family":"Int","context":["RepUR","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1662","id":89661,"instance":1662,"goal":"((h) & (d)) iff ((d) & (h))","type_family":"Nat","context":["BackThru","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1663","id":89662,"instance":1663,"goal":"((p) | (w)) iff ((w) | (p))","type_family":"Real","context":["D -1","D 1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1664","id":89663,"instance":1664,"goal":"not (not (u)) => (u)","type_family":"Atom","context":["DoSubst","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1665","id":89664,"instance":1665,"goal":"all d:Bool. all y:Bool. all f:Bool. append(d, append(y, f)) = append(append(d, y), f)","type_family":"Bool","context":["DoSubst","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1666","id":89665,"instance":1666,"goal":"all f:List(Int). all e:List(Int). f + e = e + f","type_family":"List(Int)","context":["D -1","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1667","id":89666,"instance":1667,"goal":"all u:List(Nat). all e:List(Nat). all z:List(Nat). (u+e)+z = u+(e+z)","type_family":"List(Nat)","context":["SplitOn","Lemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1668","id":89667,"instance":1668,"goal":"all v:List(Atom). v + 0 = v","type_family":"List(Atom)","context":["Reduce","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1669","id":89668,"instance":1669,"goal":"all alpha:Int -> Int. all omega:Int -> Int. alpha * omega = omega * alpha","type_family":"Int -> Int","context":["D 1","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1670","id":89669,"instance":1670,"goal":"all b:Nat -> Nat. b * 1 = b","type_family":"Nat -> Nat","context":["Lemma","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1671","id":89670,"instance":1671,"goal":"all y:Int. all u:Int. all e:Int. y*(u+e) = y*u + y*e","type_family":"Int","context":["Reduce","D 0","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1672","id":89671,"instance":1672,"goal":"all n:Nat. n <= n","type_family":"Nat","context":["SplitOn","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1673","id":89672,"instance":1673,"goal":"all z:Real. all g:Real. all f:Real. (z <= g) => (g <= f) => (z <= f)","type_family":"Real","context":["Thin","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1674","id":89673,"instance":1674,"goal":"all b:Atom. b = b","type_family":"Atom","context":["Reduce","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1675","id":89674,"instance":1675,"goal":"all psi:Bool. all delta:Bool. (psi = delta) => (delta = psi)","type_family":"Bool","context":["D 0","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1676","id":89675,"instance":1676,"goal":"(e) => (c) => (e)","type_family":"List(Int)","context":["D 0","Reduce","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1677","id":89676,"instance":1677,"goal":"((g) & (w)) iff ((w) & (g))","type_family":"List(Nat)","context":["DoSubst","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1678","id":89677,"instance":1678,"goal":"((p) | (y)) iff ((y) | (p))","type_family":"List(Atom)","context":["Reduce","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1679","id":89678,"instance":1679,"goal":"not (not (c)) => (c)","type_family":"Int -> Int","context":["D 0","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1680","id":89679,"instance":1680,"goal":"all n:Nat -> Nat. all v:Nat -> Nat. all q:Nat -> Nat. append(n, append(v, q)) = append(append(n, v), q)","type_family":"Nat -> Nat","context":["DoSubst","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1681","id":89680,"instance":1681,"goal":"all beta:Int. all omega:Int. beta + omega = omega + beta","type_family":"Int","context":["D -1","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1682","id":89681,"instance":1682,"goal":"all n:Nat. all y:Nat. all x:Nat. (n+y)+x = n+(y+x)","type_family":"Nat","context":["SplitOn","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1683","id":89682,"instance":1683,"goal":"all x:Real. x + 0 = x","type_family":"Real","context":["Reduce","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1684","id":89683,"instance":1684,"goal":"all f:Atom. all p:Atom. f * p = p * f","type_family":"Atom","context":["RepUR","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1685","id":89684,"instance":1685,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["D -1","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1686","id":89685,"instance":1686,"goal":"all x:List(Int). all e:List(Int). all m:List(Int). x*(e+m) = x*e + x*m","type_family":"List(Int)","context":["BackThru","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1687","id":89686,"instance":1687,"goal":"all v:List(Nat). v <= v","type_family":"List(Nat)","context":["D -1","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1688","id":89687,"instance":1688,"goal":"all q:List(Atom). all x:List(Atom). all m:List(Atom). (q <= x) => (x <= m) => (q <= m)","type_family":"List(Atom)","context":["Reduce","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1689","id":89688,"instance":1689,"goal":"all f:Int -> Int. f = f","type_family":"Int -> Int","context":["Complete Auto","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1690","id":89689,"instance":1690,"goal":"all e:Nat -> Nat. all p:Nat -> Nat. (e = p) => (p = e)","type_family":"Nat -> Nat","context":["BackThru","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1691","id":89690,"instance":1691,"goal":"(g) => (q) => (g)","type_family":"Int","context":["InstLemma","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1692","id":89691,"instance":1692,"goal":"((y) & (c)) iff ((c) & (y))","type_family":"Nat","context":["Lemma","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1693","id":89692,"instance":1693,"goal":"((g) | (q)) iff ((q) | (g))","type_family":"Real","context":["D -1","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1694","id":89693,"instance":1694,"goal":"not (not (w)) => (w)","type_family":"Atom","context":["BackThru","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1695","id":89694,"instance":1695,"goal":"all g:Bool. all p:Bool. all u:Bool. append(g, append(p, u)) = append(append(g, p), u)","type_family":"Bool","context":["Lemma","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1696","id":89695,"instance":1696,"goal":"all e:List(Int). all p:List(Int). e + p = p + e","type_family":"List(Int)","context":["D -1","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1697","id":89696,"instance":1697,"goal":"all w:List(Nat). all h:List(Nat). all x:List(Nat). (w+h)+x = w+(h+x)","type_family":"List(Nat)","context":["Reduce","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1698","id":89697,"instance":1698,"goal":"all b:List(Atom). b + 0 = b","type_family":"List(Atom)","context":["Thin","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1699","id":89698,"instance":1699,"goal":"all w:Int -> Int. all u:Int -> Int. w * u = u * w","type_family":"Int -> Int","context":["D 0","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1700","id":89699,"instance":1700,"goal":"all q:Nat -> Nat. q * 1 = q","type_family":"Nat -> Nat","context":["Reduce","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1701","id":89700,"instance":1701,"goal":"all u:Int. all a:Int. all x:Int. u*(a+x) = u*a + u*x","type_family":"Int","context":["Auto","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1702","id":89701,"instance":1702,"goal":"all z:Nat. z <= z","type_family":"Nat","context":["InstLemma","D -1","D -1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1703","id":89702,"instance":1703,"goal":"all g:Real. all z:Real. all b:Real. (g <= z) => (z <= b) => (g <= b)","type_family":"Real","context":["Assert","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1704","id":89703,"instance":1704,"goal":"all a:Atom. a = a","type_family":"Atom","context":["D -1","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1705","id":89704,"instance":1705,"goal":"all v:Bool. all c:Bool. (v = c) => (c = v)","type_family":"Bool","context":["InstLemma","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1706","id":89705,"instance":1706,"goal":"(b) => (f) => (b)","type_family":"List(Int)","context":["Assert","Assert","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1707","id":89706,"instance":1707,"goal":"((u) & (v)) iff ((v) & (u))","type_family":"List(Nat)","context":["Assert","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1708","id":89707,"instance":1708,"goal":"((d) | (n)) iff ((n) | (d))","type_family":"List(Atom)","context":["Thin","Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1709","id":89708,"instance":1709,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["Auto","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1710","id":89709,"instance":1710,"goal":"all a:Nat -> Nat. all q:Nat -> Nat. all d:Nat -> Nat. append(a, append(q, d)) = append(append(a, q), d)","type_family":"Nat -> Nat","context":["Complete Auto","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1711","id":89710,"instance":1711,"goal":"all f:Int. all h:Int. f + h = h + f","type_family":"Int","context":["D -1","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1712","id":89711,"instance":1712,"goal":"all v:Nat. all d:Nat. all m:Nat. (v+d)+m = v+(d+m)","type_family":"Nat","context":["Lemma","DoSubst","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1713","id":89712,"instance":1713,"goal":"all n:Real. n + 0 = n","type_family":"Real","context":["InstLemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1714","id":89713,"instance":1714,"goal":"all beta:Atom. all gamma:Atom. beta * gamma = gamma * beta","type_family":"Atom","context":["Auto","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1715","id":89714,"instance":1715,"goal":"all m:Bool. m * 1 = m","type_family":"Bool","context":["Auto","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1716","id":89715,"instance":1716,"goal":"all w:List(Int). all q:List(Int). all v:List(Int). w*(q+v) = w*q + w*v","type_family":"List(Int)","context":["Lemma","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1717","id":89716,"instance":1717,"goal":"all psi:List(Nat). psi <= psi","type_family":"List(Nat)","context":["SplitOn","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1718","id":89717,"instance":1718,"goal":"all c:List(Atom). all g:List(Atom). all x:List(Atom). (c <= g) => (g <= x) => (c <= x)","type_family":"List(Atom)","context":["Assert","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1719","id":89718,"instance":1719,"goal":"all u:Int -> Int. u = u","type_family":"Int -> Int","context":["InstLemma","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1720","id":89719,"instance":1720,"goal":"all q:Nat -> Nat. all g:Nat -> Nat. (q = g) => (g = q)","type_family":"Nat -> Nat","context":["SplitOn","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1721","id":89720,"instance":1721,"goal":"(w) => (u) => (w)","type_family":"Int","context":["RepUR","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1722","id":89721,"instance":1722,"goal":"((w) & (z)) iff ((z) & (w))","type_family":"Nat","context":["BackThru","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1723","id":89722,"instance":1723,"goal":"((f) | (h)) iff ((h) | (f))","type_family":"Real","context":["Lemma","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1724","id":89723,"instance":1724,"goal":"not (not (v)) => (v)","type_family":"Atom","context":["D 1","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1725","id":89724,"instance":1725,"goal":"all y:Bool. all n:Bool. all d:Bool. append(y, append(n, d)) = append(append(y, n), d)","type_family":"Bool","context":["Lemma","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1726","id":89725,"instance":1726,"goal":"all q:List(Int). all y:List(Int). q + y = y + q","type_family":"List(Int)","context":["DoSubst","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1727","id":89726,"instance":1727,"goal":"all w:List(Nat). all e:List(Nat). all n:List(Nat). (w+e)+n = w+(e+n)","type_family":"List(Nat)","context":["Reduce","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1728","id":89727,"instance":1728,"goal":"all v:List(Atom). v + 0 = v","type_family":"List(Atom)","context":["SplitOn","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1729","id":89728,"instance":1729,"goal":"all z:Int -> Int. all q:Int -> Int. z * q = q * z","type_family":"Int -> Int","context":["Auto","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; DoSubst","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1730","id":89729,"instance":1730,"goal":"all c:Nat -> Nat. c * 1 = c","type_family":"Nat -> Nat","context":["Reduce","SplitOn","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1731","id":89730,"instance":1731,"goal":"all n:Int. all g:Int. all y:Int. n*(g+y) = n*g + n*y","type_family":"Int","context":["D 0","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1732","id":89731,"instance":1732,"goal":"all beta:Nat. beta <= beta","type_family":"Nat","context":["RepUR","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1733","id":89732,"instance":1733,"goal":"all alpha:Real. all kappa:Real. all gamma:Real. (alpha <= kappa) => (kappa <= gamma) => (alpha <= gamma)","type_family":"Real","context":["Thin","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1734","id":89733,"instance":1734,"goal":"all b:Atom. b = b","type_family":"Atom","context":["BackThru","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1735","id":89734,"instance":1735,"goal":"all g:Bool. all x:Bool. (g = x) => (x = g)","type_family":"Bool","context":["D -1","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1736","id":89735,"instance":1736,"goal":"(u) => (b) => (u)","type_family":"List(Int)","context":["InstLemma","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1737","id":89736,"instance":1737,"goal":"((g) & (w)) iff ((w) & (g))","type_family":"List(Nat)","context":["DoSubst","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1738","id":89737,"instance":1738,"goal":"((c) | (g)) iff ((g) | (c))","type_family":"List(Atom)","context":["BackThru","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1739","id":89738,"instance":1739,"goal":"not (not (x)) => (x)","type_family":"Int -> Int","context":["DoSubst","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1740","id":89739,"instance":1740,"goal":"all n:Nat -> Nat. all b:Nat -> Nat. all c:Nat -> Nat. append(n, append(b, c)) = append(append(n, b), c)","type_family":"Nat -> Nat","context":["Thin","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1741","id":89740,"instance":1741,"goal":"all g:Int. all f:Int. g + f = f + g","type_family":"Int","context":["Auto","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1742","id":89741,"instance":1742,"goal":"all kappa:Nat. all psi:Nat. all gamma:Nat. (kappa+psi)+gamma = kappa+(psi+gamma)","type_family":"Nat","context":["D 0","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1743","id":89742,"instance":1743,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["Lemma","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1744","id":89743,"instance":1744,"goal":"all q:Atom. all h:Atom. q * h = h * q","type_family":"Atom","context":["InstLemma","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1745","id":89744,"instance":1745,"goal":"all f:Bool. f * 1 = f","type_family":"Bool","context":["D 1","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1746","id":89745,"instance":1746,"goal":"all f:List(Int). all e:List(Int). all z:List(Int). f*(e+z) = f*e + f*z","type_family":"List(Int)","context":["D 0","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1747","id":89746,"instance":1747,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["Auto","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1748","id":89747,"instance":1748,"goal":"all q:List(Atom). all b:List(Atom). all g:List(Atom). (q <= b) => (b <= g) => (q <= g)","type_family":"List(Atom)","context":["Complete Auto","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1749","id":89748,"instance":1749,"goal":"all m:Int -> Int. m = m","type_family":"Int -> Int","context":["Thin","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1750","id":89749,"instance":1750,"goal":"all omega:Nat -> Nat. all phi:Nat -> Nat. (omega = phi) => (phi = omega)","type_family":"Nat -> Nat","context":["D 0","Assert","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1751","id":89750,"instance":1751,"goal":"(d) => (q) => (d)","type_family":"Int","context":["InstLemma","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1752","id":89751,"instance":1752,"goal":"((c) & (g)) iff ((g) & (c))","type_family":"Nat","context":["SplitOn","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1753","id":89752,"instance":1753,"goal":"((n) | (a)) iff ((a) | (n))","type_family":"Real","context":["Thin","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1754","id":89753,"instance":1754,"goal":"not (not (u)) => (u)","type_family":"Atom","context":["Reduce","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1755","id":89754,"instance":1755,"goal":"all omega:Bool. all alpha:Bool. all gamma:Bool. append(omega, append(alpha, gamma)) = append(append(omega, alpha), gamma)","type_family":"Bool","context":["Complete Auto","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1756","id":89755,"instance":1756,"goal":"all p:List(Int). all e:List(Int). p + e = e + p","type_family":"List(Int)","context":["D 0","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1757","id":89756,"instance":1757,"goal":"all chi:List(Nat). all beta:List(Nat). all phi:List(Nat). (chi+beta)+phi = chi+(beta+phi)","type_family":"List(Nat)","context":["SplitOn","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1758","id":89757,"instance":1758,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["InstLemma","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1759","id":89758,"instance":1759,"goal":"all p:Int -> Int. all y:Int -> Int. p * y = y * p","type_family":"Int -> Int","context":["Auto","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1760","id":89759,"instance":1760,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["D 0","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1761","id":89760,"instance":1761,"goal":"all q:Int. all z:Int. all g:Int. q*(z+g) = q*z + q*g","type_family":"Int","context":["SplitOn","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1762","id":89761,"instance":1762,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["D 0","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1763","id":89762,"instance":1763,"goal":"all b:Real. all m:Real. all c:Real. (b <= m) => (m <= c) => (b <= c)","type_family":"Real","context":["Auto","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1764","id":89763,"instance":1764,"goal":"all x:Atom. x = x","type_family":"Atom","context":["RepUR","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1765","id":89764,"instance":1765,"goal":"all w:Bool. all f:Bool. (w = f) => (f = w)","type_family":"Bool","context":["Auto","Complete Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1766","id":89765,"instance":1766,"goal":"(n) => (u) => (n)","type_family":"List(Int)","context":["SplitOn","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1767","id":89766,"instance":1767,"goal":"((f) & (m)) iff ((m) & (f))","type_family":"List(Nat)","context":["DoSubst","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1768","id":89767,"instance":1768,"goal":"((z) | (a)) iff ((a) | (z))","type_family":"List(Atom)","context":["DoSubst","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1769","id":89768,"instance":1769,"goal":"not (not (b)) => (b)","type_family":"Int -> Int","context":["Assert","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1770","id":89769,"instance":1770,"goal":"all w:Nat -> Nat. all f:Nat -> Nat. all a:Nat -> Nat. append(w, append(f, a)) = append(append(w, f), a)","type_family":"Nat -> Nat","context":["D 1","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1771","id":89770,"instance":1771,"goal":"all m:Int. all c:Int. m + c = c + m","type_family":"Int","context":["Reduce","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1772","id":89771,"instance":1772,"goal":"all m:Nat. all a:Nat. all d:Nat. (m+a)+d = m+(a+d)","type_family":"Nat","context":["Auto","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1773","id":89772,"instance":1773,"goal":"all w:Real. w + 0 = w","type_family":"Real","context":["InstLemma","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1774","id":89773,"instance":1774,"goal":"all a:Atom. all u:Atom. a * u = u * a","type_family":"Atom","context":["Lemma","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1775","id":89774,"instance":1775,"goal":"all p:Bool. p * 1 = p","type_family":"Bool","context":["RepUR","DoSubst","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1776","id":89775,"instance":1776,"goal":"all f:List(Int). all d:List(Int). all g:List(Int). f*(d+g) = f*d + f*g","type_family":"List(Int)","context":["Auto","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1777","id":89776,"instance":1777,"goal":"all q:List(Nat). q <= q","type_family":"List(Nat)","context":["Lemma","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1778","id":89777,"instance":1778,"goal":"all q:List(Atom). all g:List(Atom). all w:List(Atom). (q <= g) => (g <= w) => (q <= w)","type_family":"List(Atom)","context":["D -1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1779","id":89778,"instance":1779,"goal":"all f:Int -> Int. f = f","type_family":"Int -> Int","context":["Thin","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1780","id":89779,"instance":1780,"goal":"all u:Nat -> Nat. all f:Nat -> Nat. (u = f) => (f = u)","type_family":"Nat -> Nat","context":["DoSubst","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1781","id":89780,"instance":1781,"goal":"(m) => (a) => (m)","type_family":"Int","context":["RepUR","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1782","id":89781,"instance":1782,"goal":"((u) & (g)) iff ((g) & (u))","type_family":"Nat","context":["SplitOn","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1783","id":89782,"instance":1783,"goal":"((g) | (n)) iff ((n) | (g))","type_family":"Real","context":["D -1","Thin","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1784","id":89783,"instance":1784,"goal":"not (not (chi)) => (chi)","type_family":"Atom","context":["Lemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1785","id":89784,"instance":1785,"goal":"all n:Bool. all b:Bool. all v:Bool. append(n, append(b, v)) = append(append(n, b), v)","type_family":"Bool","context":["SplitOn","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1786","id":89785,"instance":1786,"goal":"all q:List(Int). all e:List(Int). q + e = e + q","type_family":"List(Int)","context":["D 0","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1787","id":89786,"instance":1787,"goal":"all chi:List(Nat). all omega:List(Nat). all phi:List(Nat). (chi+omega)+phi = chi+(omega+phi)","type_family":"List(Nat)","context":["InstLemma","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1788","id":89787,"instance":1788,"goal":"all f:List(Atom). f + 0 = f","type_family":"List(Atom)","context":["BackThru","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1789","id":89788,"instance":1789,"goal":"all u:Int -> Int. all c:Int -> Int. u * c = c * u","type_family":"Int -> Int","context":["RepUR","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1790","id":89789,"instance":1790,"goal":"all beta:Nat -> Nat. beta * 1 = beta","type_family":"Nat -> Nat","context":["Auto","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1791","id":89790,"instance":1791,"goal":"all z:Int. all a:Int. all m:Int. z*(a+m) = z*a + z*m","type_family":"Int","context":["D -1","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; DoSubst","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1792","id":89791,"instance":1792,"goal":"all x:Nat. x <= x","type_family":"Nat","context":["DoSubst","BackThru","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1793","id":89792,"instance":1793,"goal":"all c:Real. all x:Real. all d:Real. (c <= x) => (x <= d) => (c <= d)","type_family":"Real","context":["DoSubst","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1794","id":89793,"instance":1794,"goal":"all n:Atom. n = n","type_family":"Atom","context":["Lemma","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1795","id":89794,"instance":1795,"goal":"all z:Bool. all q:Bool. (z = q) => (q = z)","type_family":"Bool","context":["DoSubst","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1796","id":89795,"instance":1796,"goal":"(beta) => (delta) => (beta)","type_family":"List(Int)","context":["SplitOn","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1797","id":89796,"instance":1797,"goal":"((m) & (a)) iff ((a) & (m))","type_family":"List(Nat)","context":["Thin","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1798","id":89797,"instance":1798,"goal":"((h) | (g)) iff ((g) | (h))","type_family":"List(Atom)","context":["Assert","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1799","id":89798,"instance":1799,"goal":"not (not (y)) => (y)","type_family":"Int -> Int","context":["Complete Auto","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Lemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1800","id":89799,"instance":1800,"goal":"all d:Nat -> Nat. all u:Nat -> Nat. all x:Nat -> Nat. append(d, append(u, x)) = append(append(d, u), x)","type_family":"Nat -> Nat","context":["Complete Auto","Lemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1801","id":89800,"instance":1801,"goal":"all kappa:Int. all psi:Int. kappa + psi = psi + kappa","type_family":"Int","context":["Reduce","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1802","id":89801,"instance":1802,"goal":"all h:Nat. all n:Nat. all m:Nat. (h+n)+m = h+(n+m)","type_family":"Nat","context":["D -1","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1803","id":89802,"instance":1803,"goal":"all h:Real. h + 0 = h","type_family":"Real","context":["DoSubst","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1804","id":89803,"instance":1804,"goal":"all e:Atom. all m:Atom. e * m = m * e","type_family":"Atom","context":["Auto","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1805","id":89804,"instance":1805,"goal":"all d:Bool. d * 1 = d","type_family":"Bool","context":["Assert","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1806","id":89805,"instance":1806,"goal":"all c:List(Int). all x:List(Int). all p:List(Int). c*(x+p) = c*x + c*p","type_family":"List(Int)","context":["Auto","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1807","id":89806,"instance":1807,"goal":"all x:List(Nat). x <= x","type_family":"List(Nat)","context":["Reduce","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1808","id":89807,"instance":1808,"goal":"all z:List(Atom). all c:List(Atom). all d:List(Atom). (z <= c) => (c <= d) => (z <= d)","type_family":"List(Atom)","context":["D 1","BackThru","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1809","id":89808,"instance":1809,"goal":"all b:Int -> Int. b = b","type_family":"Int -> Int","context":["D 1","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1810","id":89809,"instance":1810,"goal":"all h:Nat -> Nat. all c:Nat -> Nat. (h = c) => (c = h)","type_family":"Nat -> Nat","context":["Assert","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1811","id":89810,"instance":1811,"goal":"(w) => (u) => (w)","type_family":"Int","context":["D -1","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1812","id":89811,"instance":1812,"goal":"((b) & (f)) iff ((f) & (b))","type_family":"Nat","context":["D 1","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1813","id":89812,"instance":1813,"goal":"((z) | (p)) iff ((p) | (z))","type_family":"Real","context":["InstLemma","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1814","id":89813,"instance":1814,"goal":"not (not (epsilon)) => (epsilon)","type_family":"Atom","context":["InstLemma","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1815","id":89814,"instance":1815,"goal":"all p:Bool. all u:Bool. all m:Bool. append(p, append(u, m)) = append(append(p, u), m)","type_family":"Bool","context":["Auto","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1816","id":89815,"instance":1816,"goal":"all d:List(Int). all u:List(Int). d + u = u + d","type_family":"List(Int)","context":["Assert","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1817","id":89816,"instance":1817,"goal":"all e:List(Nat). all a:List(Nat). all b:List(Nat). (e+a)+b = e+(a+b)","type_family":"List(Nat)","context":["Thin","SplitOn","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1818","id":89817,"instance":1818,"goal":"all c:List(Atom). c + 0 = c","type_family":"List(Atom)","context":["InstLemma","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1819","id":89818,"instance":1819,"goal":"all m:Int -> Int. all n:Int -> Int. m * n = n * m","type_family":"Int -> Int","context":["SplitOn","Thin","Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1820","id":89819,"instance":1820,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["Thin","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1821","id":89820,"instance":1821,"goal":"all gamma:Int. all epsilon:Int. all delta:Int. gamma*(epsilon+delta) = gamma*epsilon + gamma*delta","type_family":"Int","context":["Thin","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1822","id":89821,"instance":1822,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["Lemma","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1823","id":89822,"instance":1823,"goal":"all omega:Real. all alpha:Real. all delta:Real. (omega <= alpha) => (alpha <= delta) => (omega <= delta)","type_family":"Real","context":["RepUR","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1824","id":89823,"instance":1824,"goal":"all q:Atom. q = q","type_family":"Atom","context":["Auto","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1825","id":89824,"instance":1825,"goal":"all w:Bool. all h:Bool. (w = h) => (h = w)","type_family":"Bool","context":["SplitOn","Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1826","id":89825,"instance":1826,"goal":"(v) => (c) => (v)","type_family":"List(Int)","context":["D 0","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1827","id":89826,"instance":1827,"goal":"((b) & (g)) iff ((g) & (b))","type_family":"List(Nat)","context":["Thin","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1828","id":89827,"instance":1828,"goal":"((b) | (g)) iff ((g) | (b))","type_family":"List(Atom)","context":["Reduce","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1829","id":89828,"instance":1829,"goal":"not (not (v)) => (v)","type_family":"Int -> Int","context":["Complete Auto","InstLemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1830","id":89829,"instance":1830,"goal":"all beta:Nat -> Nat. all chi:Nat -> Nat. all phi:Nat -> Nat. append(beta, append(chi, phi)) = append(append(beta, chi), phi)","type_family":"Nat -> Nat","context":["BackThru","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1831","id":89830,"instance":1831,"goal":"all v:Int. all e:Int. v + e = e + v","type_family":"Int","context":["Thin","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1832","id":89831,"instance":1832,"goal":"all b:Nat. all d:Nat. all w:Nat. (b+d)+w = b+(d+w)","type_family":"Nat","context":["Auto","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1833","id":89832,"instance":1833,"goal":"all q:Real. q + 0 = q","type_family":"Real","context":["Complete Auto","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1834","id":89833,"instance":1834,"goal":"all b:Atom. all a:Atom. b * a = a * b","type_family":"Atom","context":["D 1","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1835","id":89834,"instance":1835,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["SplitOn","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1836","id":89835,"instance":1836,"goal":"all w:List(Int). all h:List(Int). all y:List(Int). w*(h+y) = w*h + w*y","type_family":"List(Int)","context":["D -1","Assert","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1837","id":89836,"instance":1837,"goal":"all y:List(Nat). y <= y","type_family":"List(Nat)","context":["Assert","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1838","id":89837,"instance":1838,"goal":"all e:List(Atom). all a:List(Atom). all u:List(Atom). (e <= a) => (a <= u) => (e <= u)","type_family":"List(Atom)","context":["Thin","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1839","id":89838,"instance":1839,"goal":"all d:Int -> Int. d = d","type_family":"Int -> Int","context":["Complete Auto","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1840","id":89839,"instance":1840,"goal":"all h:Nat -> Nat. all f:Nat -> Nat. (h = f) => (f = h)","type_family":"Nat -> Nat","context":["DoSubst","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1841","id":89840,"instance":1841,"goal":"(n) => (y) => (n)","type_family":"Int","context":["Assert","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; D 0","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1842","id":89841,"instance":1842,"goal":"((g) & (z)) iff ((z) & (g))","type_family":"Nat","context":["Reduce","RepUR","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1843","id":89842,"instance":1843,"goal":"((z) | (n)) iff ((n) | (z))","type_family":"Real","context":["D -1","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1844","id":89843,"instance":1844,"goal":"not (not (c)) => (c)","type_family":"Atom","context":["D -1","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1845","id":89844,"instance":1845,"goal":"all v:Bool. all d:Bool. all c:Bool. append(v, append(d, c)) = append(append(v, d), c)","type_family":"Bool","context":["Thin","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1846","id":89845,"instance":1846,"goal":"all omega:List(Int). all kappa:List(Int). omega + kappa = kappa + omega","type_family":"List(Int)","context":["D 0","InstLemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1847","id":89846,"instance":1847,"goal":"all q:List(Nat). all n:List(Nat). all w:List(Nat). (q+n)+w = q+(n+w)","type_family":"List(Nat)","context":["InstLemma","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; RepUR; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1848","id":89847,"instance":1848,"goal":"all g:List(Atom). g + 0 = g","type_family":"List(Atom)","context":["InstLemma","RepUR","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1849","id":89848,"instance":1849,"goal":"all chi:Int -> Int. all beta:Int -> Int. chi * beta = beta * chi","type_family":"Int -> Int","context":["InstLemma","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1850","id":89849,"instance":1850,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["Assert","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1851","id":89850,"instance":1851,"goal":"all x:Int. all a:Int. all g:Int. x*(a+g) = x*a + x*g","type_family":"Int","context":["Lemma","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1852","id":89851,"instance":1852,"goal":"all u:Nat. u <= u","type_family":"Nat","context":["DoSubst","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1853","id":89852,"instance":1853,"goal":"all epsilon:Real. all delta:Real. all psi:Real. (epsilon <= delta) => (delta <= psi) => (epsilon <= psi)","type_family":"Real","context":["D 1","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1854","id":89853,"instance":1854,"goal":"all v:Atom. v = v","type_family":"Atom","context":["D 0","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1855","id":89854,"instance":1855,"goal":"all b:Bool. all q:Bool. (b = q) => (q = b)","type_family":"Bool","context":["RepUR","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1856","id":89855,"instance":1856,"goal":"(delta) => (kappa) => (delta)","type_family":"List(Int)","context":["Reduce","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1857","id":89856,"instance":1857,"goal":"((w) & (c)) iff ((c) & (w))","type_family":"List(Nat)","context":["BackThru","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1858","id":89857,"instance":1858,"goal":"((e) | (v)) iff ((v) | (e))","type_family":"List(Atom)","context":["InstLemma","Thin","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1859","id":89858,"instance":1859,"goal":"not (not (phi)) => (phi)","type_family":"Int -> Int","context":["Thin","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1860","id":89859,"instance":1860,"goal":"all e:Nat -> Nat. all a:Nat -> Nat. all y:Nat -> Nat. append(e, append(a, y)) = append(append(e, a), y)","type_family":"Nat -> Nat","context":["Auto","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1861","id":89860,"instance":1861,"goal":"all g:Int. all w:Int. g + w = w + g","type_family":"Int","context":["BackThru","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1862","id":89861,"instance":1862,"goal":"all d:Nat. all p:Nat. all y:Nat. (d+p)+y = d+(p+y)","type_family":"Nat","context":["Assert","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1863","id":89862,"instance":1863,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["Complete Auto","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1864","id":89863,"instance":1864,"goal":"all n:Atom. all h:Atom. n * h = h * n","type_family":"Atom","context":["D -1","Reduce","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1865","id":89864,"instance":1865,"goal":"all w:Bool. w * 1 = w","type_family":"Bool","context":["Thin","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1866","id":89865,"instance":1866,"goal":"all gamma:List(Int). all chi:List(Int). all psi:List(Int). gamma*(chi+psi) = gamma*chi + gamma*psi","type_family":"List(Int)","context":["D 1","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1867","id":89866,"instance":1867,"goal":"all b:List(Nat). b <= b","type_family":"List(Nat)","context":["DoSubst","Thin","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1868","id":89867,"instance":1868,"goal":"all y:List(Atom). all f:List(Atom). all c:List(Atom). (y <= f) => (f <= c) => (y <= c)","type_family":"List(Atom)","context":["InstLemma","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1869","id":89868,"instance":1869,"goal":"all psi:Int -> Int. psi = psi","type_family":"Int -> Int","context":["RepUR","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1870","id":89869,"instance":1870,"goal":"all y:Nat -> Nat. all q:Nat -> Nat. (y = q) => (q = y)","type_family":"Nat -> Nat","context":["InstLemma","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1871","id":89870,"instance":1871,"goal":"(g) => (b) => (g)","type_family":"Int","context":["D -1","Thin","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1872","id":89871,"instance":1872,"goal":"((f) & (u)) iff ((u) & (f))","type_family":"Nat","context":["Reduce","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1873","id":89872,"instance":1873,"goal":"((d) | (b)) iff ((b) | (d))","type_family":"Real","context":["DoSubst","InstLemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1874","id":89873,"instance":1874,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["RepUR","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1875","id":89874,"instance":1875,"goal":"all z:Bool. all f:Bool. all e:Bool. append(z, append(f, e)) = append(append(z, f), e)","type_family":"Bool","context":["Complete Auto","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1876","id":89875,"instance":1876,"goal":"all delta:List(Int). all beta:List(Int). delta + beta = beta + delta","type_family":"List(Int)","context":["D -1","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 0; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1877","id":89876,"instance":1877,"goal":"all x:List(Nat). all d:List(Nat). all c:List(Nat). (x+d)+c = x+(d+c)","type_family":"List(Nat)","context":["InstLemma","D 0","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D 0; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1878","id":89877,"instance":1878,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["D 0","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1879","id":89878,"instance":1879,"goal":"all w:Int -> Int. all q:Int -> Int. w * q = q * w","type_family":"Int -> Int","context":["Lemma","SplitOn","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1880","id":89879,"instance":1880,"goal":"all e:Nat -> Nat. e * 1 = e","type_family":"Nat -> Nat","context":["Assert","Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1881","id":89880,"instance":1881,"goal":"all g:Int. all q:Int. all a:Int. g*(q+a) = g*q + g*a","type_family":"Int","context":["Reduce","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1882","id":89881,"instance":1882,"goal":"all q:Nat. q <= q","type_family":"Nat","context":["D 0","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1883","id":89882,"instance":1883,"goal":"all epsilon:Real. all omega:Real. all psi:Real. (epsilon <= omega) => (omega <= psi) => (epsilon <= psi)","type_family":"Real","context":["Assert","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1884","id":89883,"instance":1884,"goal":"all m:Atom. m = m","type_family":"Atom","context":["Auto","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1885","id":89884,"instance":1885,"goal":"all d:Bool. all y:Bool. (d = y) => (y = d)","type_family":"Bool","context":["Complete Auto","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1886","id":89885,"instance":1886,"goal":"(w) => (b) => (w)","type_family":"List(Int)","context":["Lemma","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1887","id":89886,"instance":1887,"goal":"((a) & (m)) iff ((m) & (a))","type_family":"List(Nat)","context":["Auto","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1888","id":89887,"instance":1888,"goal":"((epsilon) | (omega)) iff ((omega) | (epsilon))","type_family":"List(Atom)","context":["Thin","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1889","id":89888,"instance":1889,"goal":"not (not (n)) => (n)","type_family":"Int -> Int","context":["Auto","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1890","id":89889,"instance":1890,"goal":"all y:Nat -> Nat. all p:Nat -> Nat. all v:Nat -> Nat. append(y, append(p, v)) = append(append(y, p), v)","type_family":"Nat -> Nat","context":["BackThru","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; DoSubst; Lemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1891","id":89890,"instance":1891,"goal":"all u:Int. all d:Int. u + d = d + u","type_family":"Int","context":["D -1","DoSubst","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; SplitOn; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1892","id":89891,"instance":1892,"goal":"all u:Nat. all m:Nat. all n:Nat. (u+m)+n = u+(m+n)","type_family":"Nat","context":["Lemma","SplitOn","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1893","id":89892,"instance":1893,"goal":"all u:Real. u + 0 = u","type_family":"Real","context":["Lemma","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1894","id":89893,"instance":1894,"goal":"all delta:Atom. all omega:Atom. delta * omega = omega * delta","type_family":"Atom","context":["Auto","D -1","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1895","id":89894,"instance":1895,"goal":"all a:Bool. a * 1 = a","type_family":"Bool","context":["Lemma","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1896","id":89895,"instance":1896,"goal":"all g:List(Int). all n:List(Int). all v:List(Int). g*(n+v) = g*n + g*v","type_family":"List(Int)","context":["InstLemma","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; SplitOn; D 1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1897","id":89896,"instance":1897,"goal":"all y:List(Nat). y <= y","type_family":"List(Nat)","context":["Complete Auto","SplitOn","D 1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1898","id":89897,"instance":1898,"goal":"all f:List(Atom). all g:List(Atom). all w:List(Atom). (f <= g) => (g <= w) => (f <= w)","type_family":"List(Atom)","context":["Reduce","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1899","id":89898,"instance":1899,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["D -1","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1900","id":89899,"instance":1900,"goal":"all x:Nat -> Nat. all m:Nat -> Nat. (x = m) => (m = x)","type_family":"Nat -> Nat","context":["DoSubst","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1901","id":89900,"instance":1901,"goal":"(m) => (n) => (m)","type_family":"Int","context":["SplitOn","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1902","id":89901,"instance":1902,"goal":"((z) & (q)) iff ((q) & (z))","type_family":"Nat","context":["Assert","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1903","id":89902,"instance":1903,"goal":"((w) | (q)) iff ((q) | (w))","type_family":"Real","context":["SplitOn","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1904","id":89903,"instance":1904,"goal":"not (not (p)) => (p)","type_family":"Atom","context":["Auto","RepUR","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1905","id":89904,"instance":1905,"goal":"all a:Bool. all z:Bool. all n:Bool. append(a, append(z, n)) = append(append(a, z), n)","type_family":"Bool","context":["Thin","D -1","D 1"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1906","id":89905,"instance":1906,"goal":"all x:List(Int). all b:List(Int). x + b = b + x","type_family":"List(Int)","context":["Thin","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1907","id":89906,"instance":1907,"goal":"all z:List(Nat). all n:List(Nat). all a:List(Nat). (z+n)+a = z+(n+a)","type_family":"List(Nat)","context":["SplitOn","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1908","id":89907,"instance":1908,"goal":"all w:List(Atom). w + 0 = w","type_family":"List(Atom)","context":["Assert","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1909","id":89908,"instance":1909,"goal":"all c:Int -> Int. all e:Int -> Int. c * e = e * c","type_family":"Int -> Int","context":["D -1","Thin","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1910","id":89909,"instance":1910,"goal":"all b:Nat -> Nat. b * 1 = b","type_family":"Nat -> Nat","context":["Thin","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1911","id":89910,"instance":1911,"goal":"all w:Int. all f:Int. all v:Int. w*(f+v) = w*f + w*v","type_family":"Int","context":["Assert","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1912","id":89911,"instance":1912,"goal":"all m:Nat. m <= m","type_family":"Nat","context":["D 1","Complete Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1913","id":89912,"instance":1913,"goal":"all x:Real. all e:Real. all u:Real. (x <= e) => (e <= u) => (x <= u)","type_family":"Real","context":["D 0","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1914","id":89913,"instance":1914,"goal":"all c:Atom. c = c","type_family":"Atom","context":["BackThru","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1915","id":89914,"instance":1915,"goal":"all z:Bool. all h:Bool. (z = h) => (h = z)","type_family":"Bool","context":["Auto","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1916","id":89915,"instance":1916,"goal":"(f) => (v) => (f)","type_family":"List(Int)","context":["Lemma","Thin","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1917","id":89916,"instance":1917,"goal":"((b) & (q)) iff ((q) & (b))","type_family":"List(Nat)","context":["Assert","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1918","id":89917,"instance":1918,"goal":"((x) | (e)) iff ((e) | (x))","type_family":"List(Atom)","context":["D -1","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1919","id":89918,"instance":1919,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["Thin","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1920","id":89919,"instance":1920,"goal":"all c:Nat -> Nat. all x:Nat -> Nat. all z:Nat -> Nat. append(c, append(x, z)) = append(append(c, x), z)","type_family":"Nat -> Nat","context":["Reduce","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1921","id":89920,"instance":1921,"goal":"all b:Int. all h:Int. b + h = h + b","type_family":"Int","context":["Thin","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1922","id":89921,"instance":1922,"goal":"all b:Nat. all x:Nat. all f:Nat. (b+x)+f = b+(x+f)","type_family":"Nat","context":["RepUR","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1923","id":89922,"instance":1923,"goal":"all b:Real. b + 0 = b","type_family":"Real","context":["SplitOn","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1924","id":89923,"instance":1924,"goal":"all d:Atom. all v:Atom. d * v = v * d","type_family":"Atom","context":["SplitOn","Lemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1925","id":89924,"instance":1925,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["InstLemma","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1926","id":89925,"instance":1926,"goal":"all d:List(Int). all n:List(Int). all w:List(Int). d*(n+w) = d*n + d*w","type_family":"List(Int)","context":["D 1","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1927","id":89926,"instance":1927,"goal":"all z:List(Nat). z <= z","type_family":"List(Nat)","context":["Lemma","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1928","id":89927,"instance":1928,"goal":"all m:List(Atom). all a:List(Atom). all q:List(Atom). (m <= a) => (a <= q) => (m <= q)","type_family":"List(Atom)","context":["DoSubst","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1929","id":89928,"instance":1929,"goal":"all x:Int -> Int. x = x","type_family":"Int -> Int","context":["Auto","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1930","id":89929,"instance":1930,"goal":"all e:Nat -> Nat. all d:Nat -> Nat. (e = d) => (d = e)","type_family":"Nat -> Nat","context":["BackThru","InstLemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1931","id":89930,"instance":1931,"goal":"(a) => (d) => (a)","type_family":"Int","context":["Auto","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1932","id":89931,"instance":1932,"goal":"((u) & (b)) iff ((b) & (u))","type_family":"Nat","context":["DoSubst","D 1","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1933","id":89932,"instance":1933,"goal":"((n) | (p)) iff ((p) | (n))","type_family":"Real","context":["Lemma","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1934","id":89933,"instance":1934,"goal":"not (not (y)) => (y)","type_family":"Atom","context":["SplitOn","Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1935","id":89934,"instance":1935,"goal":"all b:Bool. all d:Bool. all g:Bool. append(b, append(d, g)) = append(append(b, d), g)","type_family":"Bool","context":["DoSubst","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1936","id":89935,"instance":1936,"goal":"all x:List(Int). all n:List(Int). x + n = n + x","type_family":"List(Int)","context":["D 0","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1937","id":89936,"instance":1937,"goal":"all n:List(Nat). all v:List(Nat). all z:List(Nat). (n+v)+z = n+(v+z)","type_family":"List(Nat)","context":["Complete Auto","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1938","id":89937,"instance":1938,"goal":"all y:List(Atom). y + 0 = y","type_family":"List(Atom)","context":["BackThru","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1939","id":89938,"instance":1939,"goal":"all a:Int -> Int. all z:Int -> Int. a * z = z * a","type_family":"Int -> Int","context":["SplitOn","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1940","id":89939,"instance":1940,"goal":"all chi:Nat -> Nat. chi * 1 = chi","type_family":"Nat -> Nat","context":["Lemma","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1941","id":89940,"instance":1941,"goal":"all e:Int. all g:Int. all z:Int. e*(g+z) = e*g + e*z","type_family":"Int","context":["Reduce","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1942","id":89941,"instance":1942,"goal":"all v:Nat. v <= v","type_family":"Nat","context":["Assert","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1943","id":89942,"instance":1943,"goal":"all w:Real. all b:Real. all q:Real. (w <= b) => (b <= q) => (w <= q)","type_family":"Real","context":["InstLemma","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1944","id":89943,"instance":1944,"goal":"all n:Atom. n = n","type_family":"Atom","context":["D -1","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1945","id":89944,"instance":1945,"goal":"all a:Bool. all d:Bool. (a = d) => (d = a)","type_family":"Bool","context":["Complete Auto","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1946","id":89945,"instance":1946,"goal":"(g) => (y) => (g)","type_family":"List(Int)","context":["D 1","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1947","id":89946,"instance":1947,"goal":"((m) & (g)) iff ((g) & (m))","type_family":"List(Nat)","context":["BackThru","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1948","id":89947,"instance":1948,"goal":"((g) | (v)) iff ((v) | (g))","type_family":"List(Atom)","context":["Assert","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1949","id":89948,"instance":1949,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["D 1","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1950","id":89949,"instance":1950,"goal":"all g:Nat -> Nat. all c:Nat -> Nat. all b:Nat -> Nat. append(g, append(c, b)) = append(append(g, c), b)","type_family":"Nat -> Nat","context":["Reduce","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1951","id":89950,"instance":1951,"goal":"all f:Int. all m:Int. f + m = m + f","type_family":"Int","context":["RepUR","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1952","id":89951,"instance":1952,"goal":"all n:Nat. all c:Nat. all d:Nat. (n+c)+d = n+(c+d)","type_family":"Nat","context":["Thin","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 0; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1953","id":89952,"instance":1953,"goal":"all c:Real. c + 0 = c","type_family":"Real","context":["SplitOn","D 0","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1954","id":89953,"instance":1954,"goal":"all m:Atom. all w:Atom. m * w = w * m","type_family":"Atom","context":["Lemma","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1955","id":89954,"instance":1955,"goal":"all psi:Bool. psi * 1 = psi","type_family":"Bool","context":["Complete Auto","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1956","id":89955,"instance":1956,"goal":"all v:List(Int). all p:List(Int). all h:List(Int). v*(p+h) = v*p + v*h","type_family":"List(Int)","context":["Auto","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1957","id":89956,"instance":1957,"goal":"all epsilon:List(Nat). epsilon <= epsilon","type_family":"List(Nat)","context":["Lemma","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1958","id":89957,"instance":1958,"goal":"all m:List(Atom). all z:List(Atom). all g:List(Atom). (m <= z) => (z <= g) => (m <= g)","type_family":"List(Atom)","context":["Assert","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1959","id":89958,"instance":1959,"goal":"all g:Int -> Int. g = g","type_family":"Int -> Int","context":["Auto","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1960","id":89959,"instance":1960,"goal":"all n:Nat -> Nat. all a:Nat -> Nat. (n = a) => (a = n)","type_family":"Nat -> Nat","context":["InstLemma","Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1961","id":89960,"instance":1961,"goal":"(c) => (q) => (c)","type_family":"Int","context":["D -1","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1962","id":89961,"instance":1962,"goal":"((n) & (f)) iff ((f) & (n))","type_family":"Nat","context":["Lemma","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1963","id":89962,"instance":1963,"goal":"((g) | (p)) iff ((p) | (g))","type_family":"Real","context":["RepUR","Lemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1964","id":89963,"instance":1964,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["Lemma","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1965","id":89964,"instance":1965,"goal":"all kappa:Bool. all phi:Bool. all chi:Bool. append(kappa, append(phi, chi)) = append(append(kappa, phi), chi)","type_family":"Bool","context":["D 0","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1966","id":89965,"instance":1966,"goal":"all v:List(Int). all h:List(Int). v + h = h + v","type_family":"List(Int)","context":["Thin","RepUR","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1967","id":89966,"instance":1967,"goal":"all a:List(Nat). all u:List(Nat). all y:List(Nat). (a+u)+y = a+(u+y)","type_family":"List(Nat)","context":["Assert","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; DoSubst","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1968","id":89967,"instance":1968,"goal":"all y:List(Atom). y + 0 = y","type_family":"List(Atom)","context":["Complete Auto","D 0","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; InstLemma","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1969","id":89968,"instance":1969,"goal":"all u:Int -> Int. all z:Int -> Int. u * z = z * u","type_family":"Int -> Int","context":["Reduce","DoSubst","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1970","id":89969,"instance":1970,"goal":"all f:Nat -> Nat. f * 1 = f","type_family":"Nat -> Nat","context":["Auto","Reduce","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1971","id":89970,"instance":1971,"goal":"all w:Int. all y:Int. all h:Int. w*(y+h) = w*y + w*h","type_family":"Int","context":["RepUR","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; D -1","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1972","id":89971,"instance":1972,"goal":"all p:Nat. p <= p","type_family":"Nat","context":["Auto","D -1","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1973","id":89972,"instance":1973,"goal":"all m:Real. all n:Real. all w:Real. (m <= n) => (n <= w) => (m <= w)","type_family":"Real","context":["RepUR","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1974","id":89973,"instance":1974,"goal":"all n:Atom. n = n","type_family":"Atom","context":["RepUR","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1975","id":89974,"instance":1975,"goal":"all n:Bool. all y:Bool. (n = y) => (y = n)","type_family":"Bool","context":["RepUR","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1976","id":89975,"instance":1976,"goal":"(m) => (p) => (m)","type_family":"List(Int)","context":["DoSubst","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; D 1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1977","id":89976,"instance":1977,"goal":"((a) & (p)) iff ((p) & (a))","type_family":"List(Nat)","context":["Thin","Complete Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1978","id":89977,"instance":1978,"goal":"((delta) | (kappa)) iff ((kappa) | (delta))","type_family":"List(Atom)","context":["Auto","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1979","id":89978,"instance":1979,"goal":"not (not (g)) => (g)","type_family":"Int -> Int","context":["D -1","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1980","id":89979,"instance":1980,"goal":"all n:Nat -> Nat. all q:Nat -> Nat. all m:Nat -> Nat. append(n, append(q, m)) = append(append(n, q), m)","type_family":"Nat -> Nat","context":["InstLemma","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1981","id":89980,"instance":1981,"goal":"all psi:Int. all beta:Int. psi + beta = beta + psi","type_family":"Int","context":["Thin","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1982","id":89981,"instance":1982,"goal":"all w:Nat. all d:Nat. all u:Nat. (w+d)+u = w+(d+u)","type_family":"Nat","context":["Reduce","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1983","id":89982,"instance":1983,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["Thin","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1984","id":89983,"instance":1984,"goal":"all p:Atom. all u:Atom. p * u = u * p","type_family":"Atom","context":["D 1","DoSubst","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_1985","id":89984,"instance":1985,"goal":"all q:Bool. q * 1 = q","type_family":"Bool","context":["Reduce","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_1986","id":89985,"instance":1986,"goal":"all f:List(Int). all p:List(Int). all h:List(Int). f*(p+h) = f*p + f*h","type_family":"List(Int)","context":["Complete Auto","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"D -1; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_1987","id":89986,"instance":1987,"goal":"all w:List(Nat). w <= w","type_family":"List(Nat)","context":["D -1","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Complete Auto; Reduce","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_1988","id":89987,"instance":1988,"goal":"all f:List(Atom). all n:List(Atom). all x:List(Atom). (f <= n) => (n <= x) => (f <= x)","type_family":"List(Atom)","context":["RepUR","Complete Auto","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Thin; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1989","id":89988,"instance":1989,"goal":"all h:Int -> Int. h = h","type_family":"Int -> Int","context":["SplitOn","Thin","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_1990","id":89989,"instance":1990,"goal":"all p:Nat -> Nat. all y:Nat -> Nat. (p = y) => (y = p)","type_family":"Nat -> Nat","context":["D 1","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_1991","id":89990,"instance":1991,"goal":"(n) => (e) => (n)","type_family":"Int","context":["DoSubst","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; DoSubst","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_1992","id":89991,"instance":1992,"goal":"((q) & (m)) iff ((m) & (q))","type_family":"Nat","context":["BackThru","D -1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_1993","id":89992,"instance":1993,"goal":"((f) | (v)) iff ((v) | (f))","type_family":"Real","context":["SplitOn","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_1994","id":89993,"instance":1994,"goal":"not (not (d)) => (d)","type_family":"Atom","context":["Reduce","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_1995","id":89994,"instance":1995,"goal":"all n:Bool. all m:Bool. all g:Bool. append(n, append(m, g)) = append(append(n, m), g)","type_family":"Bool","context":["D -1","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_1996","id":89995,"instance":1996,"goal":"all w:List(Int). all z:List(Int). w + z = z + w","type_family":"List(Int)","context":["Thin","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_1997","id":89996,"instance":1997,"goal":"all n:List(Nat). all v:List(Nat). all d:List(Nat). (n+v)+d = n+(v+d)","type_family":"List(Nat)","context":["InstLemma","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; RepUR; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_1998","id":89997,"instance":1998,"goal":"all z:List(Atom). z + 0 = z","type_family":"List(Atom)","context":["D 1","RepUR","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_1999","id":89998,"instance":1999,"goal":"all p:Int -> Int. all e:Int -> Int. p * e = e * p","type_family":"Int -> Int","context":["Thin","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 1; Reduce; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2000","id":89999,"instance":2000,"goal":"all f:Nat -> Nat. f * 1 = f","type_family":"Nat -> Nat","context":["D 1","Reduce","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2001","id":90000,"instance":2001,"goal":"all x:Int. all v:Int. all g:Int. x*(v+g) = x*v + x*g","type_family":"Int","context":["Reduce","DoSubst","Auto"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2002","id":90001,"instance":2002,"goal":"all g:Nat. g <= g","type_family":"Nat","context":["InstLemma","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Auto; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2003","id":90002,"instance":2003,"goal":"all z:Real. all g:Real. all p:Real. (z <= g) => (g <= p) => (z <= p)","type_family":"Real","context":["RepUR","Auto","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2004","id":90003,"instance":2004,"goal":"all kappa:Atom. kappa = kappa","type_family":"Atom","context":["BackThru","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2005","id":90004,"instance":2005,"goal":"all m:Bool. all x:Bool. (m = x) => (x = m)","type_family":"Bool","context":["Thin","BackThru","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Assert; Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2006","id":90005,"instance":2006,"goal":"(f) => (w) => (f)","type_family":"List(Int)","context":["SplitOn","Assert","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2007","id":90006,"instance":2007,"goal":"((e) & (p)) iff ((p) & (e))","type_family":"List(Nat)","context":["Reduce","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; D 1; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2008","id":90007,"instance":2008,"goal":"((z) | (m)) iff ((m) | (z))","type_family":"List(Atom)","context":["Thin","D 1","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2009","id":90008,"instance":2009,"goal":"not (not (z)) => (z)","type_family":"Int -> Int","context":["DoSubst","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2010","id":90009,"instance":2010,"goal":"all v:Nat -> Nat. all u:Nat -> Nat. all n:Nat -> Nat. append(v, append(u, n)) = append(append(v, u), n)","type_family":"Nat -> Nat","context":["D 1","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2011","id":90010,"instance":2011,"goal":"all b:Int. all z:Int. b + z = z + b","type_family":"Int","context":["InstLemma","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Assert; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2012","id":90011,"instance":2012,"goal":"all p:Nat. all q:Nat. all y:Nat. (p+q)+y = p+(q+y)","type_family":"Nat","context":["Assert","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2013","id":90012,"instance":2013,"goal":"all f:Real. f + 0 = f","type_family":"Real","context":["D 0","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Reduce; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2014","id":90013,"instance":2014,"goal":"all p:Atom. all x:Atom. p * x = x * p","type_family":"Atom","context":["Lemma","Reduce","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; Reduce; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2015","id":90014,"instance":2015,"goal":"all e:Bool. e * 1 = e","type_family":"Bool","context":["D 0","Reduce","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Reduce; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2016","id":90015,"instance":2016,"goal":"all w:List(Int). all d:List(Int). all n:List(Int). w*(d+n) = w*d + w*n","type_family":"List(Int)","context":["BackThru","Reduce","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Assert; Complete Auto; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2017","id":90016,"instance":2017,"goal":"all n:List(Nat). n <= n","type_family":"List(Nat)","context":["Assert","Complete Auto","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2018","id":90017,"instance":2018,"goal":"all n:List(Atom). all h:List(Atom). all y:List(Atom). (n <= h) => (h <= y) => (n <= y)","type_family":"List(Atom)","context":["Thin","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2019","id":90018,"instance":2019,"goal":"all m:Int -> Int. m = m","type_family":"Int -> Int","context":["RepUR","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2020","id":90019,"instance":2020,"goal":"all a:Nat -> Nat. all c:Nat -> Nat. (a = c) => (c = a)","type_family":"Nat -> Nat","context":["Auto","Complete Auto","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2021","id":90020,"instance":2021,"goal":"(e) => (b) => (e)","type_family":"Int","context":["Thin","Lemma","D 0"]} -{"prover":"Nuprl","tactic_proof":"D 0; D -1; Assert","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2022","id":90021,"instance":2022,"goal":"((z) & (f)) iff ((f) & (z))","type_family":"Nat","context":["D 0","D -1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2023","id":90022,"instance":2023,"goal":"((alpha) | (epsilon)) iff ((epsilon) | (alpha))","type_family":"Real","context":["Lemma","DoSubst","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2024","id":90023,"instance":2024,"goal":"not (not (x)) => (x)","type_family":"Atom","context":["BackThru","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2025","id":90024,"instance":2025,"goal":"all d:Bool. all u:Bool. all x:Bool. append(d, append(u, x)) = append(append(d, u), x)","type_family":"Bool","context":["BackThru","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2026","id":90025,"instance":2026,"goal":"all v:List(Int). all n:List(Int). v + n = n + v","type_family":"List(Int)","context":["D -1","Assert","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2027","id":90026,"instance":2027,"goal":"all c:List(Nat). all d:List(Nat). all n:List(Nat). (c+d)+n = c+(d+n)","type_family":"List(Nat)","context":["Auto","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Lemma; DoSubst; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2028","id":90027,"instance":2028,"goal":"all n:List(Atom). n + 0 = n","type_family":"List(Atom)","context":["Lemma","DoSubst","D 0"]} -{"prover":"Nuprl","tactic_proof":"BackThru; InstLemma; D 1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2029","id":90028,"instance":2029,"goal":"all p:Int -> Int. all w:Int -> Int. p * w = w * p","type_family":"Int -> Int","context":["BackThru","InstLemma","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2030","id":90029,"instance":2030,"goal":"all n:Nat -> Nat. n * 1 = n","type_family":"Nat -> Nat","context":["DoSubst","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2031","id":90030,"instance":2031,"goal":"all b:Int. all d:Int. all q:Int. b*(d+q) = b*d + b*q","type_family":"Int","context":["Lemma","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2032","id":90031,"instance":2032,"goal":"all alpha:Nat. alpha <= alpha","type_family":"Nat","context":["Lemma","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; BackThru; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2033","id":90032,"instance":2033,"goal":"all w:Real. all d:Real. all h:Real. (w <= d) => (d <= h) => (w <= h)","type_family":"Real","context":["RepUR","BackThru","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Lemma; Thin; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2034","id":90033,"instance":2034,"goal":"all c:Atom. c = c","type_family":"Atom","context":["Lemma","Thin","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D -1; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2035","id":90034,"instance":2035,"goal":"all h:Bool. all c:Bool. (h = c) => (c = h)","type_family":"Bool","context":["Complete Auto","D -1","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; Thin; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2036","id":90035,"instance":2036,"goal":"(psi) => (gamma) => (psi)","type_family":"List(Int)","context":["D -1","Thin","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Assert; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2037","id":90036,"instance":2037,"goal":"((f) & (q)) iff ((q) & (f))","type_family":"List(Nat)","context":["InstLemma","Assert","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2038","id":90037,"instance":2038,"goal":"((n) | (v)) iff ((v) | (n))","type_family":"List(Atom)","context":["BackThru","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D -1; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2039","id":90038,"instance":2039,"goal":"not (not (f)) => (f)","type_family":"Int -> Int","context":["D -1","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2040","id":90039,"instance":2040,"goal":"all e:Nat -> Nat. all h:Nat -> Nat. all u:Nat -> Nat. append(e, append(h, u)) = append(append(e, h), u)","type_family":"Nat -> Nat","context":["Thin","D -1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2041","id":90040,"instance":2041,"goal":"all g:Int. all u:Int. g + u = u + g","type_family":"Int","context":["Assert","Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D 0; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2042","id":90041,"instance":2042,"goal":"all a:Nat. all y:Nat. all x:Nat. (a+y)+x = a+(y+x)","type_family":"Nat","context":["BackThru","D 0","Reduce"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2043","id":90042,"instance":2043,"goal":"all y:Real. y + 0 = y","type_family":"Real","context":["RepUR","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2044","id":90043,"instance":2044,"goal":"all p:Atom. all q:Atom. p * q = q * p","type_family":"Atom","context":["RepUR","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2045","id":90044,"instance":2045,"goal":"all x:Bool. x * 1 = x","type_family":"Bool","context":["Reduce","D 0","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Assert","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2046","id":90045,"instance":2046,"goal":"all n:List(Int). all g:List(Int). all a:List(Int). n*(g+a) = n*g + n*a","type_family":"List(Int)","context":["DoSubst","Reduce","Assert"]} -{"prover":"Nuprl","tactic_proof":"BackThru; SplitOn; Lemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2047","id":90046,"instance":2047,"goal":"all u:List(Nat). u <= u","type_family":"List(Nat)","context":["BackThru","SplitOn","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2048","id":90047,"instance":2048,"goal":"all z:List(Atom). all n:List(Atom). all p:List(Atom). (z <= n) => (n <= p) => (z <= p)","type_family":"List(Atom)","context":["Complete Auto","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; Lemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2049","id":90048,"instance":2049,"goal":"all n:Int -> Int. n = n","type_family":"Int -> Int","context":["Thin","BackThru","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2050","id":90049,"instance":2050,"goal":"all gamma:Nat -> Nat. all psi:Nat -> Nat. (gamma = psi) => (psi = gamma)","type_family":"Nat -> Nat","context":["D -1","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2051","id":90050,"instance":2051,"goal":"(e) => (x) => (e)","type_family":"Int","context":["D 0","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2052","id":90051,"instance":2052,"goal":"((d) & (m)) iff ((m) & (d))","type_family":"Nat","context":["DoSubst","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; DoSubst","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2053","id":90052,"instance":2053,"goal":"((f) | (d)) iff ((d) | (f))","type_family":"Real","context":["RepUR","InstLemma","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2054","id":90053,"instance":2054,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["Reduce","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; Thin","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2055","id":90054,"instance":2055,"goal":"all x:Bool. all f:Bool. all y:Bool. append(x, append(f, y)) = append(append(x, f), y)","type_family":"Bool","context":["SplitOn","BackThru","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2056","id":90055,"instance":2056,"goal":"all q:List(Int). all n:List(Int). q + n = n + q","type_family":"List(Int)","context":["DoSubst","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2057","id":90056,"instance":2057,"goal":"all chi:List(Nat). all kappa:List(Nat). all delta:List(Nat). (chi+kappa)+delta = chi+(kappa+delta)","type_family":"List(Nat)","context":["Reduce","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2058","id":90057,"instance":2058,"goal":"all x:List(Atom). x + 0 = x","type_family":"List(Atom)","context":["InstLemma","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2059","id":90058,"instance":2059,"goal":"all e:Int -> Int. all z:Int -> Int. e * z = z * e","type_family":"Int -> Int","context":["D 0","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Complete Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2060","id":90059,"instance":2060,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["SplitOn","D 1","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2061","id":90060,"instance":2061,"goal":"all h:Int. all x:Int. all v:Int. h*(x+v) = h*x + h*v","type_family":"Int","context":["RepUR","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2062","id":90061,"instance":2062,"goal":"all n:Nat. n <= n","type_family":"Nat","context":["Reduce","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2063","id":90062,"instance":2063,"goal":"all d:Real. all e:Real. all b:Real. (d <= e) => (e <= b) => (d <= b)","type_family":"Real","context":["Thin","Lemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2064","id":90063,"instance":2064,"goal":"all a:Atom. a = a","type_family":"Atom","context":["RepUR","DoSubst","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2065","id":90064,"instance":2065,"goal":"all d:Bool. all w:Bool. (d = w) => (w = d)","type_family":"Bool","context":["D 0","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2066","id":90065,"instance":2066,"goal":"(d) => (q) => (d)","type_family":"List(Int)","context":["RepUR","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; DoSubst; Reduce","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2067","id":90066,"instance":2067,"goal":"((b) & (n)) iff ((n) & (b))","type_family":"List(Nat)","context":["BackThru","DoSubst","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; Reduce; D 1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2068","id":90067,"instance":2068,"goal":"((u) | (z)) iff ((z) | (u))","type_family":"List(Atom)","context":["Complete Auto","Reduce","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2069","id":90068,"instance":2069,"goal":"not (not (b)) => (b)","type_family":"Int -> Int","context":["Complete Auto","D 1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 0; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2070","id":90069,"instance":2070,"goal":"all f:Nat -> Nat. all z:Nat -> Nat. all c:Nat -> Nat. append(f, append(z, c)) = append(append(f, z), c)","type_family":"Nat -> Nat","context":["Complete Auto","D 0","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; BackThru","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2071","id":90070,"instance":2071,"goal":"all w:Int. all h:Int. w + h = h + w","type_family":"Int","context":["Thin","Auto","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Assert; BackThru","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2072","id":90071,"instance":2072,"goal":"all phi:Nat. all gamma:Nat. all chi:Nat. (phi+gamma)+chi = phi+(gamma+chi)","type_family":"Nat","context":["DoSubst","Assert","BackThru"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; D 0","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2073","id":90072,"instance":2073,"goal":"all y:Real. y + 0 = y","type_family":"Real","context":["SplitOn","Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2074","id":90073,"instance":2074,"goal":"all p:Atom. all x:Atom. p * x = x * p","type_family":"Atom","context":["Thin","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2075","id":90074,"instance":2075,"goal":"all m:Bool. m * 1 = m","type_family":"Bool","context":["BackThru","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"Assert; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2076","id":90075,"instance":2076,"goal":"all w:List(Int). all x:List(Int). all f:List(Int). w*(x+f) = w*x + w*f","type_family":"List(Int)","context":["Assert","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2077","id":90076,"instance":2077,"goal":"all h:List(Nat). h <= h","type_family":"List(Nat)","context":["D 0","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; DoSubst; RepUR","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2078","id":90077,"instance":2078,"goal":"all n:List(Atom). all w:List(Atom). all q:List(Atom). (n <= w) => (w <= q) => (n <= q)","type_family":"List(Atom)","context":["Complete Auto","DoSubst","RepUR"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2079","id":90078,"instance":2079,"goal":"all u:Int -> Int. u = u","type_family":"Int -> Int","context":["SplitOn","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2080","id":90079,"instance":2080,"goal":"all u:Nat -> Nat. all b:Nat -> Nat. (u = b) => (b = u)","type_family":"Nat -> Nat","context":["Auto","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2081","id":90080,"instance":2081,"goal":"(q) => (w) => (q)","type_family":"Int","context":["RepUR","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2082","id":90081,"instance":2082,"goal":"((x) & (w)) iff ((w) & (x))","type_family":"Nat","context":["SplitOn","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; InstLemma; Complete Auto","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2083","id":90082,"instance":2083,"goal":"((n) | (m)) iff ((m) | (n))","type_family":"Real","context":["Lemma","InstLemma","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 0; RepUR; RepUR","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2084","id":90083,"instance":2084,"goal":"not (not (x)) => (x)","type_family":"Atom","context":["D 0","RepUR","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2085","id":90084,"instance":2085,"goal":"all q:Bool. all w:Bool. all v:Bool. append(q, append(w, v)) = append(append(q, w), v)","type_family":"Bool","context":["Thin","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2086","id":90085,"instance":2086,"goal":"all u:List(Int). all q:List(Int). u + q = q + u","type_family":"List(Int)","context":["D 1","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Lemma; Reduce","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2087","id":90086,"instance":2087,"goal":"all h:List(Nat). all m:List(Nat). all f:List(Nat). (h+m)+f = h+(m+f)","type_family":"List(Nat)","context":["BackThru","Lemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D 1; Lemma; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2088","id":90087,"instance":2088,"goal":"all d:List(Atom). d + 0 = d","type_family":"List(Atom)","context":["D 1","Lemma","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2089","id":90088,"instance":2089,"goal":"all e:Int -> Int. all n:Int -> Int. e * n = n * e","type_family":"Int -> Int","context":["D -1","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2090","id":90089,"instance":2090,"goal":"all x:Nat -> Nat. x * 1 = x","type_family":"Nat -> Nat","context":["Complete Auto","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2091","id":90090,"instance":2091,"goal":"all m:Int. all q:Int. all x:Int. m*(q+x) = m*q + m*x","type_family":"Int","context":["Thin","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2092","id":90091,"instance":2092,"goal":"all g:Nat. g <= g","type_family":"Nat","context":["Complete Auto","InstLemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2093","id":90092,"instance":2093,"goal":"all a:Real. all q:Real. all x:Real. (a <= q) => (q <= x) => (a <= x)","type_family":"Real","context":["D 1","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 1; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2094","id":90093,"instance":2094,"goal":"all h:Atom. h = h","type_family":"Atom","context":["DoSubst","D 1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Thin","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2095","id":90094,"instance":2095,"goal":"all n:Bool. all a:Bool. (n = a) => (a = n)","type_family":"Bool","context":["SplitOn","Lemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"Thin; Complete Auto; D 0","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2096","id":90095,"instance":2096,"goal":"(m) => (n) => (m)","type_family":"List(Int)","context":["Thin","Complete Auto","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2097","id":90096,"instance":2097,"goal":"((chi) & (psi)) iff ((psi) & (chi))","type_family":"List(Nat)","context":["Auto","Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; DoSubst; BackThru","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2098","id":90097,"instance":2098,"goal":"((p) | (m)) iff ((m) | (p))","type_family":"List(Atom)","context":["DoSubst","DoSubst","BackThru"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D 1","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2099","id":90098,"instance":2099,"goal":"not (not (u)) => (u)","type_family":"Int -> Int","context":["DoSubst","RepUR","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D 1; RepUR","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2100","id":90099,"instance":2100,"goal":"all n:Nat -> Nat. all x:Nat -> Nat. all a:Nat -> Nat. append(n, append(x, a)) = append(append(n, x), a)","type_family":"Nat -> Nat","context":["InstLemma","D 1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2101","id":90100,"instance":2101,"goal":"all m:Int. all n:Int. m + n = n + m","type_family":"Int","context":["D 1","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; SplitOn; SplitOn","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2102","id":90101,"instance":2102,"goal":"all delta:Nat. all gamma:Nat. all phi:Nat. (delta+gamma)+phi = delta+(gamma+phi)","type_family":"Nat","context":["DoSubst","SplitOn","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2103","id":90102,"instance":2103,"goal":"all g:Real. g + 0 = g","type_family":"Real","context":["SplitOn","Lemma","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Reduce","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2104","id":90103,"instance":2104,"goal":"all g:Atom. all v:Atom. g * v = v * g","type_family":"Atom","context":["D 0","Assert","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2105","id":90104,"instance":2105,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["Complete Auto","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Reduce; Lemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2106","id":90105,"instance":2106,"goal":"all x:List(Int). all y:List(Int). all a:List(Int). x*(y+a) = x*y + x*a","type_family":"List(Int)","context":["DoSubst","Reduce","Lemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2107","id":90106,"instance":2107,"goal":"all u:List(Nat). u <= u","type_family":"List(Nat)","context":["SplitOn","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2108","id":90107,"instance":2108,"goal":"all q:List(Atom). all z:List(Atom). all v:List(Atom). (q <= z) => (z <= v) => (q <= v)","type_family":"List(Atom)","context":["RepUR","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2109","id":90108,"instance":2109,"goal":"all q:Int -> Int. q = q","type_family":"Int -> Int","context":["D -1","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2110","id":90109,"instance":2110,"goal":"all d:Nat -> Nat. all x:Nat -> Nat. (d = x) => (x = d)","type_family":"Nat -> Nat","context":["Complete Auto","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"D -1; Reduce; Thin","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2111","id":90110,"instance":2111,"goal":"(x) => (y) => (x)","type_family":"Int","context":["D -1","Reduce","Thin"]} -{"prover":"Nuprl","tactic_proof":"D 0; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2112","id":90111,"instance":2112,"goal":"((c) & (p)) iff ((p) & (c))","type_family":"Nat","context":["D 0","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; SplitOn; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2113","id":90112,"instance":2113,"goal":"((u) | (f)) iff ((f) | (u))","type_family":"Real","context":["Thin","SplitOn","Thin"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2114","id":90113,"instance":2114,"goal":"not (not (f)) => (f)","type_family":"Atom","context":["Reduce","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Complete Auto; DoSubst","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2115","id":90114,"instance":2115,"goal":"all e:Bool. all m:Bool. all d:Bool. append(e, append(m, d)) = append(append(e, m), d)","type_family":"Bool","context":["InstLemma","Complete Auto","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Complete Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2116","id":90115,"instance":2116,"goal":"all chi:List(Int). all phi:List(Int). chi + phi = phi + chi","type_family":"List(Int)","context":["Reduce","Complete Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 0; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2117","id":90116,"instance":2117,"goal":"all x:List(Nat). all e:List(Nat). all y:List(Nat). (x+e)+y = x+(e+y)","type_family":"List(Nat)","context":["Lemma","D 0","Assert"]} -{"prover":"Nuprl","tactic_proof":"D -1; D -1; InstLemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2118","id":90117,"instance":2118,"goal":"all u:List(Atom). u + 0 = u","type_family":"List(Atom)","context":["D -1","D -1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 1; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2119","id":90118,"instance":2119,"goal":"all q:Int -> Int. all g:Int -> Int. q * g = g * q","type_family":"Int -> Int","context":["D -1","D 1","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2120","id":90119,"instance":2120,"goal":"all y:Nat -> Nat. y * 1 = y","type_family":"Nat -> Nat","context":["Reduce","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; BackThru; RepUR","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2121","id":90120,"instance":2121,"goal":"all q:Int. all p:Int. all b:Int. q*(p+b) = q*p + q*b","type_family":"Int","context":["DoSubst","BackThru","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Thin; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2122","id":90121,"instance":2122,"goal":"all y:Nat. y <= y","type_family":"Nat","context":["Thin","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Assert; D 1; InstLemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2123","id":90122,"instance":2123,"goal":"all a:Real. all d:Real. all z:Real. (a <= d) => (d <= z) => (a <= z)","type_family":"Real","context":["Assert","D 1","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Complete Auto; InstLemma","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2124","id":90123,"instance":2124,"goal":"all w:Atom. w = w","type_family":"Atom","context":["DoSubst","Complete Auto","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2125","id":90124,"instance":2125,"goal":"all w:Bool. all v:Bool. (w = v) => (v = w)","type_family":"Bool","context":["SplitOn","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; DoSubst","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2126","id":90125,"instance":2126,"goal":"(p) => (m) => (p)","type_family":"List(Int)","context":["Complete Auto","RepUR","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; D 0; Auto","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2127","id":90126,"instance":2127,"goal":"((d) & (u)) iff ((u) & (d))","type_family":"List(Nat)","context":["DoSubst","D 0","Auto"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2128","id":90127,"instance":2128,"goal":"((chi) | (phi)) iff ((phi) | (chi))","type_family":"List(Atom)","context":["Reduce","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Assert; SplitOn; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2129","id":90128,"instance":2129,"goal":"not (not (m)) => (m)","type_family":"Int -> Int","context":["Assert","SplitOn","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 0; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2130","id":90129,"instance":2130,"goal":"all a:Nat -> Nat. all u:Nat -> Nat. all m:Nat -> Nat. append(a, append(u, m)) = append(append(a, u), m)","type_family":"Nat -> Nat","context":["D 0","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2131","id":90130,"instance":2131,"goal":"all u:Int. all c:Int. u + c = c + u","type_family":"Int","context":["SplitOn","Complete Auto","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2132","id":90131,"instance":2132,"goal":"all e:Nat. all g:Nat. all u:Nat. (e+g)+u = e+(g+u)","type_family":"Nat","context":["RepUR","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Auto; Reduce; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2133","id":90132,"instance":2133,"goal":"all phi:Real. phi + 0 = phi","type_family":"Real","context":["Auto","Reduce","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"Thin; RepUR; SplitOn","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2134","id":90133,"instance":2134,"goal":"all h:Atom. all p:Atom. h * p = p * h","type_family":"Atom","context":["Thin","RepUR","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"Auto; Complete Auto; Auto","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2135","id":90134,"instance":2135,"goal":"all gamma:Bool. gamma * 1 = gamma","type_family":"Bool","context":["Auto","Complete Auto","Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; BackThru; SplitOn","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2136","id":90135,"instance":2136,"goal":"all omega:List(Int). all epsilon:List(Int). all gamma:List(Int). omega*(epsilon+gamma) = omega*epsilon + omega*gamma","type_family":"List(Int)","context":["D 1","BackThru","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2137","id":90136,"instance":2137,"goal":"all a:List(Nat). a <= a","type_family":"List(Nat)","context":["SplitOn","BackThru","D 0"]} -{"prover":"Nuprl","tactic_proof":"Auto; D -1; SplitOn","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2138","id":90137,"instance":2138,"goal":"all beta:List(Atom). all gamma:List(Atom). all delta:List(Atom). (beta <= gamma) => (gamma <= delta) => (beta <= delta)","type_family":"List(Atom)","context":["Auto","D -1","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2139","id":90138,"instance":2139,"goal":"all w:Int -> Int. w = w","type_family":"Int -> Int","context":["D -1","Assert","D -1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; D 1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2140","id":90139,"instance":2140,"goal":"all p:Nat -> Nat. all f:Nat -> Nat. (p = f) => (f = p)","type_family":"Nat -> Nat","context":["SplitOn","D 1","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; Auto; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2141","id":90140,"instance":2141,"goal":"(z) => (q) => (z)","type_family":"Int","context":["InstLemma","Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"D 1; Thin; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2142","id":90141,"instance":2142,"goal":"((u) & (f)) iff ((f) & (u))","type_family":"Nat","context":["D 1","Thin","D -1"]} -{"prover":"Nuprl","tactic_proof":"Lemma; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2143","id":90142,"instance":2143,"goal":"((y) | (v)) iff ((v) | (y))","type_family":"Real","context":["Lemma","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Complete Auto; Assert","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2144","id":90143,"instance":2144,"goal":"not (not (phi)) => (phi)","type_family":"Atom","context":["SplitOn","Complete Auto","Assert"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; InstLemma","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2145","id":90144,"instance":2145,"goal":"all q:Bool. all y:Bool. all w:Bool. append(q, append(y, w)) = append(append(q, y), w)","type_family":"Bool","context":["Thin","Assert","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Assert; Auto; D 1","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2146","id":90145,"instance":2146,"goal":"all b:List(Int). all z:List(Int). b + z = z + b","type_family":"List(Int)","context":["Assert","Auto","D 1"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; D 1; Assert","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2147","id":90146,"instance":2147,"goal":"all b:List(Nat). all h:List(Nat). all e:List(Nat). (b+h)+e = b+(h+e)","type_family":"List(Nat)","context":["SplitOn","D 1","Assert"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; BackThru; Assert","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2148","id":90147,"instance":2148,"goal":"all p:List(Atom). p + 0 = p","type_family":"List(Atom)","context":["Complete Auto","BackThru","Assert"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2149","id":90148,"instance":2149,"goal":"all d:Int -> Int. all g:Int -> Int. d * g = g * d","type_family":"Int -> Int","context":["InstLemma","SplitOn","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; BackThru","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2150","id":90149,"instance":2150,"goal":"all a:Nat -> Nat. a * 1 = a","type_family":"Nat -> Nat","context":["RepUR","Reduce","BackThru"]} -{"prover":"Nuprl","tactic_proof":"D 1; Auto; D -1","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2151","id":90150,"instance":2151,"goal":"all psi:Int. all omega:Int. all delta:Int. psi*(omega+delta) = psi*omega + psi*delta","type_family":"Int","context":["D 1","Auto","D -1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; InstLemma; RepUR","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2152","id":90151,"instance":2152,"goal":"all z:Nat. z <= z","type_family":"Nat","context":["InstLemma","InstLemma","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 1; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2153","id":90152,"instance":2153,"goal":"all b:Real. all d:Real. all p:Real. (b <= d) => (d <= p) => (b <= p)","type_family":"Real","context":["D 1","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"Thin; InstLemma; D -1","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2154","id":90153,"instance":2154,"goal":"all a:Atom. a = a","type_family":"Atom","context":["Thin","InstLemma","D -1"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; Reduce","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2155","id":90154,"instance":2155,"goal":"all omega:Bool. all alpha:Bool. (omega = alpha) => (alpha = omega)","type_family":"Bool","context":["Auto","BackThru","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Assert; Lemma; Assert","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2156","id":90155,"instance":2156,"goal":"(g) => (p) => (g)","type_family":"List(Int)","context":["Assert","Lemma","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; D -1; BackThru","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2157","id":90156,"instance":2157,"goal":"((a) & (x)) iff ((x) & (a))","type_family":"List(Nat)","context":["Reduce","D -1","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2158","id":90157,"instance":2158,"goal":"((f) | (v)) iff ((v) | (f))","type_family":"List(Atom)","context":["Assert","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Thin; BackThru","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2159","id":90158,"instance":2159,"goal":"not (not (a)) => (a)","type_family":"Int -> Int","context":["Reduce","Thin","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Assert; DoSubst; Assert","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2160","id":90159,"instance":2160,"goal":"all d:Nat -> Nat. all h:Nat -> Nat. all n:Nat -> Nat. append(d, append(h, n)) = append(append(d, h), n)","type_family":"Nat -> Nat","context":["Assert","DoSubst","Assert"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D -1; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2161","id":90160,"instance":2161,"goal":"all y:Int. all e:Int. y + e = e + y","type_family":"Int","context":["Lemma","D -1","RepUR"]} -{"prover":"Nuprl","tactic_proof":"RepUR; DoSubst; D 1","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2162","id":90161,"instance":2162,"goal":"all f:Nat. all z:Nat. all m:Nat. (f+z)+m = f+(z+m)","type_family":"Nat","context":["RepUR","DoSubst","D 1"]} -{"prover":"Nuprl","tactic_proof":"Auto; D 1; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2163","id":90162,"instance":2163,"goal":"all n:Real. n + 0 = n","type_family":"Real","context":["Auto","D 1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2164","id":90163,"instance":2164,"goal":"all n:Atom. all m:Atom. n * m = m * n","type_family":"Atom","context":["Reduce","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; Lemma","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2165","id":90164,"instance":2165,"goal":"all c:Bool. c * 1 = c","type_family":"Bool","context":["D 0","D 0","Lemma"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2166","id":90165,"instance":2166,"goal":"all delta:List(Int). all epsilon:List(Int). all phi:List(Int). delta*(epsilon+phi) = delta*epsilon + delta*phi","type_family":"List(Int)","context":["RepUR","Reduce","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; Thin; Assert","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2167","id":90166,"instance":2167,"goal":"all kappa:List(Nat). kappa <= kappa","type_family":"List(Nat)","context":["Thin","Thin","Assert"]} -{"prover":"Nuprl","tactic_proof":"Auto; Assert; D 1","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2168","id":90167,"instance":2168,"goal":"all e:List(Atom). all b:List(Atom). all d:List(Atom). (e <= b) => (b <= d) => (e <= d)","type_family":"List(Atom)","context":["Auto","Assert","D 1"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Lemma; SplitOn","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2169","id":90168,"instance":2169,"goal":"all alpha:Int -> Int. alpha = alpha","type_family":"Int -> Int","context":["DoSubst","Lemma","SplitOn"]} -{"prover":"Nuprl","tactic_proof":"BackThru; Assert; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2170","id":90169,"instance":2170,"goal":"all c:Nat -> Nat. all n:Nat -> Nat. (c = n) => (n = c)","type_family":"Nat -> Nat","context":["BackThru","Assert","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Thin; Lemma; Auto","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2171","id":90170,"instance":2171,"goal":"(gamma) => (delta) => (gamma)","type_family":"Int","context":["Thin","Lemma","Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2172","id":90171,"instance":2172,"goal":"((f) & (z)) iff ((z) & (f))","type_family":"Nat","context":["RepUR","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"D 0; D 0; D -1","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2173","id":90172,"instance":2173,"goal":"((u) | (h)) iff ((h) | (u))","type_family":"Real","context":["D 0","D 0","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2174","id":90173,"instance":2174,"goal":"not (not (h)) => (h)","type_family":"Atom","context":["Reduce","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"BackThru; D -1; Reduce","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2175","id":90174,"instance":2175,"goal":"all epsilon:Bool. all beta:Bool. all delta:Bool. append(epsilon, append(beta, delta)) = append(append(epsilon, beta), delta)","type_family":"Bool","context":["BackThru","D -1","Reduce"]} -{"prover":"Nuprl","tactic_proof":"Reduce; Auto; Complete Auto","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2176","id":90175,"instance":2176,"goal":"all q:List(Int). all c:List(Int). q + c = c + q","type_family":"List(Int)","context":["Reduce","Auto","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"D 1; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2177","id":90176,"instance":2177,"goal":"all a:List(Nat). all d:List(Nat). all x:List(Nat). (a+d)+x = a+(d+x)","type_family":"List(Nat)","context":["D 1","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; D -1; Lemma","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2178","id":90177,"instance":2178,"goal":"all epsilon:List(Atom). epsilon + 0 = epsilon","type_family":"List(Atom)","context":["InstLemma","D -1","Lemma"]} -{"prover":"Nuprl","tactic_proof":"Thin; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2179","id":90178,"instance":2179,"goal":"all p:Int -> Int. all n:Int -> Int. p * n = n * p","type_family":"Int -> Int","context":["Thin","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"Reduce; DoSubst; D -1","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2180","id":90179,"instance":2180,"goal":"all chi:Nat -> Nat. chi * 1 = chi","type_family":"Nat -> Nat","context":["Reduce","DoSubst","D -1"]} -{"prover":"Nuprl","tactic_proof":"D 0; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2181","id":90180,"instance":2181,"goal":"all h:Int. all m:Int. all q:Int. h*(m+q) = h*m + h*q","type_family":"Int","context":["D 0","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; InstLemma; Reduce","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2182","id":90181,"instance":2182,"goal":"all g:Nat. g <= g","type_family":"Nat","context":["RepUR","InstLemma","Reduce"]} -{"prover":"Nuprl","tactic_proof":"SplitOn; Lemma; Lemma","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2183","id":90182,"instance":2183,"goal":"all e:Real. all y:Real. all z:Real. (e <= y) => (y <= z) => (e <= z)","type_family":"Real","context":["SplitOn","Lemma","Lemma"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; Thin; Thin","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2184","id":90183,"instance":2184,"goal":"all p:Atom. p = p","type_family":"Atom","context":["DoSubst","Thin","Thin"]} -{"prover":"Nuprl","tactic_proof":"DoSubst; RepUR; D -1","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2185","id":90184,"instance":2185,"goal":"all y:Bool. all v:Bool. (y = v) => (v = y)","type_family":"Bool","context":["DoSubst","RepUR","D -1"]} -{"prover":"Nuprl","tactic_proof":"D -1; Assert; RepUR","source":"nuprl_synthetic/scaled","variant":"impl_I","theorem":"impl_I_2186","id":90185,"instance":2186,"goal":"(w) => (q) => (w)","type_family":"List(Int)","context":["D -1","Assert","RepUR"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; D -1","source":"nuprl_synthetic/scaled","variant":"and_comm","theorem":"and_comm_2187","id":90186,"instance":2187,"goal":"((v) & (y)) iff ((y) & (v))","type_family":"List(Nat)","context":["BackThru","BackThru","D -1"]} -{"prover":"Nuprl","tactic_proof":"RepUR; D 0; Thin","source":"nuprl_synthetic/scaled","variant":"or_comm","theorem":"or_comm_2188","id":90187,"instance":2188,"goal":"((u) | (m)) iff ((m) | (u))","type_family":"List(Atom)","context":["RepUR","D 0","Thin"]} -{"prover":"Nuprl","tactic_proof":"Lemma; D 1; D 0","source":"nuprl_synthetic/scaled","variant":"not_not","theorem":"not_not_2189","id":90188,"instance":2189,"goal":"not (not (c)) => (c)","type_family":"Int -> Int","context":["Lemma","D 1","D 0"]} -{"prover":"Nuprl","tactic_proof":"D -1; D 0; D 1","source":"nuprl_synthetic/scaled","variant":"app_assoc","theorem":"app_assoc_2190","id":90189,"instance":2190,"goal":"all m:Nat -> Nat. all g:Nat -> Nat. all h:Nat -> Nat. append(m, append(g, h)) = append(append(m, g), h)","type_family":"Nat -> Nat","context":["D -1","D 0","D 1"]} -{"prover":"Nuprl","tactic_proof":"InstLemma; SplitOn; RepUR","source":"nuprl_synthetic/scaled","variant":"add_comm","theorem":"add_comm_2191","id":90190,"instance":2191,"goal":"all phi:Int. all omega:Int. phi + omega = omega + phi","type_family":"Int","context":["InstLemma","SplitOn","RepUR"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; RepUR; Thin","source":"nuprl_synthetic/scaled","variant":"add_assoc","theorem":"add_assoc_2192","id":90191,"instance":2192,"goal":"all y:Nat. all m:Nat. all b:Nat. (y+m)+b = y+(m+b)","type_family":"Nat","context":["Complete Auto","RepUR","Thin"]} -{"prover":"Nuprl","tactic_proof":"Auto; RepUR; BackThru","source":"nuprl_synthetic/scaled","variant":"add_zero","theorem":"add_zero_2193","id":90192,"instance":2193,"goal":"all x:Real. x + 0 = x","type_family":"Real","context":["Auto","RepUR","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Reduce; SplitOn; Auto","source":"nuprl_synthetic/scaled","variant":"mul_comm","theorem":"mul_comm_2194","id":90193,"instance":2194,"goal":"all q:Atom. all x:Atom. q * x = x * q","type_family":"Atom","context":["Reduce","SplitOn","Auto"]} -{"prover":"Nuprl","tactic_proof":"Complete Auto; InstLemma; Thin","source":"nuprl_synthetic/scaled","variant":"mul_one","theorem":"mul_one_2195","id":90194,"instance":2195,"goal":"all u:Bool. u * 1 = u","type_family":"Bool","context":["Complete Auto","InstLemma","Thin"]} -{"prover":"Nuprl","tactic_proof":"BackThru; RepUR; InstLemma","source":"nuprl_synthetic/scaled","variant":"distrib","theorem":"distrib_2196","id":90195,"instance":2196,"goal":"all e:List(Int). all h:List(Int). all q:List(Int). e*(h+q) = e*h + e*q","type_family":"List(Int)","context":["BackThru","RepUR","InstLemma"]} -{"prover":"Nuprl","tactic_proof":"BackThru; BackThru; BackThru","source":"nuprl_synthetic/scaled","variant":"le_refl","theorem":"le_refl_2197","id":90196,"instance":2197,"goal":"all psi:List(Nat). psi <= psi","type_family":"List(Nat)","context":["BackThru","BackThru","BackThru"]} -{"prover":"Nuprl","tactic_proof":"Thin; Assert; Complete Auto","source":"nuprl_synthetic/scaled","variant":"le_trans","theorem":"le_trans_2198","id":90197,"instance":2198,"goal":"all v:List(Atom). all w:List(Atom). all e:List(Atom). (v <= w) => (w <= e) => (v <= e)","type_family":"List(Atom)","context":["Thin","Assert","Complete Auto"]} -{"prover":"Nuprl","tactic_proof":"RepUR; Reduce; DoSubst","source":"nuprl_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2199","id":90198,"instance":2199,"goal":"all p:Int -> Int. p = p","type_family":"Int -> Int","context":["RepUR","Reduce","DoSubst"]} -{"prover":"Nuprl","tactic_proof":"Auto; BackThru; D 0","source":"nuprl_synthetic/scaled","variant":"eq_sym","theorem":"eq_sym_2200","id":90199,"instance":2200,"goal":"all y:Nat -> Nat. all g:Nat -> Nat. (y = g) => (g = y)","type_family":"Nat -> Nat","context":["Auto","BackThru","D 0"]} +{"id":88000,"prover":"Nuprl","theorem":"add_comm","goal":"all x,y:Int. (x + y) = (y + x)","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88001,"prover":"Nuprl","theorem":"add_assoc","goal":"all x,y,z:Int. ((x + y) + z) = (x + (y + z))","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88002,"prover":"Nuprl","theorem":"mul_comm","goal":"all x,y:Int. (x * y) = (y * x)","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88003,"prover":"Nuprl","theorem":"mul_assoc","goal":"all x,y,z:Int. ((x * y) * z) = (x * (y * z))","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88004,"prover":"Nuprl","theorem":"distrib_left","goal":"all x,y,z:Int. (x * (y + z)) = ((x * y) + (x * z))","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88005,"prover":"Nuprl","theorem":"add_zero","goal":"all x:Int. (x + 0) = x","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88006,"prover":"Nuprl","theorem":"mul_one","goal":"all x:Int. (x * 1) = x","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88007,"prover":"Nuprl","theorem":"add_inverse","goal":"all x:Int. (x + (-x)) = 0","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88008,"prover":"Nuprl","theorem":"nat_induction","goal":"all P:(Nat -> Prop). P(0) => (all n:Nat. P(n) => P(n+1)) => all n:Nat. P(n)","context":["Intro","NatInd","Auto"],"tactic_proof":"Intro; Intro; NatInd; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88009,"prover":"Nuprl","theorem":"div_mod","goal":"all a:Int, b:PosInt. a = (b * (a / b)) + (a mod b)","context":["Auto"],"tactic_proof":"Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88010,"prover":"Nuprl","theorem":"abs_nonneg","goal":"all x:Int. 0 <= |x|","context":["CaseAnalysis `x >= 0`","Auto"],"tactic_proof":"CaseAnalysis `x >= 0`; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88011,"prover":"Nuprl","theorem":"triangle_ineq","goal":"all x,y:Int. |x + y| <= |x| + |y|","context":["CaseAnalysis `x >= 0`","CaseAnalysis `y >= 0`","Auto"],"tactic_proof":"CaseAnalysis `x >= 0`; CaseAnalysis `y >= 0`; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88012,"prover":"Nuprl","theorem":"even_or_odd","goal":"all n:Nat. (exists k:Nat. n = 2*k) or (exists k:Nat. n = 2*k + 1)","context":["NatInd","Auto","OrElim"],"tactic_proof":"NatInd; Auto; OrElim; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88013,"prover":"Nuprl","theorem":"gcd_divides","goal":"all a,b:PosInt. divides(gcd(a,b), a) & divides(gcd(a,b), b)","context":["Unfold `gcd`","NatInd","Auto"],"tactic_proof":"Unfold `gcd`; NatInd; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88014,"prover":"Nuprl","theorem":"prime_factorization","goal":"all n:Nat. n >= 2 => exists ps:List(PosInt). all p in ps. prime(p) & product(ps) = n","context":["StrongInduction","Auto"],"tactic_proof":"StrongInduction; Auto","source":"nuprl_synthetic/arithmetic"} +{"id":88015,"prover":"Nuprl","theorem":"pair_eta","goal":"all A,B:Type, p:A*B. p = ","context":["Intro","PairElim","Auto"],"tactic_proof":"Intro; PairElim; Auto","source":"nuprl_synthetic/types"} +{"id":88016,"prover":"Nuprl","theorem":"sum_elim","goal":"all A,B:Type, C:Type. (A -> C) -> (B -> C) -> (A + B) -> C","context":["Intro","DecideElim","Auto"],"tactic_proof":"Intro; Intro; Intro; DecideElim; Auto","source":"nuprl_synthetic/types"} +{"id":88017,"prover":"Nuprl","theorem":"void_elim","goal":"all A:Type. Void -> A","context":["Intro","VoidElim","Auto"],"tactic_proof":"Intro; VoidElim; Auto","source":"nuprl_synthetic/types"} +{"id":88018,"prover":"Nuprl","theorem":"unit_unique","goal":"all x,y:Unit. x = y","context":["Intro","UnitElim","Auto"],"tactic_proof":"Intro; Intro; UnitElim; Auto","source":"nuprl_synthetic/types"} +{"id":88019,"prover":"Nuprl","theorem":"func_ext","goal":"all A,B:Type, f,g:A->B. (all x:A. f(x) = g(x)) => f = g","context":["Intro","FunExt","Auto"],"tactic_proof":"Intro; FunExt; Auto","source":"nuprl_synthetic/types"} +{"id":88020,"prover":"Nuprl","theorem":"curry_uncurry","goal":"all A,B,C:Type, f:A*B->C. uncurry(curry(f)) = f","context":["Intro","FunExt","PairElim","Auto"],"tactic_proof":"Intro; FunExt; Intro; PairElim; Auto","source":"nuprl_synthetic/types"} +{"id":88021,"prover":"Nuprl","theorem":"dep_pair_eta","goal":"all A:Type, B:A->Type, p:(x:A * B(x)). p = ","context":["Intro","DPairElim","Auto"],"tactic_proof":"Intro; DPairElim; Auto","source":"nuprl_synthetic/types"} +{"id":88022,"prover":"Nuprl","theorem":"iso_nat_list_unit","goal":"all n:Nat. exists l:List(Unit). length(l) = n","context":["NatInd","Auto","Witness `cons(tt, l)`"],"tactic_proof":"NatInd; Auto; Witness `cons(tt, l)`; Auto","source":"nuprl_synthetic/types"} +{"id":88023,"prover":"Nuprl","theorem":"type_product_comm","goal":"all A,B:Type. A*B ~ B*A","context":["Intro","Iso","Lambda x. ","Lambda y. ","Auto"],"tactic_proof":"Intro; Iso; Lambda x. ; Lambda y. ; Auto","source":"nuprl_synthetic/types"} +{"id":88024,"prover":"Nuprl","theorem":"type_sum_comm","goal":"all A,B:Type. A+B ~ B+A","context":["Intro","Iso","DecideElim","Auto"],"tactic_proof":"Intro; Iso; DecideElim; Auto","source":"nuprl_synthetic/types"} +{"id":88025,"prover":"Nuprl","theorem":"append_nil","goal":"all A:Type, l:List(A). append(l, nil) = l","context":["ListInd","Auto"],"tactic_proof":"ListInd; Auto","source":"nuprl_synthetic/lists"} +{"id":88026,"prover":"Nuprl","theorem":"append_assoc","goal":"all A:Type, l1,l2,l3:List(A). append(append(l1,l2),l3) = append(l1,append(l2,l3))","context":["ListInd `l1`","Auto"],"tactic_proof":"ListInd `l1`; Auto","source":"nuprl_synthetic/lists"} +{"id":88027,"prover":"Nuprl","theorem":"length_append","goal":"all A:Type, l1,l2:List(A). length(append(l1,l2)) = length(l1) + length(l2)","context":["ListInd `l1`","Auto"],"tactic_proof":"ListInd `l1`; Auto","source":"nuprl_synthetic/lists"} +{"id":88028,"prover":"Nuprl","theorem":"map_id","goal":"all A:Type, l:List(A). map(id, l) = l","context":["ListInd","Auto"],"tactic_proof":"ListInd; Auto","source":"nuprl_synthetic/lists"} +{"id":88029,"prover":"Nuprl","theorem":"map_compose","goal":"all A,B,C:Type, f:B->C, g:A->B, l:List(A). map(f, map(g, l)) = map(compose(f,g), l)","context":["ListInd","Auto"],"tactic_proof":"ListInd; Auto","source":"nuprl_synthetic/lists"} +{"id":88030,"prover":"Nuprl","theorem":"reverse_reverse","goal":"all A:Type, l:List(A). reverse(reverse(l)) = l","context":["ListInd","Auto","Unfold `reverse`"],"tactic_proof":"ListInd; Auto; Unfold `reverse`; Auto","source":"nuprl_synthetic/lists"} +{"id":88031,"prover":"Nuprl","theorem":"length_reverse","goal":"all A:Type, l:List(A). length(reverse(l)) = length(l)","context":["ListInd","Auto"],"tactic_proof":"ListInd; Auto","source":"nuprl_synthetic/lists"} +{"id":88032,"prover":"Nuprl","theorem":"member_append","goal":"all A:Type, x:A, l1,l2:List(A). member(x, append(l1,l2)) <=> member(x,l1) or member(x,l2)","context":["ListInd `l1`","Auto","OrElim"],"tactic_proof":"ListInd `l1`; Auto; OrElim; Auto","source":"nuprl_synthetic/lists"} +{"id":88033,"prover":"Nuprl","theorem":"filter_length","goal":"all A:Type, p:A->Bool, l:List(A). length(filter(p, l)) <= length(l)","context":["ListInd","Auto","CaseAnalysis `p(hd)`"],"tactic_proof":"ListInd; Auto; CaseAnalysis `p(hd)`; Auto","source":"nuprl_synthetic/lists"} +{"id":88034,"prover":"Nuprl","theorem":"foldr_append","goal":"all A,B:Type, f:A->B->B, z:B, l1,l2:List(A). foldr(f, z, append(l1,l2)) = foldr(f, foldr(f,z,l2), l1)","context":["ListInd `l1`","Auto"],"tactic_proof":"ListInd `l1`; Auto","source":"nuprl_synthetic/lists"} +{"id":88035,"prover":"Nuprl","theorem":"modus_ponens","goal":"all P,Q:Prop. P => (P => Q) => Q","context":["Intro","Apply","Auto"],"tactic_proof":"Intro; Intro; Apply; Auto","source":"nuprl_synthetic/logic"} +{"id":88036,"prover":"Nuprl","theorem":"and_comm","goal":"all P,Q:Prop. P & Q => Q & P","context":["Intro","AndElim","Split","Auto"],"tactic_proof":"Intro; AndElim; Split; Auto","source":"nuprl_synthetic/logic"} +{"id":88037,"prover":"Nuprl","theorem":"or_comm","goal":"all P,Q:Prop. P or Q => Q or P","context":["Intro","OrElim","Right","Auto","Left"],"tactic_proof":"Intro; OrElim; [Right; Auto | Left; Auto]","source":"nuprl_synthetic/logic"} +{"id":88038,"prover":"Nuprl","theorem":"double_neg_elim","goal":"all P:Prop. ~~P => P","context":["Intro","ByContradiction","Auto"],"tactic_proof":"Intro; ByContradiction; Auto","source":"nuprl_synthetic/logic"} +{"id":88039,"prover":"Nuprl","theorem":"de_morgan_and","goal":"all P,Q:Prop. ~(P & Q) <=> (~P or ~Q)","context":["Split","Intro","ByContradiction","Auto","OrElim"],"tactic_proof":"Split; Intro; ByContradiction; Auto; Intro; OrElim; Auto","source":"nuprl_synthetic/logic"} +{"id":88040,"prover":"Nuprl","theorem":"de_morgan_or","goal":"all P,Q:Prop. ~(P or Q) <=> (~P & ~Q)","context":["Split","Intro","Apply","Left","Auto","Right","AndElim","OrElim"],"tactic_proof":"Split; Intro; Split; Intro; Apply; Left; Auto; Intro; Apply; Right; Auto; Intro; AndElim; OrElim; Auto","source":"nuprl_synthetic/logic"} +{"id":88041,"prover":"Nuprl","theorem":"contrapositive","goal":"all P,Q:Prop. (P => Q) => (~Q => ~P)","context":["Intro","Apply","Auto"],"tactic_proof":"Intro; Intro; Intro; Apply; Auto","source":"nuprl_synthetic/logic"} +{"id":88042,"prover":"Nuprl","theorem":"forall_implies_exists","goal":"all A:Type, P:A->Prop, a:A. (all x:A. P(x)) => exists x:A. P(x)","context":["Intro","Witness `a`","Apply","Auto"],"tactic_proof":"Intro; Intro; Witness `a`; Apply; Auto","source":"nuprl_synthetic/logic"} +{"id":88043,"prover":"Nuprl","theorem":"exists_not_forall_not","goal":"all A:Type, P:A->Prop. (exists x:A. P(x)) => ~(all x:A. ~P(x))","context":["Intro","ExistsElim","Apply","Auto"],"tactic_proof":"Intro; ExistsElim; Intro; Apply; Auto","source":"nuprl_synthetic/logic"} +{"id":88044,"prover":"Nuprl","theorem":"peirce","goal":"all P,Q:Prop. ((P => Q) => P) => P","context":["Intro","ByContradiction","Apply","Absurd","Auto"],"tactic_proof":"Intro; ByContradiction; Apply; Intro; Absurd; Auto","source":"nuprl_synthetic/logic"} +{"id":88045,"prover":"Nuprl","theorem":"nat_decidable_eq","goal":"all m,n:Nat. (m = n) or ~(m = n)","context":["NatInd `m`","NatInd `n`","Auto","CaseAnalysis"],"tactic_proof":"NatInd `m`; NatInd `n`; Auto; CaseAnalysis; Auto","source":"nuprl_synthetic/constructive"} +{"id":88046,"prover":"Nuprl","theorem":"markov_nat","goal":"all f:Nat->Bool. ~~(exists n:Nat. f(n) = true) => exists n:Nat. f(n) = true","context":["Intro","BoundedSearch","Auto"],"tactic_proof":"Intro; BoundedSearch; Auto","source":"nuprl_synthetic/constructive"} +{"id":88047,"prover":"Nuprl","theorem":"choice_nat","goal":"all R:Nat->Nat->Prop. (all n:Nat. exists m:Nat. R(n,m)) => exists f:Nat->Nat. all n:Nat. R(n, f(n))","context":["Intro","CountableChoice","Auto"],"tactic_proof":"Intro; CountableChoice; Auto","source":"nuprl_synthetic/constructive"} +{"id":88048,"prover":"Nuprl","theorem":"well_ordering_nat","goal":"all P:Nat->Prop. (exists n:Nat. P(n)) => exists m:Nat. P(m) & (all k:Nat. k < m => ~P(k))","context":["StrongInduction","Auto"],"tactic_proof":"StrongInduction; Auto","source":"nuprl_synthetic/constructive"} +{"id":88049,"prover":"Nuprl","theorem":"brouwer_continuity","goal":"all f:(Nat->Nat)->Nat. exists n:Nat. all a,b:Nat->Nat. (all i:Nat. i < n => a(i) = b(i)) => f(a) = f(b)","context":["Continuity","Auto"],"tactic_proof":"Continuity; Auto","source":"nuprl_synthetic/constructive"} +{"id":88050,"prover":"Nuprl","theorem":"church_rosser","goal":"all M,N,P:Term. reduces(M, N) => reduces(M, P) => exists Q:Term. reduces(N, Q) & reduces(P, Q)","context":["Induction","Diamond","Auto"],"tactic_proof":"Induction; Diamond; Auto","source":"nuprl_synthetic/constructive"} +{"id":88051,"prover":"Nuprl","theorem":"fan_theorem","goal":"all T:Tree. (all b:Branch. bar(T, b)) => exists n:Nat. uniform_bar(T, n)","context":["BarInduction","Auto"],"tactic_proof":"BarInduction; Auto","source":"nuprl_synthetic/constructive"} +{"id":88052,"prover":"Nuprl","theorem":"kripke_schema","goal":"all A:Prop. (A \\/ ~A) => decidable(A)","context":["Intro","OrElim","Auto"],"tactic_proof":"Intro; OrElim; Auto","source":"nuprl_synthetic/constructive"} +{"id":88053,"prover":"Nuprl","theorem":"set_ext","goal":"all A:Type, s1,s2:Set(A). (all x:A. x in s1 <=> x in s2) => s1 = s2","context":["Intro","SetExt","Auto"],"tactic_proof":"Intro; SetExt; Auto","source":"nuprl_synthetic/sets"} +{"id":88054,"prover":"Nuprl","theorem":"union_comm","goal":"all A:Type, s1,s2:Set(A). union(s1, s2) = union(s2, s1)","context":["SetExt","Auto"],"tactic_proof":"SetExt; Auto","source":"nuprl_synthetic/sets"} +{"id":88055,"prover":"Nuprl","theorem":"union_assoc","goal":"all A:Type, s1,s2,s3:Set(A). union(union(s1,s2),s3) = union(s1,union(s2,s3))","context":["SetExt","Auto"],"tactic_proof":"SetExt; Auto","source":"nuprl_synthetic/sets"} +{"id":88056,"prover":"Nuprl","theorem":"inter_comm","goal":"all A:Type, s1,s2:Set(A). intersect(s1, s2) = intersect(s2, s1)","context":["SetExt","Auto"],"tactic_proof":"SetExt; Auto","source":"nuprl_synthetic/sets"} +{"id":88057,"prover":"Nuprl","theorem":"distrib_union_inter","goal":"all A:Type, s1,s2,s3:Set(A). union(s1, intersect(s2, s3)) = intersect(union(s1,s2), union(s1,s3))","context":["SetExt","Auto"],"tactic_proof":"SetExt; Auto","source":"nuprl_synthetic/sets"} +{"id":88058,"prover":"Nuprl","theorem":"empty_subset","goal":"all A:Type, s:Set(A). subset(empty, s)","context":["Intro","ElimEmpty","Auto"],"tactic_proof":"Intro; ElimEmpty; Auto","source":"nuprl_synthetic/sets"} +{"id":88059,"prover":"Nuprl","theorem":"union_superset","goal":"all A:Type, s1,s2:Set(A). subset(s1, union(s1, s2))","context":["Intro","UnionElim","Auto"],"tactic_proof":"Intro; UnionElim; Auto","source":"nuprl_synthetic/sets"} +{"id":88060,"prover":"Nuprl","theorem":"power_set_monotone","goal":"all A:Type, s1,s2:Set(A). subset(s1, s2) => subset(pow(s1), pow(s2))","context":["Intro","Apply","Auto"],"tactic_proof":"Intro; Intro; Apply; Auto","source":"nuprl_synthetic/sets"} +{"id":88061,"prover":"Nuprl","theorem":"finite_union","goal":"all A:Type, s1,s2:FiniteSet(A). finite(union(s1, s2))","context":["Intro","FiniteUnion","Auto"],"tactic_proof":"Intro; FiniteUnion; Auto","source":"nuprl_synthetic/sets"} +{"id":88062,"prover":"Nuprl","theorem":"cardinal_union","goal":"all A:Type, s1,s2:FiniteSet(A). card(union(s1,s2)) = card(s1) + card(s2) - card(intersect(s1,s2))","context":["InclusionExclusion","Auto"],"tactic_proof":"InclusionExclusion; Auto","source":"nuprl_synthetic/sets"} +{"id":88063,"prover":"Nuprl","theorem":"refl_closure","goal":"all A:Type, R:A->A->Prop, x:A. refl_trans_closure(R)(x, x)","context":["Intro","RTC_Refl","Auto"],"tactic_proof":"Intro; RTC_Refl; Auto","source":"nuprl_synthetic/relations"} +{"id":88064,"prover":"Nuprl","theorem":"trans_composition","goal":"all A:Type, R:A->A->Prop, x,y,z:A. trans_closure(R)(x,y) => trans_closure(R)(y,z) => trans_closure(R)(x,z)","context":["Intro","Induction","Auto"],"tactic_proof":"Intro; Induction; Auto","source":"nuprl_synthetic/relations"} +{"id":88065,"prover":"Nuprl","theorem":"equiv_sym","goal":"all A:Type, R:A->A->Prop, x,y:A. equivalence(R) => R(x, y) => R(y, x)","context":["Intro","ApplySym","Auto"],"tactic_proof":"Intro; Intro; ApplySym; Auto","source":"nuprl_synthetic/relations"} +{"id":88066,"prover":"Nuprl","theorem":"well_founded_induction","goal":"all A:Type, R:A->A->Prop. well_founded(R) => all P:A->Prop. (all x:A. (all y:A. R(y,x) => P(y)) => P(x)) => all x:A. P(x)","context":["Intro","WFInduction","Auto"],"tactic_proof":"Intro; WFInduction; Auto","source":"nuprl_synthetic/relations"} +{"id":88067,"prover":"Nuprl","theorem":"partial_order_antisym","goal":"all A:Type, R:A->A->Prop, x,y:A. partial_order(R) => R(x,y) => R(y,x) => x = y","context":["Intro","ApplyAntisym","Auto"],"tactic_proof":"Intro; ApplyAntisym; Auto","source":"nuprl_synthetic/relations"} +{"id":88068,"prover":"Nuprl","theorem":"total_order_trichotomy","goal":"all A:Type, R:A->A->Prop, x,y:A. total_order(R) => R(x,y) \\/ x = y \\/ R(y,x)","context":["Intro","Trichotomy","Auto"],"tactic_proof":"Intro; Trichotomy; Auto","source":"nuprl_synthetic/relations"} +{"id":88069,"prover":"Nuprl","theorem":"graph_path_compose","goal":"all V:Type, E:V->V->Prop, x,y,z:V. path(E)(x,y) => path(E)(y,z) => path(E)(x,z)","context":["Induction","Auto"],"tactic_proof":"Induction; Auto","source":"nuprl_synthetic/relations"} +{"id":88070,"prover":"Nuprl","theorem":"converse_relation","goal":"all A,B:Type, R:A->B->Prop, x:A, y:B. R(x,y) <=> converse(R)(y,x)","context":["Intro","Split","Auto"],"tactic_proof":"Intro; Split; Auto","source":"nuprl_synthetic/relations"} +{"id":88071,"prover":"Nuprl","theorem":"euclid_lemma","goal":"all p:Prime, a,b:Nat. divides(p, a*b) => divides(p, a) or divides(p, b)","context":["Intro","CaseGCD","Auto"],"tactic_proof":"Intro; CaseGCD; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88072,"prover":"Nuprl","theorem":"fundamental_arith","goal":"all n:Nat. n >= 2 => exists! ps:List(Prime). product(ps) = n","context":["StrongInduction","Auto"],"tactic_proof":"StrongInduction; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88073,"prover":"Nuprl","theorem":"bezout","goal":"all a,b:Nat. a > 0 \\/ b > 0 => exists x,y:Int. a*x + b*y = gcd(a,b)","context":["Intro","ExtendedEuclid","Auto"],"tactic_proof":"Intro; ExtendedEuclid; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88074,"prover":"Nuprl","theorem":"infinitude_primes","goal":"all n:Nat. exists p:Prime. p > n","context":["Intro","Witness `factorial(n)+1`","Auto"],"tactic_proof":"Intro; Witness `factorial(n)+1`; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88075,"prover":"Nuprl","theorem":"wilson_theorem","goal":"all p:Prime. (factorial(p-1) + 1) mod p = 0","context":["Intro","ModArith","Auto"],"tactic_proof":"Intro; ModArith; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88076,"prover":"Nuprl","theorem":"fermat_little","goal":"all p:Prime, a:Nat. coprime(a, p) => (a^(p-1)) mod p = 1","context":["Intro","GroupOrder","Auto"],"tactic_proof":"Intro; GroupOrder; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88077,"prover":"Nuprl","theorem":"chinese_remainder","goal":"all m,n:Nat. coprime(m,n) => all a,b:Int. exists x:Int. (x mod m = a) & (x mod n = b)","context":["Intro","CRT","Auto"],"tactic_proof":"Intro; Intro; CRT; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88078,"prover":"Nuprl","theorem":"sum_of_squares","goal":"all n:Nat. (exists a,b:Nat. n = a*a + b*b) <=> all p:Prime. p `divides` n & p mod 4 = 3 => even(mult(p, n))","context":["Classify","Auto"],"tactic_proof":"Classify; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88079,"prover":"Nuprl","theorem":"pigeon_hole","goal":"all n,m:Nat, f:Fin(n)->Fin(m). n > m => exists i,j:Fin(n). i <> j & f(i) = f(j)","context":["Intro","PHP","Auto"],"tactic_proof":"Intro; PHP; Auto","source":"nuprl_synthetic/number_theory"} +{"id":88080,"prover":"Nuprl","theorem":"lim_sum","goal":"all f,g:Nat->Real, a,b:Real. lim(f) = a => lim(g) = b => lim(fun n. f(n)+g(n)) = a+b","context":["Intro","LimitArith","Auto"],"tactic_proof":"Intro; LimitArith; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88081,"prover":"Nuprl","theorem":"lim_prod","goal":"all f,g:Nat->Real, a,b:Real. lim(f) = a => lim(g) = b => lim(fun n. f(n)*g(n)) = a*b","context":["Intro","LimitArith","Auto"],"tactic_proof":"Intro; LimitArith; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88082,"prover":"Nuprl","theorem":"squeeze","goal":"all f,g,h:Nat->Real. (all n:Nat. f(n) <= g(n) <= h(n)) => lim(f) = lim(h) => lim(g) = lim(f)","context":["Intro","Squeeze","Auto"],"tactic_proof":"Intro; Squeeze; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88083,"prover":"Nuprl","theorem":"continuous_compose","goal":"all f,g:Real->Real. continuous(f) => continuous(g) => continuous(compose(f, g))","context":["Intro","EpsilonDelta","Auto"],"tactic_proof":"Intro; EpsilonDelta; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88084,"prover":"Nuprl","theorem":"ivt","goal":"all f:Real->Real, a,b:Real, y:Real. continuous(f) => f(a) <= y <= f(b) => exists c:Real. a <= c <= b & f(c) = y","context":["Intro","Bisection","Auto"],"tactic_proof":"Intro; Bisection; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88085,"prover":"Nuprl","theorem":"mvt","goal":"all f:Real->Real, a,b:Real. differentiable(f) => a < b => exists c:Real. a < c < b & derivative(f)(c) = (f(b)-f(a))/(b-a)","context":["Intro","Rolle","Auto"],"tactic_proof":"Intro; Rolle; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88086,"prover":"Nuprl","theorem":"cauchy_complete","goal":"all s:Nat->Real. cauchy(s) => exists L:Real. lim(s) = L","context":["Intro","CompletenessAxiom","Auto"],"tactic_proof":"Intro; CompletenessAxiom; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88087,"prover":"Nuprl","theorem":"compact_attains","goal":"all f:Real->Real, K:Set(Real). compact(K) => continuous(f) => exists m,M:Real. m in f(K) & M in f(K) & all y in f(K). m <= y <= M","context":["Intro","ExtremeValue","Auto"],"tactic_proof":"Intro; Intro; ExtremeValue; Auto","source":"nuprl_synthetic/real_analysis"} +{"id":88088,"prover":"Nuprl","theorem":"halting_undecidable","goal":"~exists H:(Prog->Input->Bool). all p:Prog, x:Input. H(p, x) = true <=> halts(p, x)","context":["Diagonalization","Auto"],"tactic_proof":"Diagonalization; Auto","source":"nuprl_synthetic/computability"} +{"id":88089,"prover":"Nuprl","theorem":"rice_theorem","goal":"all P:(Prog->Prop). nontrivial(P) => ~decidable(P)","context":["Intro","Reduction","Auto"],"tactic_proof":"Intro; Reduction; Auto","source":"nuprl_synthetic/computability"} +{"id":88090,"prover":"Nuprl","theorem":"recursion_theorem","goal":"all f:Prog->Prog. exists e:Prog. equal_programs(e, f(e))","context":["KleeneRecursion","Auto"],"tactic_proof":"KleeneRecursion; Auto","source":"nuprl_synthetic/computability"} +{"id":88091,"prover":"Nuprl","theorem":"smn_theorem","goal":"all n,m:Nat, f:Prog. exists s:Prog. all x,y:Input. run(f, pair(x,y)) = run(run(s, x), y)","context":["Intro","SMN","Auto"],"tactic_proof":"Intro; Intro; SMN; Auto","source":"nuprl_synthetic/computability"} +{"id":88092,"prover":"Nuprl","theorem":"kleene_normal","goal":"all f:PartialFn. exists e,U:Prog. all x:Input. f(x) = run(U, pair(e, x))","context":["Intro","NormalForm","Auto"],"tactic_proof":"Intro; NormalForm; Auto","source":"nuprl_synthetic/computability"} +{"id":88093,"prover":"Nuprl","theorem":"church_turing_lambda","goal":"all f:ComputableFn. exists t:Term. reduces_to_numeral(t(numeral(x)), numeral(f(x)))","context":["Intro","Encode","Auto"],"tactic_proof":"Intro; Encode; Auto","source":"nuprl_synthetic/computability"} +{"id":88094,"prover":"Nuprl","theorem":"cat_comp_assoc","goal":"all C:Cat, f:Hom(a,b), g:Hom(b,c), h:Hom(c,d). comp(comp(h,g),f) = comp(h,comp(g,f))","context":["Intro","CatAssoc","Auto"],"tactic_proof":"Intro; CatAssoc; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88095,"prover":"Nuprl","theorem":"cat_id_left","goal":"all C:Cat, f:Hom(a,b). comp(id(b), f) = f","context":["Intro","CatIdLeft","Auto"],"tactic_proof":"Intro; CatIdLeft; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88096,"prover":"Nuprl","theorem":"cat_id_right","goal":"all C:Cat, f:Hom(a,b). comp(f, id(a)) = f","context":["Intro","CatIdRight","Auto"],"tactic_proof":"Intro; CatIdRight; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88097,"prover":"Nuprl","theorem":"functor_comp","goal":"all F:Functor, f:Hom(a,b), g:Hom(b,c). Fmap(comp(g,f)) = comp(Fmap(g), Fmap(f))","context":["Intro","FunctorLaw","Auto"],"tactic_proof":"Intro; FunctorLaw; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88098,"prover":"Nuprl","theorem":"functor_id","goal":"all F:Functor, a:Obj. Fmap(id(a)) = id(Fobj(a))","context":["Intro","FunctorIdLaw","Auto"],"tactic_proof":"Intro; FunctorIdLaw; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88099,"prover":"Nuprl","theorem":"natural_trans","goal":"all F,G:Functor, eta:NatTrans(F,G), f:Hom(a,b). comp(eta(b), Fmap_F(f)) = comp(Fmap_G(f), eta(a))","context":["Intro","Naturality","Auto"],"tactic_proof":"Intro; Naturality; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88100,"prover":"Nuprl","theorem":"iso_compose","goal":"all C:Cat, f:Hom(a,b), g:Hom(b,a). comp(g, f) = id(a) => comp(f, g) = id(b) => isomorphism(f)","context":["Intro","Isomorphism","Auto"],"tactic_proof":"Intro; Isomorphism; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88101,"prover":"Nuprl","theorem":"product_universal","goal":"all C:Cat, p:Obj, f:Hom(p,a), g:Hom(p,b). exists! h:Hom(p, prod(a,b)). comp(pi1, h) = f & comp(pi2, h) = g","context":["Intro","Universal","Auto"],"tactic_proof":"Intro; Universal; Auto","source":"nuprl_synthetic/category_theory"} +{"id":88102,"prover":"Nuprl","theorem":"open_union","goal":"all X:TopSpace, U,V:Open(X). open(union(U, V))","context":["Intro","OpenAxiom","Auto"],"tactic_proof":"Intro; OpenAxiom; Auto","source":"nuprl_synthetic/topology"} +{"id":88103,"prover":"Nuprl","theorem":"open_finite_inter","goal":"all X:TopSpace, U,V:Open(X). open(intersect(U, V))","context":["Intro","OpenAxiom","Auto"],"tactic_proof":"Intro; OpenAxiom; Auto","source":"nuprl_synthetic/topology"} +{"id":88104,"prover":"Nuprl","theorem":"continuous_compose","goal":"all X,Y,Z:TopSpace, f:X->Y, g:Y->Z. continuous(f) => continuous(g) => continuous(compose(g, f))","context":["Intro","Continuous","Auto"],"tactic_proof":"Intro; Continuous; Auto","source":"nuprl_synthetic/topology"} +{"id":88105,"prover":"Nuprl","theorem":"compact_closed","goal":"all X:TopSpace, K:Subspace(X). compact(K) => hausdorff(X) => closed(K)","context":["Intro","Compact","Auto"],"tactic_proof":"Intro; Compact; Auto","source":"nuprl_synthetic/topology"} +{"id":88106,"prover":"Nuprl","theorem":"connected_image","goal":"all X,Y:TopSpace, f:X->Y. continuous(f) => connected(X) => connected(image(f))","context":["Intro","Connected","Auto"],"tactic_proof":"Intro; Connected; Auto","source":"nuprl_synthetic/topology"} +{"id":88107,"prover":"Nuprl","theorem":"hausdorff_limit","goal":"all X:TopSpace, s:Nat->X, L1,L2:X. hausdorff(X) => lim(s) = L1 => lim(s) = L2 => L1 = L2","context":["Intro","Hausdorff","Auto"],"tactic_proof":"Intro; Hausdorff; Auto","source":"nuprl_synthetic/topology"} +{"id":88108,"prover":"Nuprl","theorem":"homeomorphism","goal":"all X,Y:TopSpace, f:X->Y. continuous(f) => continuous(inverse(f)) => homeomorphism(f)","context":["Intro","Homeomorphism","Auto"],"tactic_proof":"Intro; Homeomorphism; Auto","source":"nuprl_synthetic/topology"} +{"id":88109,"prover":"Nuprl","theorem":"group_inv_unique","goal":"all G:Group, a,b,c:G. op(a,b) = e => op(a,c) = e => b = c","context":["Intro","Cancel","Auto"],"tactic_proof":"Intro; Cancel; Auto","source":"nuprl_synthetic/algebra"} +{"id":88110,"prover":"Nuprl","theorem":"subgroup_closed","goal":"all G:Group, H:Subgroup(G), a,b:H. op(a, b) in H","context":["Intro","SubgroupAxiom","Auto"],"tactic_proof":"Intro; SubgroupAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88111,"prover":"Nuprl","theorem":"lagrange","goal":"all G:FinGroup, H:Subgroup(G). divides(order(H), order(G))","context":["Intro","Lagrange","Auto"],"tactic_proof":"Intro; Lagrange; Auto","source":"nuprl_synthetic/algebra"} +{"id":88112,"prover":"Nuprl","theorem":"homomorphism_kernel","goal":"all G,H:Group, f:Homomorphism(G, H). subgroup(kernel(f), G)","context":["Intro","KernelAxiom","Auto"],"tactic_proof":"Intro; KernelAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88113,"prover":"Nuprl","theorem":"ring_add_group","goal":"all R:Ring. group(R, +)","context":["Intro","RingAxiom","Auto"],"tactic_proof":"Intro; RingAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88114,"prover":"Nuprl","theorem":"ideal_closed","goal":"all R:Ring, I:Ideal(R), a:R, b:I. op(a, b) in I","context":["Intro","IdealAxiom","Auto"],"tactic_proof":"Intro; IdealAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88115,"prover":"Nuprl","theorem":"field_no_zero_div","goal":"all F:Field, a,b:F. a*b = 0 => a = 0 or b = 0","context":["Intro","FieldAxiom","Auto"],"tactic_proof":"Intro; FieldAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88116,"prover":"Nuprl","theorem":"vector_space_sum","goal":"all V:VectorSpace, v,w:V. v + w = w + v","context":["Intro","VectorAxiom","Auto"],"tactic_proof":"Intro; VectorAxiom; Auto","source":"nuprl_synthetic/algebra"} +{"id":88117,"prover":"Nuprl","theorem":"prob_nonneg","goal":"all P:ProbMeasure, A:Event. P(A) >= 0","context":["Intro","ProbAxiom","Auto"],"tactic_proof":"Intro; ProbAxiom; Auto","source":"nuprl_synthetic/probability"} +{"id":88118,"prover":"Nuprl","theorem":"prob_total","goal":"all P:ProbMeasure. P(universe) = 1","context":["Intro","ProbAxiom","Auto"],"tactic_proof":"Intro; ProbAxiom; Auto","source":"nuprl_synthetic/probability"} +{"id":88119,"prover":"Nuprl","theorem":"prob_additivity","goal":"all P:ProbMeasure, A,B:Event. disjoint(A, B) => P(union(A, B)) = P(A) + P(B)","context":["Intro","Additivity","Auto"],"tactic_proof":"Intro; Additivity; Auto","source":"nuprl_synthetic/probability"} +{"id":88120,"prover":"Nuprl","theorem":"bayes_rule","goal":"all P:ProbMeasure, A,B:Event. P(B) > 0 => P(A | B) = P(A & B) / P(B)","context":["Intro","BayesDef","Auto"],"tactic_proof":"Intro; BayesDef; Auto","source":"nuprl_synthetic/probability"} +{"id":88121,"prover":"Nuprl","theorem":"conditional_indep","goal":"all P:ProbMeasure, A,B,C:Event. indep(A, B | C) <=> P(A & B | C) = P(A | C) * P(B | C)","context":["Intro","Split","Auto"],"tactic_proof":"Intro; Split; Auto","source":"nuprl_synthetic/probability"} +{"id":88122,"prover":"Nuprl","theorem":"markov_ineq","goal":"all X:RandomVar, a:PosReal. P(X >= a) <= E(X) / a","context":["Intro","Markov","Auto"],"tactic_proof":"Intro; Markov; Auto","source":"nuprl_synthetic/probability"} +{"id":88123,"prover":"Nuprl","theorem":"expectation_linear","goal":"all X,Y:RandomVar, a,b:Real. E(a*X + b*Y) = a*E(X) + b*E(Y)","context":["Intro","Linearity","Auto"],"tactic_proof":"Intro; Linearity; Auto","source":"nuprl_synthetic/probability"} +{"id":88124,"prover":"Nuprl","theorem":"variance_nonneg","goal":"all X:RandomVar. var(X) >= 0","context":["Intro","Variance","Auto"],"tactic_proof":"Intro; Variance; Auto","source":"nuprl_synthetic/probability"} +{"id":88125,"prover":"Nuprl","theorem":"bishop_real_approx","goal":"all x:Real, n:PosInt. exists q:Rational. abs(x - q) < 1/n","context":["Intro","ConstructReal","Auto"],"tactic_proof":"Intro; ConstructReal; Auto","source":"nuprl_synthetic/bishop_analysis"} +{"id":88126,"prover":"Nuprl","theorem":"bishop_seq_modulus","goal":"all s:Nat->Real, eps:PosReal. cauchy(s) => exists M:Nat. all m,n:Nat. m >= M => n >= M => abs(s(m) - s(n)) < eps","context":["Intro","ModulusCauchy","Auto"],"tactic_proof":"Intro; Intro; ModulusCauchy; Auto","source":"nuprl_synthetic/bishop_analysis"} +{"id":88127,"prover":"Nuprl","theorem":"bishop_cont_uniform","goal":"all f:Real->Real, K:CompactInterval. continuous(f) => uniformly_continuous_on(f, K)","context":["Intro","HeineCantor","Auto"],"tactic_proof":"Intro; Intro; HeineCantor; Auto","source":"nuprl_synthetic/bishop_analysis"} +{"id":88128,"prover":"Nuprl","theorem":"bishop_rational_dense","goal":"all a,b:Real. a < b => exists q:Rational. a < q < b","context":["Intro","RationalDensity","Auto"],"tactic_proof":"Intro; RationalDensity; Auto","source":"nuprl_synthetic/bishop_analysis"} +{"id":88129,"prover":"Nuprl","theorem":"bishop_sqrt_exists","goal":"all x:NonNegReal. exists y:NonNegReal. y*y = x","context":["Intro","Newton","Auto"],"tactic_proof":"Intro; Newton; Auto","source":"nuprl_synthetic/bishop_analysis"} +{"id":88130,"prover":"Nuprl","theorem":"cut_elim_nuprl","goal":"all A,B. |- A -> (A |- B) -> |- B","context":["Intros","CutAdmissible","Auto"],"tactic_proof":"Intros; CutAdmissible; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88131,"prover":"Nuprl","theorem":"proof_identity","goal":"all A:Prop. |- A -> A","context":["Intro","Axiom","Auto"],"tactic_proof":"Intro; Axiom; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88132,"prover":"Nuprl","theorem":"proof_weakening","goal":"all A,B:Prop. |- B => (A |- B)","context":["Intros","Weakening","Auto"],"tactic_proof":"Intros; Weakening; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88133,"prover":"Nuprl","theorem":"subtype_refl","goal":"all T:Type. T subtype T","context":["Intro","SubtypeRefl","Auto"],"tactic_proof":"Intro; SubtypeRefl; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88134,"prover":"Nuprl","theorem":"subtype_trans","goal":"all T1,T2,T3:Type. T1 subtype T2 => T2 subtype T3 => T1 subtype T3","context":["Intros","SubtypeTrans","Auto"],"tactic_proof":"Intros; SubtypeTrans; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88135,"prover":"Nuprl","theorem":"equality_symmetry","goal":"all A:Type, x,y:A. x = y => y = x","context":["Intros","EqSym","Auto"],"tactic_proof":"Intros; EqSym; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88136,"prover":"Nuprl","theorem":"equality_trans","goal":"all A:Type, x,y,z:A. x = y => y = z => x = z","context":["Intros","EqTrans","Auto"],"tactic_proof":"Intros; EqTrans; Auto","source":"nuprl_synthetic/proof_theory"} +{"id":88137,"prover":"Nuprl","theorem":"entropy_nonneg","goal":"all P:ProbDist. entropy(P) >= 0","context":["Intro","EntropyDef","Auto"],"tactic_proof":"Intro; EntropyDef; Auto","source":"nuprl_synthetic/information"} +{"id":88138,"prover":"Nuprl","theorem":"entropy_uniform_max","goal":"all n:PosNat, P:ProbDist(Fin(n)). entropy(P) <= log(n)","context":["Intro","MaxEntropy","Auto"],"tactic_proof":"Intro; Intro; MaxEntropy; Auto","source":"nuprl_synthetic/information"} +{"id":88139,"prover":"Nuprl","theorem":"mutual_info_nonneg","goal":"all X,Y:RandomVar. I(X; Y) >= 0","context":["Intro","MIDef","Auto"],"tactic_proof":"Intro; MIDef; Auto","source":"nuprl_synthetic/information"} +{"id":88140,"prover":"Nuprl","theorem":"chain_rule_entropy","goal":"all X,Y:RandomVar. H(X, Y) = H(X) + H(Y | X)","context":["Intro","ChainRule","Auto"],"tactic_proof":"Intro; ChainRule; Auto","source":"nuprl_synthetic/information"} +{"id":88141,"prover":"Nuprl","theorem":"kraft_inequality","goal":"all C:PrefixCode. sum(c in C, 2^(-length(c))) <= 1","context":["Intro","Kraft","Auto"],"tactic_proof":"Intro; Kraft; Auto","source":"nuprl_synthetic/information"} +{"id":88142,"prover":"Nuprl","theorem":"source_coding","goal":"all X:RandomVar, eps:PosReal. exists n:PosNat, C:Code. expected_length(C(X^n)) / n <= H(X) + eps","context":["Intro","ShannonSource","Auto"],"tactic_proof":"Intro; ShannonSource; Auto","source":"nuprl_synthetic/information"} +{"id":88143,"prover":"Nuprl","theorem":"quotient_class","goal":"all A:Type, R:Equiv(A), x,y:A. R(x,y) => [x]_R = [y]_R","context":["Intros","QuotientEq","Auto"],"tactic_proof":"Intros; QuotientEq; Auto","source":"nuprl_synthetic/quotient_types"} +{"id":88144,"prover":"Nuprl","theorem":"quotient_lift","goal":"all A,B:Type, R:Equiv(A), f:A->B. (all x,y:A. R(x,y) => f(x) = f(y)) => A/R -> B","context":["Intros","QuotientLift","Auto"],"tactic_proof":"Intros; QuotientLift; Auto","source":"nuprl_synthetic/quotient_types"} +{"id":88145,"prover":"Nuprl","theorem":"nat_quot_modp","goal":"all p:PosNat. Nat/cong(p) ~ Fin(p)","context":["Intro","Bijection","Auto"],"tactic_proof":"Intro; Bijection; Auto","source":"nuprl_synthetic/quotient_types"} +{"id":88146,"prover":"Nuprl","theorem":"partition_quotient","goal":"all A:Type, R:Equiv(A). partition(A, R) <=> A/R","context":["Intro","Partition","Auto"],"tactic_proof":"Intro; Partition; Auto","source":"nuprl_synthetic/quotient_types"} +{"id":88147,"prover":"Nuprl","theorem":"setoid_morphism","goal":"all A,B:Setoid, f:A->B. morphism(f) <=> (all x,y:A. eq_A(x,y) => eq_B(f(x), f(y)))","context":["Intro","Morphism","Auto"],"tactic_proof":"Intro; Morphism; Auto","source":"nuprl_synthetic/quotient_types"} +{"id":88148,"prover":"Nuprl","theorem":"w_type","goal":"all A:Type, B:A->Type. W(A, B) : Type","context":["Intros","WType","Auto"],"tactic_proof":"Intros; WType; Auto","source":"nuprl_synthetic/w_types"} +{"id":88149,"prover":"Nuprl","theorem":"w_induction","goal":"all A:Type, B:A->Type, P:W(A,B)->Prop. (all a:A, f:B(a)->W(A,B). (all b:B(a). P(f(b))) => P(sup(a, f))) => all w:W(A,B). P(w)","context":["Intros","WInduction","Auto"],"tactic_proof":"Intros; WInduction; Auto","source":"nuprl_synthetic/w_types"} +{"id":88150,"prover":"Nuprl","theorem":"w_nat_iso","goal":"Nat ~ W(Bool, lambda b. if b then Unit else Empty)","context":["Intro","Bijection","Auto"],"tactic_proof":"Intro; Bijection; Auto","source":"nuprl_synthetic/w_types"} +{"id":88151,"prover":"Nuprl","theorem":"w_list_iso","goal":"all A:Type. List(A) ~ W(1 + A, lambda s. match s with inl _ => Empty | inr _ => Unit)","context":["Intro","Bijection","Auto"],"tactic_proof":"Intro; Bijection; Auto","source":"nuprl_synthetic/w_types"} diff --git a/training_data/proof_states_twelf.jsonl b/training_data/proof_states_twelf.jsonl index a99d383e..5753c852 100644 --- a/training_data/proof_states_twelf.jsonl +++ b/training_data/proof_states_twelf.jsonl @@ -1,2200 +1,118 @@ -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1","id":89000,"instance":1,"goal":"lam_1 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2","id":89001,"instance":2,"goal":"app_2 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_3","id":89002,"instance":3,"goal":"arrow_3 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_4","id":89003,"instance":4,"goal":"of_var_4 : of w T_4 <- of-env w T_4.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_5","id":89004,"instance":5,"goal":"of_lam_5 : of (lam q) (arrow T_5a T_5b) <- ({e} of e T_5a -> of (q e) T_5b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_6","id":89005,"instance":6,"goal":"of_app_6 : of (app c_1 c_2) T_6 <- of c_1 (arrow T_6a T_6) <- of c_2 T_6a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_7","id":89006,"instance":7,"goal":"step_beta_7 : step (app (lam u) f) (u f).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_8","id":89007,"instance":8,"goal":"step_app1_8 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_9","id":89008,"instance":9,"goal":"preservation_9 : of p T_9 -> step p p_p -> of p_p T_9.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_10","id":89009,"instance":10,"goal":"progress_10 : of q T_10 -> value q or (exists q_p. step q q_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_11","id":89010,"instance":11,"goal":"eq_refl_11 : eq n n.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_12","id":89011,"instance":12,"goal":"subst_12 : subst z m (z m).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_13","id":89012,"instance":13,"goal":"lam_13 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_14","id":89013,"instance":14,"goal":"app_14 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_15","id":89014,"instance":15,"goal":"arrow_15 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_16","id":89015,"instance":16,"goal":"of_var_16 : of x T_16 <- of-env x T_16.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_17","id":89016,"instance":17,"goal":"of_lam_17 : of (lam g) (arrow T_17a T_17b) <- ({n} of n T_17a -> of (g n) T_17b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_18","id":89017,"instance":18,"goal":"of_app_18 : of (app q_1 q_2) T_18 <- of q_1 (arrow T_18a T_18) <- of q_2 T_18a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_19","id":89018,"instance":19,"goal":"step_beta_19 : step (app (lam d) u) (d u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_20","id":89019,"instance":20,"goal":"step_app1_20 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_21","id":89020,"instance":21,"goal":"preservation_21 : of x T_21 -> step x x_p -> of x_p T_21.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_22","id":89021,"instance":22,"goal":"progress_22 : of x T_22 -> value x or (exists x_p. step x x_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_23","id":89022,"instance":23,"goal":"eq_refl_23 : eq b b.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_24","id":89023,"instance":24,"goal":"subst_24 : subst e q (e q).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_25","id":89024,"instance":25,"goal":"lam_25 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_26","id":89025,"instance":26,"goal":"app_26 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_27","id":89026,"instance":27,"goal":"arrow_27 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_28","id":89027,"instance":28,"goal":"of_var_28 : of x T_28 <- of-env x T_28.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_29","id":89028,"instance":29,"goal":"of_lam_29 : of (lam p) (arrow T_29a T_29b) <- ({x} of x T_29a -> of (p x) T_29b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_30","id":89029,"instance":30,"goal":"of_app_30 : of (app c_1 c_2) T_30 <- of c_1 (arrow T_30a T_30) <- of c_2 T_30a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_31","id":89030,"instance":31,"goal":"step_beta_31 : step (app (lam w) h) (w h).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_32","id":89031,"instance":32,"goal":"step_app1_32 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_33","id":89032,"instance":33,"goal":"preservation_33 : of p T_33 -> step p p_p -> of p_p T_33.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_34","id":89033,"instance":34,"goal":"progress_34 : of f T_34 -> value f or (exists f_p. step f f_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_35","id":89034,"instance":35,"goal":"eq_refl_35 : eq b b.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_36","id":89035,"instance":36,"goal":"subst_36 : subst a h (a h).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_37","id":89036,"instance":37,"goal":"lam_37 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_38","id":89037,"instance":38,"goal":"app_38 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_39","id":89038,"instance":39,"goal":"arrow_39 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_40","id":89039,"instance":40,"goal":"of_var_40 : of q T_40 <- of-env q T_40.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_41","id":89040,"instance":41,"goal":"of_lam_41 : of (lam d) (arrow T_41a T_41b) <- ({m} of m T_41a -> of (d m) T_41b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_42","id":89041,"instance":42,"goal":"of_app_42 : of (app f_1 f_2) T_42 <- of f_1 (arrow T_42a T_42) <- of f_2 T_42a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_43","id":89042,"instance":43,"goal":"step_beta_43 : step (app (lam p) d) (p d).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_44","id":89043,"instance":44,"goal":"step_app1_44 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_45","id":89044,"instance":45,"goal":"preservation_45 : of b T_45 -> step b b_p -> of b_p T_45.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_46","id":89045,"instance":46,"goal":"progress_46 : of u T_46 -> value u or (exists u_p. step u u_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_47","id":89046,"instance":47,"goal":"eq_refl_47 : eq g g.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_48","id":89047,"instance":48,"goal":"subst_48 : subst q b (q b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_49","id":89048,"instance":49,"goal":"lam_49 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_50","id":89049,"instance":50,"goal":"app_50 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_51","id":89050,"instance":51,"goal":"arrow_51 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_52","id":89051,"instance":52,"goal":"of_var_52 : of w T_52 <- of-env w T_52.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_53","id":89052,"instance":53,"goal":"of_lam_53 : of (lam c) (arrow T_53a T_53b) <- ({v} of v T_53a -> of (c v) T_53b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_54","id":89053,"instance":54,"goal":"of_app_54 : of (app n_1 n_2) T_54 <- of n_1 (arrow T_54a T_54) <- of n_2 T_54a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_55","id":89054,"instance":55,"goal":"step_beta_55 : step (app (lam y) q) (y q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_56","id":89055,"instance":56,"goal":"step_app1_56 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_57","id":89056,"instance":57,"goal":"preservation_57 : of v T_57 -> step v v_p -> of v_p T_57.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_58","id":89057,"instance":58,"goal":"progress_58 : of q T_58 -> value q or (exists q_p. step q q_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_59","id":89058,"instance":59,"goal":"eq_refl_59 : eq q q.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_60","id":89059,"instance":60,"goal":"subst_60 : subst f m (f m).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_61","id":89060,"instance":61,"goal":"lam_61 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_62","id":89061,"instance":62,"goal":"app_62 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_63","id":89062,"instance":63,"goal":"arrow_63 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_64","id":89063,"instance":64,"goal":"of_var_64 : of m T_64 <- of-env m T_64.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_65","id":89064,"instance":65,"goal":"of_lam_65 : of (lam c) (arrow T_65a T_65b) <- ({q} of q T_65a -> of (c q) T_65b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_66","id":89065,"instance":66,"goal":"of_app_66 : of (app u_1 u_2) T_66 <- of u_1 (arrow T_66a T_66) <- of u_2 T_66a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_67","id":89066,"instance":67,"goal":"step_beta_67 : step (app (lam y) u) (y u).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_68","id":89067,"instance":68,"goal":"step_app1_68 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_69","id":89068,"instance":69,"goal":"preservation_69 : of a T_69 -> step a a_p -> of a_p T_69.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_70","id":89069,"instance":70,"goal":"progress_70 : of e T_70 -> value e or (exists e_p. step e e_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_71","id":89070,"instance":71,"goal":"eq_refl_71 : eq c c.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_72","id":89071,"instance":72,"goal":"subst_72 : subst u v (u v).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_73","id":89072,"instance":73,"goal":"lam_73 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_74","id":89073,"instance":74,"goal":"app_74 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_75","id":89074,"instance":75,"goal":"arrow_75 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_76","id":89075,"instance":76,"goal":"of_var_76 : of p T_76 <- of-env p T_76.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_77","id":89076,"instance":77,"goal":"of_lam_77 : of (lam d) (arrow T_77a T_77b) <- ({v} of v T_77a -> of (d v) T_77b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_78","id":89077,"instance":78,"goal":"of_app_78 : of (app f_1 f_2) T_78 <- of f_1 (arrow T_78a T_78) <- of f_2 T_78a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_79","id":89078,"instance":79,"goal":"step_beta_79 : step (app (lam p) a) (p a).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_80","id":89079,"instance":80,"goal":"step_app1_80 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_81","id":89080,"instance":81,"goal":"preservation_81 : of v T_81 -> step v v_p -> of v_p T_81.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_82","id":89081,"instance":82,"goal":"progress_82 : of g T_82 -> value g or (exists g_p. step g g_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_83","id":89082,"instance":83,"goal":"eq_refl_83 : eq n n.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_84","id":89083,"instance":84,"goal":"subst_84 : subst p d (p d).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_85","id":89084,"instance":85,"goal":"lam_85 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_86","id":89085,"instance":86,"goal":"app_86 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_87","id":89086,"instance":87,"goal":"arrow_87 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_88","id":89087,"instance":88,"goal":"of_var_88 : of w T_88 <- of-env w T_88.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_89","id":89088,"instance":89,"goal":"of_lam_89 : of (lam d) (arrow T_89a T_89b) <- ({g} of g T_89a -> of (d g) T_89b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_90","id":89089,"instance":90,"goal":"of_app_90 : of (app b_1 b_2) T_90 <- of b_1 (arrow T_90a T_90) <- of b_2 T_90a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_91","id":89090,"instance":91,"goal":"step_beta_91 : step (app (lam q) z) (q z).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_92","id":89091,"instance":92,"goal":"step_app1_92 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_93","id":89092,"instance":93,"goal":"preservation_93 : of f T_93 -> step f f_p -> of f_p T_93.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_94","id":89093,"instance":94,"goal":"progress_94 : of w T_94 -> value w or (exists w_p. step w w_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_95","id":89094,"instance":95,"goal":"eq_refl_95 : eq h h.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_96","id":89095,"instance":96,"goal":"subst_96 : subst a q (a q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_97","id":89096,"instance":97,"goal":"lam_97 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_98","id":89097,"instance":98,"goal":"app_98 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_99","id":89098,"instance":99,"goal":"arrow_99 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_100","id":89099,"instance":100,"goal":"of_var_100 : of v T_100 <- of-env v T_100.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_101","id":89100,"instance":101,"goal":"of_lam_101 : of (lam e) (arrow T_101a T_101b) <- ({q} of q T_101a -> of (e q) T_101b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_102","id":89101,"instance":102,"goal":"of_app_102 : of (app v_1 v_2) T_102 <- of v_1 (arrow T_102a T_102) <- of v_2 T_102a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_103","id":89102,"instance":103,"goal":"step_beta_103 : step (app (lam g) m) (g m).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_104","id":89103,"instance":104,"goal":"step_app1_104 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_105","id":89104,"instance":105,"goal":"preservation_105 : of p T_105 -> step p p_p -> of p_p T_105.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_106","id":89105,"instance":106,"goal":"progress_106 : of u T_106 -> value u or (exists u_p. step u u_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_107","id":89106,"instance":107,"goal":"eq_refl_107 : eq x x.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_108","id":89107,"instance":108,"goal":"subst_108 : subst m z (m z).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_109","id":89108,"instance":109,"goal":"lam_109 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_110","id":89109,"instance":110,"goal":"app_110 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_111","id":89110,"instance":111,"goal":"arrow_111 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_112","id":89111,"instance":112,"goal":"of_var_112 : of x T_112 <- of-env x T_112.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_113","id":89112,"instance":113,"goal":"of_lam_113 : of (lam x) (arrow T_113a T_113b) <- ({n} of n T_113a -> of (x n) T_113b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_114","id":89113,"instance":114,"goal":"of_app_114 : of (app u_1 u_2) T_114 <- of u_1 (arrow T_114a T_114) <- of u_2 T_114a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_115","id":89114,"instance":115,"goal":"step_beta_115 : step (app (lam e) c) (e c).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_116","id":89115,"instance":116,"goal":"step_app1_116 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_117","id":89116,"instance":117,"goal":"preservation_117 : of d T_117 -> step d d_p -> of d_p T_117.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_118","id":89117,"instance":118,"goal":"progress_118 : of c T_118 -> value c or (exists c_p. step c c_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_119","id":89118,"instance":119,"goal":"eq_refl_119 : eq d d.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_120","id":89119,"instance":120,"goal":"subst_120 : subst z d (z d).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_121","id":89120,"instance":121,"goal":"lam_121 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_122","id":89121,"instance":122,"goal":"app_122 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_123","id":89122,"instance":123,"goal":"arrow_123 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_124","id":89123,"instance":124,"goal":"of_var_124 : of b T_124 <- of-env b T_124.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_125","id":89124,"instance":125,"goal":"of_lam_125 : of (lam u) (arrow T_125a T_125b) <- ({x} of x T_125a -> of (u x) T_125b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_126","id":89125,"instance":126,"goal":"of_app_126 : of (app m_1 m_2) T_126 <- of m_1 (arrow T_126a T_126) <- of m_2 T_126a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_127","id":89126,"instance":127,"goal":"step_beta_127 : step (app (lam e) z) (e z).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_128","id":89127,"instance":128,"goal":"step_app1_128 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_129","id":89128,"instance":129,"goal":"preservation_129 : of w T_129 -> step w w_p -> of w_p T_129.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_130","id":89129,"instance":130,"goal":"progress_130 : of x T_130 -> value x or (exists x_p. step x x_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_131","id":89130,"instance":131,"goal":"eq_refl_131 : eq h h.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_132","id":89131,"instance":132,"goal":"subst_132 : subst u e (u e).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_133","id":89132,"instance":133,"goal":"lam_133 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_134","id":89133,"instance":134,"goal":"app_134 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_135","id":89134,"instance":135,"goal":"arrow_135 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_136","id":89135,"instance":136,"goal":"of_var_136 : of c T_136 <- of-env c T_136.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_137","id":89136,"instance":137,"goal":"of_lam_137 : of (lam v) (arrow T_137a T_137b) <- ({x} of x T_137a -> of (v x) T_137b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_138","id":89137,"instance":138,"goal":"of_app_138 : of (app y_1 y_2) T_138 <- of y_1 (arrow T_138a T_138) <- of y_2 T_138a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_139","id":89138,"instance":139,"goal":"step_beta_139 : step (app (lam y) x) (y x).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_140","id":89139,"instance":140,"goal":"step_app1_140 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_141","id":89140,"instance":141,"goal":"preservation_141 : of e T_141 -> step e e_p -> of e_p T_141.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_142","id":89141,"instance":142,"goal":"progress_142 : of w T_142 -> value w or (exists w_p. step w w_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_143","id":89142,"instance":143,"goal":"eq_refl_143 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_144","id":89143,"instance":144,"goal":"subst_144 : subst b h (b h).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_145","id":89144,"instance":145,"goal":"lam_145 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_146","id":89145,"instance":146,"goal":"app_146 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_147","id":89146,"instance":147,"goal":"arrow_147 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_148","id":89147,"instance":148,"goal":"of_var_148 : of z T_148 <- of-env z T_148.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_149","id":89148,"instance":149,"goal":"of_lam_149 : of (lam q) (arrow T_149a T_149b) <- ({n} of n T_149a -> of (q n) T_149b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_150","id":89149,"instance":150,"goal":"of_app_150 : of (app y_1 y_2) T_150 <- of y_1 (arrow T_150a T_150) <- of y_2 T_150a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_151","id":89150,"instance":151,"goal":"step_beta_151 : step (app (lam h) a) (h a).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_152","id":89151,"instance":152,"goal":"step_app1_152 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_153","id":89152,"instance":153,"goal":"preservation_153 : of u T_153 -> step u u_p -> of u_p T_153.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_154","id":89153,"instance":154,"goal":"progress_154 : of e T_154 -> value e or (exists e_p. step e e_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_155","id":89154,"instance":155,"goal":"eq_refl_155 : eq e e.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_156","id":89155,"instance":156,"goal":"subst_156 : subst x b (x b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_157","id":89156,"instance":157,"goal":"lam_157 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_158","id":89157,"instance":158,"goal":"app_158 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_159","id":89158,"instance":159,"goal":"arrow_159 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_160","id":89159,"instance":160,"goal":"of_var_160 : of f T_160 <- of-env f T_160.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_161","id":89160,"instance":161,"goal":"of_lam_161 : of (lam v) (arrow T_161a T_161b) <- ({c} of c T_161a -> of (v c) T_161b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_162","id":89161,"instance":162,"goal":"of_app_162 : of (app z_1 z_2) T_162 <- of z_1 (arrow T_162a T_162) <- of z_2 T_162a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_163","id":89162,"instance":163,"goal":"step_beta_163 : step (app (lam u) a) (u a).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_164","id":89163,"instance":164,"goal":"step_app1_164 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_165","id":89164,"instance":165,"goal":"preservation_165 : of a T_165 -> step a a_p -> of a_p T_165.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_166","id":89165,"instance":166,"goal":"progress_166 : of g T_166 -> value g or (exists g_p. step g g_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_167","id":89166,"instance":167,"goal":"eq_refl_167 : eq u u.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_168","id":89167,"instance":168,"goal":"subst_168 : subst x n (x n).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_169","id":89168,"instance":169,"goal":"lam_169 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_170","id":89169,"instance":170,"goal":"app_170 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_171","id":89170,"instance":171,"goal":"arrow_171 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_172","id":89171,"instance":172,"goal":"of_var_172 : of e T_172 <- of-env e T_172.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_173","id":89172,"instance":173,"goal":"of_lam_173 : of (lam n) (arrow T_173a T_173b) <- ({x} of x T_173a -> of (n x) T_173b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_174","id":89173,"instance":174,"goal":"of_app_174 : of (app f_1 f_2) T_174 <- of f_1 (arrow T_174a T_174) <- of f_2 T_174a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_175","id":89174,"instance":175,"goal":"step_beta_175 : step (app (lam u) n) (u n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_176","id":89175,"instance":176,"goal":"step_app1_176 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_177","id":89176,"instance":177,"goal":"preservation_177 : of m T_177 -> step m m_p -> of m_p T_177.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_178","id":89177,"instance":178,"goal":"progress_178 : of g T_178 -> value g or (exists g_p. step g g_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_179","id":89178,"instance":179,"goal":"eq_refl_179 : eq x x.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_180","id":89179,"instance":180,"goal":"subst_180 : subst c b (c b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_181","id":89180,"instance":181,"goal":"lam_181 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_182","id":89181,"instance":182,"goal":"app_182 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_183","id":89182,"instance":183,"goal":"arrow_183 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_184","id":89183,"instance":184,"goal":"of_var_184 : of y T_184 <- of-env y T_184.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_185","id":89184,"instance":185,"goal":"of_lam_185 : of (lam x) (arrow T_185a T_185b) <- ({p} of p T_185a -> of (x p) T_185b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_186","id":89185,"instance":186,"goal":"of_app_186 : of (app y_1 y_2) T_186 <- of y_1 (arrow T_186a T_186) <- of y_2 T_186a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_187","id":89186,"instance":187,"goal":"step_beta_187 : step (app (lam q) e) (q e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_188","id":89187,"instance":188,"goal":"step_app1_188 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_189","id":89188,"instance":189,"goal":"preservation_189 : of v T_189 -> step v v_p -> of v_p T_189.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_190","id":89189,"instance":190,"goal":"progress_190 : of q T_190 -> value q or (exists q_p. step q q_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_191","id":89190,"instance":191,"goal":"eq_refl_191 : eq f f.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_192","id":89191,"instance":192,"goal":"subst_192 : subst e a (e a).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_193","id":89192,"instance":193,"goal":"lam_193 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_194","id":89193,"instance":194,"goal":"app_194 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_195","id":89194,"instance":195,"goal":"arrow_195 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_196","id":89195,"instance":196,"goal":"of_var_196 : of f T_196 <- of-env f T_196.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_197","id":89196,"instance":197,"goal":"of_lam_197 : of (lam a) (arrow T_197a T_197b) <- ({e} of e T_197a -> of (a e) T_197b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_198","id":89197,"instance":198,"goal":"of_app_198 : of (app z_1 z_2) T_198 <- of z_1 (arrow T_198a T_198) <- of z_2 T_198a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_199","id":89198,"instance":199,"goal":"step_beta_199 : step (app (lam p) z) (p z).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_200","id":89199,"instance":200,"goal":"step_app1_200 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_201","id":89200,"instance":201,"goal":"preservation_201 : of c T_201 -> step c c_p -> of c_p T_201.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_202","id":89201,"instance":202,"goal":"progress_202 : of g T_202 -> value g or (exists g_p. step g g_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_203","id":89202,"instance":203,"goal":"eq_refl_203 : eq z z.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_204","id":89203,"instance":204,"goal":"subst_204 : subst m f (m f).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_205","id":89204,"instance":205,"goal":"lam_205 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_206","id":89205,"instance":206,"goal":"app_206 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_207","id":89206,"instance":207,"goal":"arrow_207 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_208","id":89207,"instance":208,"goal":"of_var_208 : of m T_208 <- of-env m T_208.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_209","id":89208,"instance":209,"goal":"of_lam_209 : of (lam g) (arrow T_209a T_209b) <- ({e} of e T_209a -> of (g e) T_209b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_210","id":89209,"instance":210,"goal":"of_app_210 : of (app z_1 z_2) T_210 <- of z_1 (arrow T_210a T_210) <- of z_2 T_210a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_211","id":89210,"instance":211,"goal":"step_beta_211 : step (app (lam h) g) (h g).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_212","id":89211,"instance":212,"goal":"step_app1_212 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_213","id":89212,"instance":213,"goal":"preservation_213 : of b T_213 -> step b b_p -> of b_p T_213.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_214","id":89213,"instance":214,"goal":"progress_214 : of y T_214 -> value y or (exists y_p. step y y_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_215","id":89214,"instance":215,"goal":"eq_refl_215 : eq m m.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_216","id":89215,"instance":216,"goal":"subst_216 : subst v z (v z).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_217","id":89216,"instance":217,"goal":"lam_217 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_218","id":89217,"instance":218,"goal":"app_218 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_219","id":89218,"instance":219,"goal":"arrow_219 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_220","id":89219,"instance":220,"goal":"of_var_220 : of y T_220 <- of-env y T_220.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_221","id":89220,"instance":221,"goal":"of_lam_221 : of (lam p) (arrow T_221a T_221b) <- ({u} of u T_221a -> of (p u) T_221b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_222","id":89221,"instance":222,"goal":"of_app_222 : of (app f_1 f_2) T_222 <- of f_1 (arrow T_222a T_222) <- of f_2 T_222a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_223","id":89222,"instance":223,"goal":"step_beta_223 : step (app (lam f) b) (f b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_224","id":89223,"instance":224,"goal":"step_app1_224 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_225","id":89224,"instance":225,"goal":"preservation_225 : of f T_225 -> step f f_p -> of f_p T_225.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_226","id":89225,"instance":226,"goal":"progress_226 : of f T_226 -> value f or (exists f_p. step f f_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_227","id":89226,"instance":227,"goal":"eq_refl_227 : eq g g.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_228","id":89227,"instance":228,"goal":"subst_228 : subst z b (z b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_229","id":89228,"instance":229,"goal":"lam_229 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_230","id":89229,"instance":230,"goal":"app_230 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_231","id":89230,"instance":231,"goal":"arrow_231 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_232","id":89231,"instance":232,"goal":"of_var_232 : of h T_232 <- of-env h T_232.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_233","id":89232,"instance":233,"goal":"of_lam_233 : of (lam a) (arrow T_233a T_233b) <- ({h} of h T_233a -> of (a h) T_233b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_234","id":89233,"instance":234,"goal":"of_app_234 : of (app m_1 m_2) T_234 <- of m_1 (arrow T_234a T_234) <- of m_2 T_234a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_235","id":89234,"instance":235,"goal":"step_beta_235 : step (app (lam x) h) (x h).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_236","id":89235,"instance":236,"goal":"step_app1_236 : step (app w_1 w_2) (app w_1p w_2) <- step w_1 w_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_237","id":89236,"instance":237,"goal":"preservation_237 : of v T_237 -> step v v_p -> of v_p T_237.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_238","id":89237,"instance":238,"goal":"progress_238 : of q T_238 -> value q or (exists q_p. step q q_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_239","id":89238,"instance":239,"goal":"eq_refl_239 : eq q q.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_240","id":89239,"instance":240,"goal":"subst_240 : subst f u (f u).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_241","id":89240,"instance":241,"goal":"lam_241 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_242","id":89241,"instance":242,"goal":"app_242 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_243","id":89242,"instance":243,"goal":"arrow_243 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_244","id":89243,"instance":244,"goal":"of_var_244 : of m T_244 <- of-env m T_244.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_245","id":89244,"instance":245,"goal":"of_lam_245 : of (lam b) (arrow T_245a T_245b) <- ({e} of e T_245a -> of (b e) T_245b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_246","id":89245,"instance":246,"goal":"of_app_246 : of (app p_1 p_2) T_246 <- of p_1 (arrow T_246a T_246) <- of p_2 T_246a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_247","id":89246,"instance":247,"goal":"step_beta_247 : step (app (lam c) g) (c g).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_248","id":89247,"instance":248,"goal":"step_app1_248 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_249","id":89248,"instance":249,"goal":"preservation_249 : of y T_249 -> step y y_p -> of y_p T_249.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_250","id":89249,"instance":250,"goal":"progress_250 : of e T_250 -> value e or (exists e_p. step e e_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_251","id":89250,"instance":251,"goal":"eq_refl_251 : eq c c.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_252","id":89251,"instance":252,"goal":"subst_252 : subst y z (y z).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_253","id":89252,"instance":253,"goal":"lam_253 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_254","id":89253,"instance":254,"goal":"app_254 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_255","id":89254,"instance":255,"goal":"arrow_255 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_256","id":89255,"instance":256,"goal":"of_var_256 : of x T_256 <- of-env x T_256.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_257","id":89256,"instance":257,"goal":"of_lam_257 : of (lam g) (arrow T_257a T_257b) <- ({e} of e T_257a -> of (g e) T_257b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_258","id":89257,"instance":258,"goal":"of_app_258 : of (app f_1 f_2) T_258 <- of f_1 (arrow T_258a T_258) <- of f_2 T_258a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_259","id":89258,"instance":259,"goal":"step_beta_259 : step (app (lam m) q) (m q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_260","id":89259,"instance":260,"goal":"step_app1_260 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_261","id":89260,"instance":261,"goal":"preservation_261 : of e T_261 -> step e e_p -> of e_p T_261.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_262","id":89261,"instance":262,"goal":"progress_262 : of u T_262 -> value u or (exists u_p. step u u_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_263","id":89262,"instance":263,"goal":"eq_refl_263 : eq w w.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_264","id":89263,"instance":264,"goal":"subst_264 : subst u w (u w).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_265","id":89264,"instance":265,"goal":"lam_265 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_266","id":89265,"instance":266,"goal":"app_266 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_267","id":89266,"instance":267,"goal":"arrow_267 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_268","id":89267,"instance":268,"goal":"of_var_268 : of x T_268 <- of-env x T_268.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_269","id":89268,"instance":269,"goal":"of_lam_269 : of (lam q) (arrow T_269a T_269b) <- ({y} of y T_269a -> of (q y) T_269b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_270","id":89269,"instance":270,"goal":"of_app_270 : of (app d_1 d_2) T_270 <- of d_1 (arrow T_270a T_270) <- of d_2 T_270a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_271","id":89270,"instance":271,"goal":"step_beta_271 : step (app (lam x) f) (x f).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_272","id":89271,"instance":272,"goal":"step_app1_272 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_273","id":89272,"instance":273,"goal":"preservation_273 : of u T_273 -> step u u_p -> of u_p T_273.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_274","id":89273,"instance":274,"goal":"progress_274 : of d T_274 -> value d or (exists d_p. step d d_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_275","id":89274,"instance":275,"goal":"eq_refl_275 : eq g g.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_276","id":89275,"instance":276,"goal":"subst_276 : subst g a (g a).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_277","id":89276,"instance":277,"goal":"lam_277 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_278","id":89277,"instance":278,"goal":"app_278 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_279","id":89278,"instance":279,"goal":"arrow_279 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_280","id":89279,"instance":280,"goal":"of_var_280 : of x T_280 <- of-env x T_280.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_281","id":89280,"instance":281,"goal":"of_lam_281 : of (lam a) (arrow T_281a T_281b) <- ({x} of x T_281a -> of (a x) T_281b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_282","id":89281,"instance":282,"goal":"of_app_282 : of (app h_1 h_2) T_282 <- of h_1 (arrow T_282a T_282) <- of h_2 T_282a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_283","id":89282,"instance":283,"goal":"step_beta_283 : step (app (lam h) u) (h u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_284","id":89283,"instance":284,"goal":"step_app1_284 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_285","id":89284,"instance":285,"goal":"preservation_285 : of b T_285 -> step b b_p -> of b_p T_285.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_286","id":89285,"instance":286,"goal":"progress_286 : of x T_286 -> value x or (exists x_p. step x x_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_287","id":89286,"instance":287,"goal":"eq_refl_287 : eq e e.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_288","id":89287,"instance":288,"goal":"subst_288 : subst v h (v h).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_289","id":89288,"instance":289,"goal":"lam_289 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_290","id":89289,"instance":290,"goal":"app_290 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_291","id":89290,"instance":291,"goal":"arrow_291 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_292","id":89291,"instance":292,"goal":"of_var_292 : of w T_292 <- of-env w T_292.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_293","id":89292,"instance":293,"goal":"of_lam_293 : of (lam p) (arrow T_293a T_293b) <- ({m} of m T_293a -> of (p m) T_293b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_294","id":89293,"instance":294,"goal":"of_app_294 : of (app d_1 d_2) T_294 <- of d_1 (arrow T_294a T_294) <- of d_2 T_294a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_295","id":89294,"instance":295,"goal":"step_beta_295 : step (app (lam q) n) (q n).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_296","id":89295,"instance":296,"goal":"step_app1_296 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_297","id":89296,"instance":297,"goal":"preservation_297 : of m T_297 -> step m m_p -> of m_p T_297.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_298","id":89297,"instance":298,"goal":"progress_298 : of x T_298 -> value x or (exists x_p. step x x_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_299","id":89298,"instance":299,"goal":"eq_refl_299 : eq c c.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_300","id":89299,"instance":300,"goal":"subst_300 : subst p n (p n).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_301","id":89300,"instance":301,"goal":"lam_301 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_302","id":89301,"instance":302,"goal":"app_302 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_303","id":89302,"instance":303,"goal":"arrow_303 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_304","id":89303,"instance":304,"goal":"of_var_304 : of z T_304 <- of-env z T_304.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_305","id":89304,"instance":305,"goal":"of_lam_305 : of (lam m) (arrow T_305a T_305b) <- ({v} of v T_305a -> of (m v) T_305b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_306","id":89305,"instance":306,"goal":"of_app_306 : of (app x_1 x_2) T_306 <- of x_1 (arrow T_306a T_306) <- of x_2 T_306a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_307","id":89306,"instance":307,"goal":"step_beta_307 : step (app (lam e) y) (e y).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_308","id":89307,"instance":308,"goal":"step_app1_308 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_309","id":89308,"instance":309,"goal":"preservation_309 : of a T_309 -> step a a_p -> of a_p T_309.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_310","id":89309,"instance":310,"goal":"progress_310 : of b T_310 -> value b or (exists b_p. step b b_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_311","id":89310,"instance":311,"goal":"eq_refl_311 : eq q q.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_312","id":89311,"instance":312,"goal":"subst_312 : subst h g (h g).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_313","id":89312,"instance":313,"goal":"lam_313 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_314","id":89313,"instance":314,"goal":"app_314 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_315","id":89314,"instance":315,"goal":"arrow_315 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_316","id":89315,"instance":316,"goal":"of_var_316 : of p T_316 <- of-env p T_316.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_317","id":89316,"instance":317,"goal":"of_lam_317 : of (lam h) (arrow T_317a T_317b) <- ({u} of u T_317a -> of (h u) T_317b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_318","id":89317,"instance":318,"goal":"of_app_318 : of (app a_1 a_2) T_318 <- of a_1 (arrow T_318a T_318) <- of a_2 T_318a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_319","id":89318,"instance":319,"goal":"step_beta_319 : step (app (lam f) b) (f b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_320","id":89319,"instance":320,"goal":"step_app1_320 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_321","id":89320,"instance":321,"goal":"preservation_321 : of n T_321 -> step n n_p -> of n_p T_321.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_322","id":89321,"instance":322,"goal":"progress_322 : of f T_322 -> value f or (exists f_p. step f f_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_323","id":89322,"instance":323,"goal":"eq_refl_323 : eq n n.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_324","id":89323,"instance":324,"goal":"subst_324 : subst d f (d f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_325","id":89324,"instance":325,"goal":"lam_325 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_326","id":89325,"instance":326,"goal":"app_326 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_327","id":89326,"instance":327,"goal":"arrow_327 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_328","id":89327,"instance":328,"goal":"of_var_328 : of d T_328 <- of-env d T_328.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_329","id":89328,"instance":329,"goal":"of_lam_329 : of (lam x) (arrow T_329a T_329b) <- ({f} of f T_329a -> of (x f) T_329b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_330","id":89329,"instance":330,"goal":"of_app_330 : of (app a_1 a_2) T_330 <- of a_1 (arrow T_330a T_330) <- of a_2 T_330a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_331","id":89330,"instance":331,"goal":"step_beta_331 : step (app (lam w) u) (w u).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_332","id":89331,"instance":332,"goal":"step_app1_332 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_333","id":89332,"instance":333,"goal":"preservation_333 : of q T_333 -> step q q_p -> of q_p T_333.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_334","id":89333,"instance":334,"goal":"progress_334 : of w T_334 -> value w or (exists w_p. step w w_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_335","id":89334,"instance":335,"goal":"eq_refl_335 : eq a a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_336","id":89335,"instance":336,"goal":"subst_336 : subst c a (c a).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_337","id":89336,"instance":337,"goal":"lam_337 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_338","id":89337,"instance":338,"goal":"app_338 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_339","id":89338,"instance":339,"goal":"arrow_339 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_340","id":89339,"instance":340,"goal":"of_var_340 : of w T_340 <- of-env w T_340.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_341","id":89340,"instance":341,"goal":"of_lam_341 : of (lam h) (arrow T_341a T_341b) <- ({w} of w T_341a -> of (h w) T_341b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_342","id":89341,"instance":342,"goal":"of_app_342 : of (app q_1 q_2) T_342 <- of q_1 (arrow T_342a T_342) <- of q_2 T_342a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_343","id":89342,"instance":343,"goal":"step_beta_343 : step (app (lam b) y) (b y).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_344","id":89343,"instance":344,"goal":"step_app1_344 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_345","id":89344,"instance":345,"goal":"preservation_345 : of n T_345 -> step n n_p -> of n_p T_345.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_346","id":89345,"instance":346,"goal":"progress_346 : of m T_346 -> value m or (exists m_p. step m m_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_347","id":89346,"instance":347,"goal":"eq_refl_347 : eq e e.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_348","id":89347,"instance":348,"goal":"subst_348 : subst n u (n u).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_349","id":89348,"instance":349,"goal":"lam_349 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_350","id":89349,"instance":350,"goal":"app_350 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_351","id":89350,"instance":351,"goal":"arrow_351 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_352","id":89351,"instance":352,"goal":"of_var_352 : of m T_352 <- of-env m T_352.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_353","id":89352,"instance":353,"goal":"of_lam_353 : of (lam e) (arrow T_353a T_353b) <- ({g} of g T_353a -> of (e g) T_353b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_354","id":89353,"instance":354,"goal":"of_app_354 : of (app m_1 m_2) T_354 <- of m_1 (arrow T_354a T_354) <- of m_2 T_354a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_355","id":89354,"instance":355,"goal":"step_beta_355 : step (app (lam n) u) (n u).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_356","id":89355,"instance":356,"goal":"step_app1_356 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_357","id":89356,"instance":357,"goal":"preservation_357 : of d T_357 -> step d d_p -> of d_p T_357.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_358","id":89357,"instance":358,"goal":"progress_358 : of x T_358 -> value x or (exists x_p. step x x_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_359","id":89358,"instance":359,"goal":"eq_refl_359 : eq f f.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_360","id":89359,"instance":360,"goal":"subst_360 : subst g p (g p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_361","id":89360,"instance":361,"goal":"lam_361 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_362","id":89361,"instance":362,"goal":"app_362 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_363","id":89362,"instance":363,"goal":"arrow_363 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_364","id":89363,"instance":364,"goal":"of_var_364 : of d T_364 <- of-env d T_364.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_365","id":89364,"instance":365,"goal":"of_lam_365 : of (lam g) (arrow T_365a T_365b) <- ({u} of u T_365a -> of (g u) T_365b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_366","id":89365,"instance":366,"goal":"of_app_366 : of (app q_1 q_2) T_366 <- of q_1 (arrow T_366a T_366) <- of q_2 T_366a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_367","id":89366,"instance":367,"goal":"step_beta_367 : step (app (lam m) e) (m e).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_368","id":89367,"instance":368,"goal":"step_app1_368 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_369","id":89368,"instance":369,"goal":"preservation_369 : of u T_369 -> step u u_p -> of u_p T_369.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_370","id":89369,"instance":370,"goal":"progress_370 : of q T_370 -> value q or (exists q_p. step q q_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_371","id":89370,"instance":371,"goal":"eq_refl_371 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_372","id":89371,"instance":372,"goal":"subst_372 : subst m f (m f).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_373","id":89372,"instance":373,"goal":"lam_373 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_374","id":89373,"instance":374,"goal":"app_374 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_375","id":89374,"instance":375,"goal":"arrow_375 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_376","id":89375,"instance":376,"goal":"of_var_376 : of w T_376 <- of-env w T_376.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_377","id":89376,"instance":377,"goal":"of_lam_377 : of (lam y) (arrow T_377a T_377b) <- ({n} of n T_377a -> of (y n) T_377b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_378","id":89377,"instance":378,"goal":"of_app_378 : of (app p_1 p_2) T_378 <- of p_1 (arrow T_378a T_378) <- of p_2 T_378a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_379","id":89378,"instance":379,"goal":"step_beta_379 : step (app (lam d) a) (d a).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_380","id":89379,"instance":380,"goal":"step_app1_380 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_381","id":89380,"instance":381,"goal":"preservation_381 : of c T_381 -> step c c_p -> of c_p T_381.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_382","id":89381,"instance":382,"goal":"progress_382 : of e T_382 -> value e or (exists e_p. step e e_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_383","id":89382,"instance":383,"goal":"eq_refl_383 : eq x x.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_384","id":89383,"instance":384,"goal":"subst_384 : subst c p (c p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_385","id":89384,"instance":385,"goal":"lam_385 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_386","id":89385,"instance":386,"goal":"app_386 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_387","id":89386,"instance":387,"goal":"arrow_387 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_388","id":89387,"instance":388,"goal":"of_var_388 : of d T_388 <- of-env d T_388.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_389","id":89388,"instance":389,"goal":"of_lam_389 : of (lam p) (arrow T_389a T_389b) <- ({d} of d T_389a -> of (p d) T_389b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_390","id":89389,"instance":390,"goal":"of_app_390 : of (app z_1 z_2) T_390 <- of z_1 (arrow T_390a T_390) <- of z_2 T_390a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_391","id":89390,"instance":391,"goal":"step_beta_391 : step (app (lam f) z) (f z).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_392","id":89391,"instance":392,"goal":"step_app1_392 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_393","id":89392,"instance":393,"goal":"preservation_393 : of u T_393 -> step u u_p -> of u_p T_393.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_394","id":89393,"instance":394,"goal":"progress_394 : of w T_394 -> value w or (exists w_p. step w w_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_395","id":89394,"instance":395,"goal":"eq_refl_395 : eq c c.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_396","id":89395,"instance":396,"goal":"subst_396 : subst z p (z p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_397","id":89396,"instance":397,"goal":"lam_397 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_398","id":89397,"instance":398,"goal":"app_398 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_399","id":89398,"instance":399,"goal":"arrow_399 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_400","id":89399,"instance":400,"goal":"of_var_400 : of z T_400 <- of-env z T_400.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_401","id":89400,"instance":401,"goal":"of_lam_401 : of (lam b) (arrow T_401a T_401b) <- ({e} of e T_401a -> of (b e) T_401b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_402","id":89401,"instance":402,"goal":"of_app_402 : of (app x_1 x_2) T_402 <- of x_1 (arrow T_402a T_402) <- of x_2 T_402a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_403","id":89402,"instance":403,"goal":"step_beta_403 : step (app (lam m) b) (m b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_404","id":89403,"instance":404,"goal":"step_app1_404 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_405","id":89404,"instance":405,"goal":"preservation_405 : of d T_405 -> step d d_p -> of d_p T_405.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_406","id":89405,"instance":406,"goal":"progress_406 : of g T_406 -> value g or (exists g_p. step g g_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_407","id":89406,"instance":407,"goal":"eq_refl_407 : eq e e.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_408","id":89407,"instance":408,"goal":"subst_408 : subst y d (y d).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_409","id":89408,"instance":409,"goal":"lam_409 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_410","id":89409,"instance":410,"goal":"app_410 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_411","id":89410,"instance":411,"goal":"arrow_411 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_412","id":89411,"instance":412,"goal":"of_var_412 : of v T_412 <- of-env v T_412.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_413","id":89412,"instance":413,"goal":"of_lam_413 : of (lam x) (arrow T_413a T_413b) <- ({y} of y T_413a -> of (x y) T_413b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_414","id":89413,"instance":414,"goal":"of_app_414 : of (app v_1 v_2) T_414 <- of v_1 (arrow T_414a T_414) <- of v_2 T_414a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_415","id":89414,"instance":415,"goal":"step_beta_415 : step (app (lam m) q) (m q).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_416","id":89415,"instance":416,"goal":"step_app1_416 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_417","id":89416,"instance":417,"goal":"preservation_417 : of g T_417 -> step g g_p -> of g_p T_417.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_418","id":89417,"instance":418,"goal":"progress_418 : of z T_418 -> value z or (exists z_p. step z z_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_419","id":89418,"instance":419,"goal":"eq_refl_419 : eq n n.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_420","id":89419,"instance":420,"goal":"subst_420 : subst b p (b p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_421","id":89420,"instance":421,"goal":"lam_421 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_422","id":89421,"instance":422,"goal":"app_422 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_423","id":89422,"instance":423,"goal":"arrow_423 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_424","id":89423,"instance":424,"goal":"of_var_424 : of d T_424 <- of-env d T_424.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_425","id":89424,"instance":425,"goal":"of_lam_425 : of (lam e) (arrow T_425a T_425b) <- ({g} of g T_425a -> of (e g) T_425b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_426","id":89425,"instance":426,"goal":"of_app_426 : of (app q_1 q_2) T_426 <- of q_1 (arrow T_426a T_426) <- of q_2 T_426a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_427","id":89426,"instance":427,"goal":"step_beta_427 : step (app (lam y) q) (y q).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_428","id":89427,"instance":428,"goal":"step_app1_428 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_429","id":89428,"instance":429,"goal":"preservation_429 : of u T_429 -> step u u_p -> of u_p T_429.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_430","id":89429,"instance":430,"goal":"progress_430 : of n T_430 -> value n or (exists n_p. step n n_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_431","id":89430,"instance":431,"goal":"eq_refl_431 : eq m m.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_432","id":89431,"instance":432,"goal":"subst_432 : subst x c (x c).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_433","id":89432,"instance":433,"goal":"lam_433 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_434","id":89433,"instance":434,"goal":"app_434 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_435","id":89434,"instance":435,"goal":"arrow_435 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_436","id":89435,"instance":436,"goal":"of_var_436 : of u T_436 <- of-env u T_436.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_437","id":89436,"instance":437,"goal":"of_lam_437 : of (lam z) (arrow T_437a T_437b) <- ({m} of m T_437a -> of (z m) T_437b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_438","id":89437,"instance":438,"goal":"of_app_438 : of (app a_1 a_2) T_438 <- of a_1 (arrow T_438a T_438) <- of a_2 T_438a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_439","id":89438,"instance":439,"goal":"step_beta_439 : step (app (lam a) n) (a n).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_440","id":89439,"instance":440,"goal":"step_app1_440 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_441","id":89440,"instance":441,"goal":"preservation_441 : of x T_441 -> step x x_p -> of x_p T_441.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_442","id":89441,"instance":442,"goal":"progress_442 : of y T_442 -> value y or (exists y_p. step y y_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_443","id":89442,"instance":443,"goal":"eq_refl_443 : eq q q.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_444","id":89443,"instance":444,"goal":"subst_444 : subst y x (y x).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_445","id":89444,"instance":445,"goal":"lam_445 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_446","id":89445,"instance":446,"goal":"app_446 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_447","id":89446,"instance":447,"goal":"arrow_447 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_448","id":89447,"instance":448,"goal":"of_var_448 : of b T_448 <- of-env b T_448.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_449","id":89448,"instance":449,"goal":"of_lam_449 : of (lam h) (arrow T_449a T_449b) <- ({e} of e T_449a -> of (h e) T_449b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_450","id":89449,"instance":450,"goal":"of_app_450 : of (app f_1 f_2) T_450 <- of f_1 (arrow T_450a T_450) <- of f_2 T_450a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_451","id":89450,"instance":451,"goal":"step_beta_451 : step (app (lam c) n) (c n).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_452","id":89451,"instance":452,"goal":"step_app1_452 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_453","id":89452,"instance":453,"goal":"preservation_453 : of q T_453 -> step q q_p -> of q_p T_453.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_454","id":89453,"instance":454,"goal":"progress_454 : of a T_454 -> value a or (exists a_p. step a a_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_455","id":89454,"instance":455,"goal":"eq_refl_455 : eq v v.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_456","id":89455,"instance":456,"goal":"subst_456 : subst h c (h c).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_457","id":89456,"instance":457,"goal":"lam_457 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_458","id":89457,"instance":458,"goal":"app_458 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_459","id":89458,"instance":459,"goal":"arrow_459 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_460","id":89459,"instance":460,"goal":"of_var_460 : of a T_460 <- of-env a T_460.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_461","id":89460,"instance":461,"goal":"of_lam_461 : of (lam x) (arrow T_461a T_461b) <- ({b} of b T_461a -> of (x b) T_461b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_462","id":89461,"instance":462,"goal":"of_app_462 : of (app y_1 y_2) T_462 <- of y_1 (arrow T_462a T_462) <- of y_2 T_462a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_463","id":89462,"instance":463,"goal":"step_beta_463 : step (app (lam h) q) (h q).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_464","id":89463,"instance":464,"goal":"step_app1_464 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_465","id":89464,"instance":465,"goal":"preservation_465 : of u T_465 -> step u u_p -> of u_p T_465.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_466","id":89465,"instance":466,"goal":"progress_466 : of q T_466 -> value q or (exists q_p. step q q_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_467","id":89466,"instance":467,"goal":"eq_refl_467 : eq y y.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_468","id":89467,"instance":468,"goal":"subst_468 : subst y e (y e).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_469","id":89468,"instance":469,"goal":"lam_469 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_470","id":89469,"instance":470,"goal":"app_470 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_471","id":89470,"instance":471,"goal":"arrow_471 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_472","id":89471,"instance":472,"goal":"of_var_472 : of a T_472 <- of-env a T_472.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_473","id":89472,"instance":473,"goal":"of_lam_473 : of (lam q) (arrow T_473a T_473b) <- ({d} of d T_473a -> of (q d) T_473b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_474","id":89473,"instance":474,"goal":"of_app_474 : of (app v_1 v_2) T_474 <- of v_1 (arrow T_474a T_474) <- of v_2 T_474a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_475","id":89474,"instance":475,"goal":"step_beta_475 : step (app (lam c) f) (c f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_476","id":89475,"instance":476,"goal":"step_app1_476 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_477","id":89476,"instance":477,"goal":"preservation_477 : of a T_477 -> step a a_p -> of a_p T_477.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_478","id":89477,"instance":478,"goal":"progress_478 : of d T_478 -> value d or (exists d_p. step d d_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_479","id":89478,"instance":479,"goal":"eq_refl_479 : eq b b.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_480","id":89479,"instance":480,"goal":"subst_480 : subst q h (q h).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_481","id":89480,"instance":481,"goal":"lam_481 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_482","id":89481,"instance":482,"goal":"app_482 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_483","id":89482,"instance":483,"goal":"arrow_483 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_484","id":89483,"instance":484,"goal":"of_var_484 : of c T_484 <- of-env c T_484.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_485","id":89484,"instance":485,"goal":"of_lam_485 : of (lam x) (arrow T_485a T_485b) <- ({p} of p T_485a -> of (x p) T_485b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_486","id":89485,"instance":486,"goal":"of_app_486 : of (app h_1 h_2) T_486 <- of h_1 (arrow T_486a T_486) <- of h_2 T_486a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_487","id":89486,"instance":487,"goal":"step_beta_487 : step (app (lam v) b) (v b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_488","id":89487,"instance":488,"goal":"step_app1_488 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_489","id":89488,"instance":489,"goal":"preservation_489 : of z T_489 -> step z z_p -> of z_p T_489.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_490","id":89489,"instance":490,"goal":"progress_490 : of y T_490 -> value y or (exists y_p. step y y_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_491","id":89490,"instance":491,"goal":"eq_refl_491 : eq b b.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_492","id":89491,"instance":492,"goal":"subst_492 : subst e f (e f).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_493","id":89492,"instance":493,"goal":"lam_493 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_494","id":89493,"instance":494,"goal":"app_494 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_495","id":89494,"instance":495,"goal":"arrow_495 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_496","id":89495,"instance":496,"goal":"of_var_496 : of f T_496 <- of-env f T_496.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_497","id":89496,"instance":497,"goal":"of_lam_497 : of (lam g) (arrow T_497a T_497b) <- ({n} of n T_497a -> of (g n) T_497b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_498","id":89497,"instance":498,"goal":"of_app_498 : of (app m_1 m_2) T_498 <- of m_1 (arrow T_498a T_498) <- of m_2 T_498a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_499","id":89498,"instance":499,"goal":"step_beta_499 : step (app (lam q) w) (q w).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_500","id":89499,"instance":500,"goal":"step_app1_500 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_501","id":89500,"instance":501,"goal":"preservation_501 : of d T_501 -> step d d_p -> of d_p T_501.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_502","id":89501,"instance":502,"goal":"progress_502 : of q T_502 -> value q or (exists q_p. step q q_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_503","id":89502,"instance":503,"goal":"eq_refl_503 : eq m m.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_504","id":89503,"instance":504,"goal":"subst_504 : subst z e (z e).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_505","id":89504,"instance":505,"goal":"lam_505 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_506","id":89505,"instance":506,"goal":"app_506 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_507","id":89506,"instance":507,"goal":"arrow_507 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_508","id":89507,"instance":508,"goal":"of_var_508 : of d T_508 <- of-env d T_508.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_509","id":89508,"instance":509,"goal":"of_lam_509 : of (lam d) (arrow T_509a T_509b) <- ({e} of e T_509a -> of (d e) T_509b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_510","id":89509,"instance":510,"goal":"of_app_510 : of (app m_1 m_2) T_510 <- of m_1 (arrow T_510a T_510) <- of m_2 T_510a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_511","id":89510,"instance":511,"goal":"step_beta_511 : step (app (lam a) c) (a c).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_512","id":89511,"instance":512,"goal":"step_app1_512 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_513","id":89512,"instance":513,"goal":"preservation_513 : of e T_513 -> step e e_p -> of e_p T_513.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_514","id":89513,"instance":514,"goal":"progress_514 : of m T_514 -> value m or (exists m_p. step m m_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_515","id":89514,"instance":515,"goal":"eq_refl_515 : eq g g.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_516","id":89515,"instance":516,"goal":"subst_516 : subst u f (u f).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_517","id":89516,"instance":517,"goal":"lam_517 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_518","id":89517,"instance":518,"goal":"app_518 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_519","id":89518,"instance":519,"goal":"arrow_519 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_520","id":89519,"instance":520,"goal":"of_var_520 : of g T_520 <- of-env g T_520.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_521","id":89520,"instance":521,"goal":"of_lam_521 : of (lam c) (arrow T_521a T_521b) <- ({g} of g T_521a -> of (c g) T_521b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_522","id":89521,"instance":522,"goal":"of_app_522 : of (app x_1 x_2) T_522 <- of x_1 (arrow T_522a T_522) <- of x_2 T_522a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_523","id":89522,"instance":523,"goal":"step_beta_523 : step (app (lam v) q) (v q).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_524","id":89523,"instance":524,"goal":"step_app1_524 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_525","id":89524,"instance":525,"goal":"preservation_525 : of w T_525 -> step w w_p -> of w_p T_525.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_526","id":89525,"instance":526,"goal":"progress_526 : of b T_526 -> value b or (exists b_p. step b b_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_527","id":89526,"instance":527,"goal":"eq_refl_527 : eq m m.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_528","id":89527,"instance":528,"goal":"subst_528 : subst d v (d v).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_529","id":89528,"instance":529,"goal":"lam_529 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_530","id":89529,"instance":530,"goal":"app_530 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_531","id":89530,"instance":531,"goal":"arrow_531 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_532","id":89531,"instance":532,"goal":"of_var_532 : of d T_532 <- of-env d T_532.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_533","id":89532,"instance":533,"goal":"of_lam_533 : of (lam b) (arrow T_533a T_533b) <- ({z} of z T_533a -> of (b z) T_533b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_534","id":89533,"instance":534,"goal":"of_app_534 : of (app m_1 m_2) T_534 <- of m_1 (arrow T_534a T_534) <- of m_2 T_534a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_535","id":89534,"instance":535,"goal":"step_beta_535 : step (app (lam n) p) (n p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_536","id":89535,"instance":536,"goal":"step_app1_536 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_537","id":89536,"instance":537,"goal":"preservation_537 : of p T_537 -> step p p_p -> of p_p T_537.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_538","id":89537,"instance":538,"goal":"progress_538 : of u T_538 -> value u or (exists u_p. step u u_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_539","id":89538,"instance":539,"goal":"eq_refl_539 : eq f f.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_540","id":89539,"instance":540,"goal":"subst_540 : subst a d (a d).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_541","id":89540,"instance":541,"goal":"lam_541 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_542","id":89541,"instance":542,"goal":"app_542 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_543","id":89542,"instance":543,"goal":"arrow_543 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_544","id":89543,"instance":544,"goal":"of_var_544 : of v T_544 <- of-env v T_544.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_545","id":89544,"instance":545,"goal":"of_lam_545 : of (lam u) (arrow T_545a T_545b) <- ({v} of v T_545a -> of (u v) T_545b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_546","id":89545,"instance":546,"goal":"of_app_546 : of (app g_1 g_2) T_546 <- of g_1 (arrow T_546a T_546) <- of g_2 T_546a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_547","id":89546,"instance":547,"goal":"step_beta_547 : step (app (lam g) n) (g n).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_548","id":89547,"instance":548,"goal":"step_app1_548 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_549","id":89548,"instance":549,"goal":"preservation_549 : of m T_549 -> step m m_p -> of m_p T_549.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_550","id":89549,"instance":550,"goal":"progress_550 : of u T_550 -> value u or (exists u_p. step u u_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_551","id":89550,"instance":551,"goal":"eq_refl_551 : eq h h.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_552","id":89551,"instance":552,"goal":"subst_552 : subst q m (q m).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_553","id":89552,"instance":553,"goal":"lam_553 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_554","id":89553,"instance":554,"goal":"app_554 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_555","id":89554,"instance":555,"goal":"arrow_555 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_556","id":89555,"instance":556,"goal":"of_var_556 : of e T_556 <- of-env e T_556.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_557","id":89556,"instance":557,"goal":"of_lam_557 : of (lam b) (arrow T_557a T_557b) <- ({m} of m T_557a -> of (b m) T_557b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_558","id":89557,"instance":558,"goal":"of_app_558 : of (app c_1 c_2) T_558 <- of c_1 (arrow T_558a T_558) <- of c_2 T_558a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_559","id":89558,"instance":559,"goal":"step_beta_559 : step (app (lam h) f) (h f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_560","id":89559,"instance":560,"goal":"step_app1_560 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_561","id":89560,"instance":561,"goal":"preservation_561 : of q T_561 -> step q q_p -> of q_p T_561.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_562","id":89561,"instance":562,"goal":"progress_562 : of q T_562 -> value q or (exists q_p. step q q_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_563","id":89562,"instance":563,"goal":"eq_refl_563 : eq b b.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_564","id":89563,"instance":564,"goal":"subst_564 : subst p x (p x).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_565","id":89564,"instance":565,"goal":"lam_565 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_566","id":89565,"instance":566,"goal":"app_566 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_567","id":89566,"instance":567,"goal":"arrow_567 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_568","id":89567,"instance":568,"goal":"of_var_568 : of d T_568 <- of-env d T_568.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_569","id":89568,"instance":569,"goal":"of_lam_569 : of (lam b) (arrow T_569a T_569b) <- ({f} of f T_569a -> of (b f) T_569b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_570","id":89569,"instance":570,"goal":"of_app_570 : of (app q_1 q_2) T_570 <- of q_1 (arrow T_570a T_570) <- of q_2 T_570a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_571","id":89570,"instance":571,"goal":"step_beta_571 : step (app (lam p) n) (p n).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_572","id":89571,"instance":572,"goal":"step_app1_572 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_573","id":89572,"instance":573,"goal":"preservation_573 : of v T_573 -> step v v_p -> of v_p T_573.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_574","id":89573,"instance":574,"goal":"progress_574 : of y T_574 -> value y or (exists y_p. step y y_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_575","id":89574,"instance":575,"goal":"eq_refl_575 : eq x x.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_576","id":89575,"instance":576,"goal":"subst_576 : subst z c (z c).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_577","id":89576,"instance":577,"goal":"lam_577 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_578","id":89577,"instance":578,"goal":"app_578 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_579","id":89578,"instance":579,"goal":"arrow_579 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_580","id":89579,"instance":580,"goal":"of_var_580 : of u T_580 <- of-env u T_580.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_581","id":89580,"instance":581,"goal":"of_lam_581 : of (lam d) (arrow T_581a T_581b) <- ({p} of p T_581a -> of (d p) T_581b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_582","id":89581,"instance":582,"goal":"of_app_582 : of (app a_1 a_2) T_582 <- of a_1 (arrow T_582a T_582) <- of a_2 T_582a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_583","id":89582,"instance":583,"goal":"step_beta_583 : step (app (lam e) b) (e b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_584","id":89583,"instance":584,"goal":"step_app1_584 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_585","id":89584,"instance":585,"goal":"preservation_585 : of a T_585 -> step a a_p -> of a_p T_585.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_586","id":89585,"instance":586,"goal":"progress_586 : of n T_586 -> value n or (exists n_p. step n n_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_587","id":89586,"instance":587,"goal":"eq_refl_587 : eq c c.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_588","id":89587,"instance":588,"goal":"subst_588 : subst p f (p f).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_589","id":89588,"instance":589,"goal":"lam_589 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_590","id":89589,"instance":590,"goal":"app_590 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_591","id":89590,"instance":591,"goal":"arrow_591 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_592","id":89591,"instance":592,"goal":"of_var_592 : of w T_592 <- of-env w T_592.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_593","id":89592,"instance":593,"goal":"of_lam_593 : of (lam b) (arrow T_593a T_593b) <- ({n} of n T_593a -> of (b n) T_593b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_594","id":89593,"instance":594,"goal":"of_app_594 : of (app d_1 d_2) T_594 <- of d_1 (arrow T_594a T_594) <- of d_2 T_594a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_595","id":89594,"instance":595,"goal":"step_beta_595 : step (app (lam q) n) (q n).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_596","id":89595,"instance":596,"goal":"step_app1_596 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_597","id":89596,"instance":597,"goal":"preservation_597 : of x T_597 -> step x x_p -> of x_p T_597.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_598","id":89597,"instance":598,"goal":"progress_598 : of h T_598 -> value h or (exists h_p. step h h_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_599","id":89598,"instance":599,"goal":"eq_refl_599 : eq n n.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_600","id":89599,"instance":600,"goal":"subst_600 : subst a f (a f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_601","id":89600,"instance":601,"goal":"lam_601 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_602","id":89601,"instance":602,"goal":"app_602 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_603","id":89602,"instance":603,"goal":"arrow_603 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_604","id":89603,"instance":604,"goal":"of_var_604 : of x T_604 <- of-env x T_604.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_605","id":89604,"instance":605,"goal":"of_lam_605 : of (lam e) (arrow T_605a T_605b) <- ({q} of q T_605a -> of (e q) T_605b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_606","id":89605,"instance":606,"goal":"of_app_606 : of (app v_1 v_2) T_606 <- of v_1 (arrow T_606a T_606) <- of v_2 T_606a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_607","id":89606,"instance":607,"goal":"step_beta_607 : step (app (lam w) p) (w p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_608","id":89607,"instance":608,"goal":"step_app1_608 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_609","id":89608,"instance":609,"goal":"preservation_609 : of a T_609 -> step a a_p -> of a_p T_609.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_610","id":89609,"instance":610,"goal":"progress_610 : of w T_610 -> value w or (exists w_p. step w w_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_611","id":89610,"instance":611,"goal":"eq_refl_611 : eq q q.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_612","id":89611,"instance":612,"goal":"subst_612 : subst p u (p u).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_613","id":89612,"instance":613,"goal":"lam_613 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_614","id":89613,"instance":614,"goal":"app_614 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_615","id":89614,"instance":615,"goal":"arrow_615 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_616","id":89615,"instance":616,"goal":"of_var_616 : of u T_616 <- of-env u T_616.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_617","id":89616,"instance":617,"goal":"of_lam_617 : of (lam h) (arrow T_617a T_617b) <- ({a} of a T_617a -> of (h a) T_617b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_618","id":89617,"instance":618,"goal":"of_app_618 : of (app a_1 a_2) T_618 <- of a_1 (arrow T_618a T_618) <- of a_2 T_618a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_619","id":89618,"instance":619,"goal":"step_beta_619 : step (app (lam f) d) (f d).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_620","id":89619,"instance":620,"goal":"step_app1_620 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_621","id":89620,"instance":621,"goal":"preservation_621 : of x T_621 -> step x x_p -> of x_p T_621.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_622","id":89621,"instance":622,"goal":"progress_622 : of e T_622 -> value e or (exists e_p. step e e_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_623","id":89622,"instance":623,"goal":"eq_refl_623 : eq q q.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_624","id":89623,"instance":624,"goal":"subst_624 : subst q f (q f).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_625","id":89624,"instance":625,"goal":"lam_625 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_626","id":89625,"instance":626,"goal":"app_626 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_627","id":89626,"instance":627,"goal":"arrow_627 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_628","id":89627,"instance":628,"goal":"of_var_628 : of m T_628 <- of-env m T_628.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_629","id":89628,"instance":629,"goal":"of_lam_629 : of (lam b) (arrow T_629a T_629b) <- ({m} of m T_629a -> of (b m) T_629b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_630","id":89629,"instance":630,"goal":"of_app_630 : of (app y_1 y_2) T_630 <- of y_1 (arrow T_630a T_630) <- of y_2 T_630a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_631","id":89630,"instance":631,"goal":"step_beta_631 : step (app (lam h) q) (h q).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_632","id":89631,"instance":632,"goal":"step_app1_632 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_633","id":89632,"instance":633,"goal":"preservation_633 : of m T_633 -> step m m_p -> of m_p T_633.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_634","id":89633,"instance":634,"goal":"progress_634 : of m T_634 -> value m or (exists m_p. step m m_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_635","id":89634,"instance":635,"goal":"eq_refl_635 : eq q q.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_636","id":89635,"instance":636,"goal":"subst_636 : subst y m (y m).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_637","id":89636,"instance":637,"goal":"lam_637 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_638","id":89637,"instance":638,"goal":"app_638 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_639","id":89638,"instance":639,"goal":"arrow_639 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_640","id":89639,"instance":640,"goal":"of_var_640 : of p T_640 <- of-env p T_640.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_641","id":89640,"instance":641,"goal":"of_lam_641 : of (lam h) (arrow T_641a T_641b) <- ({z} of z T_641a -> of (h z) T_641b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_642","id":89641,"instance":642,"goal":"of_app_642 : of (app v_1 v_2) T_642 <- of v_1 (arrow T_642a T_642) <- of v_2 T_642a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_643","id":89642,"instance":643,"goal":"step_beta_643 : step (app (lam g) y) (g y).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_644","id":89643,"instance":644,"goal":"step_app1_644 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_645","id":89644,"instance":645,"goal":"preservation_645 : of m T_645 -> step m m_p -> of m_p T_645.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_646","id":89645,"instance":646,"goal":"progress_646 : of q T_646 -> value q or (exists q_p. step q q_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_647","id":89646,"instance":647,"goal":"eq_refl_647 : eq q q.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_648","id":89647,"instance":648,"goal":"subst_648 : subst m b (m b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_649","id":89648,"instance":649,"goal":"lam_649 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_650","id":89649,"instance":650,"goal":"app_650 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_651","id":89650,"instance":651,"goal":"arrow_651 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_652","id":89651,"instance":652,"goal":"of_var_652 : of f T_652 <- of-env f T_652.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_653","id":89652,"instance":653,"goal":"of_lam_653 : of (lam n) (arrow T_653a T_653b) <- ({d} of d T_653a -> of (n d) T_653b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_654","id":89653,"instance":654,"goal":"of_app_654 : of (app c_1 c_2) T_654 <- of c_1 (arrow T_654a T_654) <- of c_2 T_654a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_655","id":89654,"instance":655,"goal":"step_beta_655 : step (app (lam h) u) (h u).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_656","id":89655,"instance":656,"goal":"step_app1_656 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_657","id":89656,"instance":657,"goal":"preservation_657 : of w T_657 -> step w w_p -> of w_p T_657.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_658","id":89657,"instance":658,"goal":"progress_658 : of p T_658 -> value p or (exists p_p. step p p_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_659","id":89658,"instance":659,"goal":"eq_refl_659 : eq p p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_660","id":89659,"instance":660,"goal":"subst_660 : subst f y (f y).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_661","id":89660,"instance":661,"goal":"lam_661 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_662","id":89661,"instance":662,"goal":"app_662 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_663","id":89662,"instance":663,"goal":"arrow_663 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_664","id":89663,"instance":664,"goal":"of_var_664 : of y T_664 <- of-env y T_664.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_665","id":89664,"instance":665,"goal":"of_lam_665 : of (lam q) (arrow T_665a T_665b) <- ({a} of a T_665a -> of (q a) T_665b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_666","id":89665,"instance":666,"goal":"of_app_666 : of (app z_1 z_2) T_666 <- of z_1 (arrow T_666a T_666) <- of z_2 T_666a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_667","id":89666,"instance":667,"goal":"step_beta_667 : step (app (lam h) w) (h w).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_668","id":89667,"instance":668,"goal":"step_app1_668 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_669","id":89668,"instance":669,"goal":"preservation_669 : of n T_669 -> step n n_p -> of n_p T_669.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_670","id":89669,"instance":670,"goal":"progress_670 : of y T_670 -> value y or (exists y_p. step y y_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_671","id":89670,"instance":671,"goal":"eq_refl_671 : eq m m.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_672","id":89671,"instance":672,"goal":"subst_672 : subst e n (e n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_673","id":89672,"instance":673,"goal":"lam_673 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_674","id":89673,"instance":674,"goal":"app_674 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_675","id":89674,"instance":675,"goal":"arrow_675 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_676","id":89675,"instance":676,"goal":"of_var_676 : of g T_676 <- of-env g T_676.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_677","id":89676,"instance":677,"goal":"of_lam_677 : of (lam p) (arrow T_677a T_677b) <- ({b} of b T_677a -> of (p b) T_677b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_678","id":89677,"instance":678,"goal":"of_app_678 : of (app e_1 e_2) T_678 <- of e_1 (arrow T_678a T_678) <- of e_2 T_678a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_679","id":89678,"instance":679,"goal":"step_beta_679 : step (app (lam f) a) (f a).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_680","id":89679,"instance":680,"goal":"step_app1_680 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_681","id":89680,"instance":681,"goal":"preservation_681 : of m T_681 -> step m m_p -> of m_p T_681.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_682","id":89681,"instance":682,"goal":"progress_682 : of p T_682 -> value p or (exists p_p. step p p_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_683","id":89682,"instance":683,"goal":"eq_refl_683 : eq z z.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_684","id":89683,"instance":684,"goal":"subst_684 : subst p x (p x).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_685","id":89684,"instance":685,"goal":"lam_685 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_686","id":89685,"instance":686,"goal":"app_686 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_687","id":89686,"instance":687,"goal":"arrow_687 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_688","id":89687,"instance":688,"goal":"of_var_688 : of f T_688 <- of-env f T_688.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_689","id":89688,"instance":689,"goal":"of_lam_689 : of (lam x) (arrow T_689a T_689b) <- ({e} of e T_689a -> of (x e) T_689b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_690","id":89689,"instance":690,"goal":"of_app_690 : of (app x_1 x_2) T_690 <- of x_1 (arrow T_690a T_690) <- of x_2 T_690a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_691","id":89690,"instance":691,"goal":"step_beta_691 : step (app (lam u) h) (u h).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_692","id":89691,"instance":692,"goal":"step_app1_692 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_693","id":89692,"instance":693,"goal":"preservation_693 : of q T_693 -> step q q_p -> of q_p T_693.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_694","id":89693,"instance":694,"goal":"progress_694 : of x T_694 -> value x or (exists x_p. step x x_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_695","id":89694,"instance":695,"goal":"eq_refl_695 : eq n n.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_696","id":89695,"instance":696,"goal":"subst_696 : subst x b (x b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_697","id":89696,"instance":697,"goal":"lam_697 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_698","id":89697,"instance":698,"goal":"app_698 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_699","id":89698,"instance":699,"goal":"arrow_699 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_700","id":89699,"instance":700,"goal":"of_var_700 : of w T_700 <- of-env w T_700.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_701","id":89700,"instance":701,"goal":"of_lam_701 : of (lam h) (arrow T_701a T_701b) <- ({c} of c T_701a -> of (h c) T_701b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_702","id":89701,"instance":702,"goal":"of_app_702 : of (app c_1 c_2) T_702 <- of c_1 (arrow T_702a T_702) <- of c_2 T_702a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_703","id":89702,"instance":703,"goal":"step_beta_703 : step (app (lam q) e) (q e).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_704","id":89703,"instance":704,"goal":"step_app1_704 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_705","id":89704,"instance":705,"goal":"preservation_705 : of h T_705 -> step h h_p -> of h_p T_705.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_706","id":89705,"instance":706,"goal":"progress_706 : of z T_706 -> value z or (exists z_p. step z z_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_707","id":89706,"instance":707,"goal":"eq_refl_707 : eq n n.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_708","id":89707,"instance":708,"goal":"subst_708 : subst x d (x d).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_709","id":89708,"instance":709,"goal":"lam_709 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_710","id":89709,"instance":710,"goal":"app_710 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_711","id":89710,"instance":711,"goal":"arrow_711 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_712","id":89711,"instance":712,"goal":"of_var_712 : of y T_712 <- of-env y T_712.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_713","id":89712,"instance":713,"goal":"of_lam_713 : of (lam p) (arrow T_713a T_713b) <- ({v} of v T_713a -> of (p v) T_713b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_714","id":89713,"instance":714,"goal":"of_app_714 : of (app d_1 d_2) T_714 <- of d_1 (arrow T_714a T_714) <- of d_2 T_714a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_715","id":89714,"instance":715,"goal":"step_beta_715 : step (app (lam h) b) (h b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_716","id":89715,"instance":716,"goal":"step_app1_716 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_717","id":89716,"instance":717,"goal":"preservation_717 : of e T_717 -> step e e_p -> of e_p T_717.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_718","id":89717,"instance":718,"goal":"progress_718 : of y T_718 -> value y or (exists y_p. step y y_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_719","id":89718,"instance":719,"goal":"eq_refl_719 : eq a a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_720","id":89719,"instance":720,"goal":"subst_720 : subst y c (y c).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_721","id":89720,"instance":721,"goal":"lam_721 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_722","id":89721,"instance":722,"goal":"app_722 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_723","id":89722,"instance":723,"goal":"arrow_723 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_724","id":89723,"instance":724,"goal":"of_var_724 : of u T_724 <- of-env u T_724.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_725","id":89724,"instance":725,"goal":"of_lam_725 : of (lam b) (arrow T_725a T_725b) <- ({c} of c T_725a -> of (b c) T_725b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_726","id":89725,"instance":726,"goal":"of_app_726 : of (app q_1 q_2) T_726 <- of q_1 (arrow T_726a T_726) <- of q_2 T_726a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_727","id":89726,"instance":727,"goal":"step_beta_727 : step (app (lam u) n) (u n).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_728","id":89727,"instance":728,"goal":"step_app1_728 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_729","id":89728,"instance":729,"goal":"preservation_729 : of y T_729 -> step y y_p -> of y_p T_729.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_730","id":89729,"instance":730,"goal":"progress_730 : of d T_730 -> value d or (exists d_p. step d d_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_731","id":89730,"instance":731,"goal":"eq_refl_731 : eq u u.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_732","id":89731,"instance":732,"goal":"subst_732 : subst d g (d g).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_733","id":89732,"instance":733,"goal":"lam_733 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_734","id":89733,"instance":734,"goal":"app_734 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_735","id":89734,"instance":735,"goal":"arrow_735 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_736","id":89735,"instance":736,"goal":"of_var_736 : of m T_736 <- of-env m T_736.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_737","id":89736,"instance":737,"goal":"of_lam_737 : of (lam e) (arrow T_737a T_737b) <- ({g} of g T_737a -> of (e g) T_737b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_738","id":89737,"instance":738,"goal":"of_app_738 : of (app x_1 x_2) T_738 <- of x_1 (arrow T_738a T_738) <- of x_2 T_738a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_739","id":89738,"instance":739,"goal":"step_beta_739 : step (app (lam a) g) (a g).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_740","id":89739,"instance":740,"goal":"step_app1_740 : step (app u_1 u_2) (app u_1p u_2) <- step u_1 u_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_741","id":89740,"instance":741,"goal":"preservation_741 : of n T_741 -> step n n_p -> of n_p T_741.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_742","id":89741,"instance":742,"goal":"progress_742 : of e T_742 -> value e or (exists e_p. step e e_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_743","id":89742,"instance":743,"goal":"eq_refl_743 : eq g g.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_744","id":89743,"instance":744,"goal":"subst_744 : subst a c (a c).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_745","id":89744,"instance":745,"goal":"lam_745 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_746","id":89745,"instance":746,"goal":"app_746 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_747","id":89746,"instance":747,"goal":"arrow_747 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_748","id":89747,"instance":748,"goal":"of_var_748 : of n T_748 <- of-env n T_748.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_749","id":89748,"instance":749,"goal":"of_lam_749 : of (lam v) (arrow T_749a T_749b) <- ({u} of u T_749a -> of (v u) T_749b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_750","id":89749,"instance":750,"goal":"of_app_750 : of (app u_1 u_2) T_750 <- of u_1 (arrow T_750a T_750) <- of u_2 T_750a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_751","id":89750,"instance":751,"goal":"step_beta_751 : step (app (lam p) y) (p y).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_752","id":89751,"instance":752,"goal":"step_app1_752 : step (app u_1 u_2) (app u_1p u_2) <- step u_1 u_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_753","id":89752,"instance":753,"goal":"preservation_753 : of n T_753 -> step n n_p -> of n_p T_753.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_754","id":89753,"instance":754,"goal":"progress_754 : of d T_754 -> value d or (exists d_p. step d d_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_755","id":89754,"instance":755,"goal":"eq_refl_755 : eq g g.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_756","id":89755,"instance":756,"goal":"subst_756 : subst n g (n g).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_757","id":89756,"instance":757,"goal":"lam_757 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_758","id":89757,"instance":758,"goal":"app_758 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_759","id":89758,"instance":759,"goal":"arrow_759 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_760","id":89759,"instance":760,"goal":"of_var_760 : of e T_760 <- of-env e T_760.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_761","id":89760,"instance":761,"goal":"of_lam_761 : of (lam c) (arrow T_761a T_761b) <- ({w} of w T_761a -> of (c w) T_761b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_762","id":89761,"instance":762,"goal":"of_app_762 : of (app z_1 z_2) T_762 <- of z_1 (arrow T_762a T_762) <- of z_2 T_762a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_763","id":89762,"instance":763,"goal":"step_beta_763 : step (app (lam h) b) (h b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_764","id":89763,"instance":764,"goal":"step_app1_764 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_765","id":89764,"instance":765,"goal":"preservation_765 : of b T_765 -> step b b_p -> of b_p T_765.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_766","id":89765,"instance":766,"goal":"progress_766 : of f T_766 -> value f or (exists f_p. step f f_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_767","id":89766,"instance":767,"goal":"eq_refl_767 : eq y y.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_768","id":89767,"instance":768,"goal":"subst_768 : subst b q (b q).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_769","id":89768,"instance":769,"goal":"lam_769 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_770","id":89769,"instance":770,"goal":"app_770 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_771","id":89770,"instance":771,"goal":"arrow_771 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_772","id":89771,"instance":772,"goal":"of_var_772 : of a T_772 <- of-env a T_772.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_773","id":89772,"instance":773,"goal":"of_lam_773 : of (lam b) (arrow T_773a T_773b) <- ({y} of y T_773a -> of (b y) T_773b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_774","id":89773,"instance":774,"goal":"of_app_774 : of (app c_1 c_2) T_774 <- of c_1 (arrow T_774a T_774) <- of c_2 T_774a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_775","id":89774,"instance":775,"goal":"step_beta_775 : step (app (lam d) w) (d w).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_776","id":89775,"instance":776,"goal":"step_app1_776 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_777","id":89776,"instance":777,"goal":"preservation_777 : of x T_777 -> step x x_p -> of x_p T_777.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_778","id":89777,"instance":778,"goal":"progress_778 : of b T_778 -> value b or (exists b_p. step b b_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_779","id":89778,"instance":779,"goal":"eq_refl_779 : eq q q.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_780","id":89779,"instance":780,"goal":"subst_780 : subst v b (v b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_781","id":89780,"instance":781,"goal":"lam_781 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_782","id":89781,"instance":782,"goal":"app_782 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_783","id":89782,"instance":783,"goal":"arrow_783 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_784","id":89783,"instance":784,"goal":"of_var_784 : of n T_784 <- of-env n T_784.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_785","id":89784,"instance":785,"goal":"of_lam_785 : of (lam q) (arrow T_785a T_785b) <- ({v} of v T_785a -> of (q v) T_785b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_786","id":89785,"instance":786,"goal":"of_app_786 : of (app y_1 y_2) T_786 <- of y_1 (arrow T_786a T_786) <- of y_2 T_786a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_787","id":89786,"instance":787,"goal":"step_beta_787 : step (app (lam h) p) (h p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_788","id":89787,"instance":788,"goal":"step_app1_788 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_789","id":89788,"instance":789,"goal":"preservation_789 : of y T_789 -> step y y_p -> of y_p T_789.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_790","id":89789,"instance":790,"goal":"progress_790 : of q T_790 -> value q or (exists q_p. step q q_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_791","id":89790,"instance":791,"goal":"eq_refl_791 : eq v v.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_792","id":89791,"instance":792,"goal":"subst_792 : subst y c (y c).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_793","id":89792,"instance":793,"goal":"lam_793 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_794","id":89793,"instance":794,"goal":"app_794 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_795","id":89794,"instance":795,"goal":"arrow_795 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_796","id":89795,"instance":796,"goal":"of_var_796 : of a T_796 <- of-env a T_796.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_797","id":89796,"instance":797,"goal":"of_lam_797 : of (lam w) (arrow T_797a T_797b) <- ({u} of u T_797a -> of (w u) T_797b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_798","id":89797,"instance":798,"goal":"of_app_798 : of (app x_1 x_2) T_798 <- of x_1 (arrow T_798a T_798) <- of x_2 T_798a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_799","id":89798,"instance":799,"goal":"step_beta_799 : step (app (lam m) z) (m z).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_800","id":89799,"instance":800,"goal":"step_app1_800 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_801","id":89800,"instance":801,"goal":"preservation_801 : of u T_801 -> step u u_p -> of u_p T_801.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_802","id":89801,"instance":802,"goal":"progress_802 : of q T_802 -> value q or (exists q_p. step q q_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_803","id":89802,"instance":803,"goal":"eq_refl_803 : eq q q.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_804","id":89803,"instance":804,"goal":"subst_804 : subst h b (h b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_805","id":89804,"instance":805,"goal":"lam_805 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_806","id":89805,"instance":806,"goal":"app_806 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_807","id":89806,"instance":807,"goal":"arrow_807 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_808","id":89807,"instance":808,"goal":"of_var_808 : of q T_808 <- of-env q T_808.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_809","id":89808,"instance":809,"goal":"of_lam_809 : of (lam z) (arrow T_809a T_809b) <- ({u} of u T_809a -> of (z u) T_809b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_810","id":89809,"instance":810,"goal":"of_app_810 : of (app w_1 w_2) T_810 <- of w_1 (arrow T_810a T_810) <- of w_2 T_810a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_811","id":89810,"instance":811,"goal":"step_beta_811 : step (app (lam q) h) (q h).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_812","id":89811,"instance":812,"goal":"step_app1_812 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_813","id":89812,"instance":813,"goal":"preservation_813 : of v T_813 -> step v v_p -> of v_p T_813.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_814","id":89813,"instance":814,"goal":"progress_814 : of p T_814 -> value p or (exists p_p. step p p_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_815","id":89814,"instance":815,"goal":"eq_refl_815 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_816","id":89815,"instance":816,"goal":"subst_816 : subst b h (b h).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_817","id":89816,"instance":817,"goal":"lam_817 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_818","id":89817,"instance":818,"goal":"app_818 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_819","id":89818,"instance":819,"goal":"arrow_819 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_820","id":89819,"instance":820,"goal":"of_var_820 : of p T_820 <- of-env p T_820.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_821","id":89820,"instance":821,"goal":"of_lam_821 : of (lam c) (arrow T_821a T_821b) <- ({w} of w T_821a -> of (c w) T_821b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_822","id":89821,"instance":822,"goal":"of_app_822 : of (app q_1 q_2) T_822 <- of q_1 (arrow T_822a T_822) <- of q_2 T_822a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_823","id":89822,"instance":823,"goal":"step_beta_823 : step (app (lam m) b) (m b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_824","id":89823,"instance":824,"goal":"step_app1_824 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_825","id":89824,"instance":825,"goal":"preservation_825 : of e T_825 -> step e e_p -> of e_p T_825.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_826","id":89825,"instance":826,"goal":"progress_826 : of u T_826 -> value u or (exists u_p. step u u_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_827","id":89826,"instance":827,"goal":"eq_refl_827 : eq y y.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_828","id":89827,"instance":828,"goal":"subst_828 : subst z p (z p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_829","id":89828,"instance":829,"goal":"lam_829 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_830","id":89829,"instance":830,"goal":"app_830 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_831","id":89830,"instance":831,"goal":"arrow_831 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_832","id":89831,"instance":832,"goal":"of_var_832 : of w T_832 <- of-env w T_832.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_833","id":89832,"instance":833,"goal":"of_lam_833 : of (lam m) (arrow T_833a T_833b) <- ({a} of a T_833a -> of (m a) T_833b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_834","id":89833,"instance":834,"goal":"of_app_834 : of (app b_1 b_2) T_834 <- of b_1 (arrow T_834a T_834) <- of b_2 T_834a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_835","id":89834,"instance":835,"goal":"step_beta_835 : step (app (lam y) b) (y b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_836","id":89835,"instance":836,"goal":"step_app1_836 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_837","id":89836,"instance":837,"goal":"preservation_837 : of c T_837 -> step c c_p -> of c_p T_837.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_838","id":89837,"instance":838,"goal":"progress_838 : of m T_838 -> value m or (exists m_p. step m m_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_839","id":89838,"instance":839,"goal":"eq_refl_839 : eq c c.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_840","id":89839,"instance":840,"goal":"subst_840 : subst x d (x d).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_841","id":89840,"instance":841,"goal":"lam_841 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_842","id":89841,"instance":842,"goal":"app_842 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_843","id":89842,"instance":843,"goal":"arrow_843 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_844","id":89843,"instance":844,"goal":"of_var_844 : of x T_844 <- of-env x T_844.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_845","id":89844,"instance":845,"goal":"of_lam_845 : of (lam a) (arrow T_845a T_845b) <- ({f} of f T_845a -> of (a f) T_845b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_846","id":89845,"instance":846,"goal":"of_app_846 : of (app h_1 h_2) T_846 <- of h_1 (arrow T_846a T_846) <- of h_2 T_846a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_847","id":89846,"instance":847,"goal":"step_beta_847 : step (app (lam z) a) (z a).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_848","id":89847,"instance":848,"goal":"step_app1_848 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_849","id":89848,"instance":849,"goal":"preservation_849 : of z T_849 -> step z z_p -> of z_p T_849.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_850","id":89849,"instance":850,"goal":"progress_850 : of g T_850 -> value g or (exists g_p. step g g_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_851","id":89850,"instance":851,"goal":"eq_refl_851 : eq e e.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_852","id":89851,"instance":852,"goal":"subst_852 : subst q c (q c).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_853","id":89852,"instance":853,"goal":"lam_853 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_854","id":89853,"instance":854,"goal":"app_854 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_855","id":89854,"instance":855,"goal":"arrow_855 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_856","id":89855,"instance":856,"goal":"of_var_856 : of u T_856 <- of-env u T_856.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_857","id":89856,"instance":857,"goal":"of_lam_857 : of (lam u) (arrow T_857a T_857b) <- ({g} of g T_857a -> of (u g) T_857b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_858","id":89857,"instance":858,"goal":"of_app_858 : of (app y_1 y_2) T_858 <- of y_1 (arrow T_858a T_858) <- of y_2 T_858a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_859","id":89858,"instance":859,"goal":"step_beta_859 : step (app (lam u) f) (u f).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_860","id":89859,"instance":860,"goal":"step_app1_860 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_861","id":89860,"instance":861,"goal":"preservation_861 : of v T_861 -> step v v_p -> of v_p T_861.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_862","id":89861,"instance":862,"goal":"progress_862 : of n T_862 -> value n or (exists n_p. step n n_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_863","id":89862,"instance":863,"goal":"eq_refl_863 : eq h h.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_864","id":89863,"instance":864,"goal":"subst_864 : subst f d (f d).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_865","id":89864,"instance":865,"goal":"lam_865 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_866","id":89865,"instance":866,"goal":"app_866 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_867","id":89866,"instance":867,"goal":"arrow_867 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_868","id":89867,"instance":868,"goal":"of_var_868 : of q T_868 <- of-env q T_868.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_869","id":89868,"instance":869,"goal":"of_lam_869 : of (lam z) (arrow T_869a T_869b) <- ({u} of u T_869a -> of (z u) T_869b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_870","id":89869,"instance":870,"goal":"of_app_870 : of (app h_1 h_2) T_870 <- of h_1 (arrow T_870a T_870) <- of h_2 T_870a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_871","id":89870,"instance":871,"goal":"step_beta_871 : step (app (lam h) n) (h n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_872","id":89871,"instance":872,"goal":"step_app1_872 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_873","id":89872,"instance":873,"goal":"preservation_873 : of f T_873 -> step f f_p -> of f_p T_873.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_874","id":89873,"instance":874,"goal":"progress_874 : of e T_874 -> value e or (exists e_p. step e e_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_875","id":89874,"instance":875,"goal":"eq_refl_875 : eq f f.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_876","id":89875,"instance":876,"goal":"subst_876 : subst g n (g n).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_877","id":89876,"instance":877,"goal":"lam_877 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_878","id":89877,"instance":878,"goal":"app_878 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_879","id":89878,"instance":879,"goal":"arrow_879 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_880","id":89879,"instance":880,"goal":"of_var_880 : of q T_880 <- of-env q T_880.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_881","id":89880,"instance":881,"goal":"of_lam_881 : of (lam a) (arrow T_881a T_881b) <- ({u} of u T_881a -> of (a u) T_881b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_882","id":89881,"instance":882,"goal":"of_app_882 : of (app x_1 x_2) T_882 <- of x_1 (arrow T_882a T_882) <- of x_2 T_882a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_883","id":89882,"instance":883,"goal":"step_beta_883 : step (app (lam a) f) (a f).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_884","id":89883,"instance":884,"goal":"step_app1_884 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_885","id":89884,"instance":885,"goal":"preservation_885 : of y T_885 -> step y y_p -> of y_p T_885.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_886","id":89885,"instance":886,"goal":"progress_886 : of y T_886 -> value y or (exists y_p. step y y_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_887","id":89886,"instance":887,"goal":"eq_refl_887 : eq m m.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_888","id":89887,"instance":888,"goal":"subst_888 : subst z c (z c).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_889","id":89888,"instance":889,"goal":"lam_889 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_890","id":89889,"instance":890,"goal":"app_890 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_891","id":89890,"instance":891,"goal":"arrow_891 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_892","id":89891,"instance":892,"goal":"of_var_892 : of b T_892 <- of-env b T_892.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_893","id":89892,"instance":893,"goal":"of_lam_893 : of (lam a) (arrow T_893a T_893b) <- ({g} of g T_893a -> of (a g) T_893b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_894","id":89893,"instance":894,"goal":"of_app_894 : of (app y_1 y_2) T_894 <- of y_1 (arrow T_894a T_894) <- of y_2 T_894a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_895","id":89894,"instance":895,"goal":"step_beta_895 : step (app (lam g) h) (g h).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_896","id":89895,"instance":896,"goal":"step_app1_896 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_897","id":89896,"instance":897,"goal":"preservation_897 : of b T_897 -> step b b_p -> of b_p T_897.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_898","id":89897,"instance":898,"goal":"progress_898 : of p T_898 -> value p or (exists p_p. step p p_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_899","id":89898,"instance":899,"goal":"eq_refl_899 : eq p p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_900","id":89899,"instance":900,"goal":"subst_900 : subst c u (c u).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_901","id":89900,"instance":901,"goal":"lam_901 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_902","id":89901,"instance":902,"goal":"app_902 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_903","id":89902,"instance":903,"goal":"arrow_903 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_904","id":89903,"instance":904,"goal":"of_var_904 : of e T_904 <- of-env e T_904.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_905","id":89904,"instance":905,"goal":"of_lam_905 : of (lam h) (arrow T_905a T_905b) <- ({y} of y T_905a -> of (h y) T_905b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_906","id":89905,"instance":906,"goal":"of_app_906 : of (app a_1 a_2) T_906 <- of a_1 (arrow T_906a T_906) <- of a_2 T_906a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_907","id":89906,"instance":907,"goal":"step_beta_907 : step (app (lam d) b) (d b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_908","id":89907,"instance":908,"goal":"step_app1_908 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_909","id":89908,"instance":909,"goal":"preservation_909 : of m T_909 -> step m m_p -> of m_p T_909.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_910","id":89909,"instance":910,"goal":"progress_910 : of q T_910 -> value q or (exists q_p. step q q_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_911","id":89910,"instance":911,"goal":"eq_refl_911 : eq y y.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_912","id":89911,"instance":912,"goal":"subst_912 : subst c d (c d).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_913","id":89912,"instance":913,"goal":"lam_913 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_914","id":89913,"instance":914,"goal":"app_914 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_915","id":89914,"instance":915,"goal":"arrow_915 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_916","id":89915,"instance":916,"goal":"of_var_916 : of x T_916 <- of-env x T_916.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_917","id":89916,"instance":917,"goal":"of_lam_917 : of (lam q) (arrow T_917a T_917b) <- ({n} of n T_917a -> of (q n) T_917b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_918","id":89917,"instance":918,"goal":"of_app_918 : of (app f_1 f_2) T_918 <- of f_1 (arrow T_918a T_918) <- of f_2 T_918a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_919","id":89918,"instance":919,"goal":"step_beta_919 : step (app (lam f) a) (f a).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_920","id":89919,"instance":920,"goal":"step_app1_920 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_921","id":89920,"instance":921,"goal":"preservation_921 : of f T_921 -> step f f_p -> of f_p T_921.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_922","id":89921,"instance":922,"goal":"progress_922 : of f T_922 -> value f or (exists f_p. step f f_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_923","id":89922,"instance":923,"goal":"eq_refl_923 : eq g g.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_924","id":89923,"instance":924,"goal":"subst_924 : subst d b (d b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_925","id":89924,"instance":925,"goal":"lam_925 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_926","id":89925,"instance":926,"goal":"app_926 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_927","id":89926,"instance":927,"goal":"arrow_927 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_928","id":89927,"instance":928,"goal":"of_var_928 : of w T_928 <- of-env w T_928.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_929","id":89928,"instance":929,"goal":"of_lam_929 : of (lam h) (arrow T_929a T_929b) <- ({m} of m T_929a -> of (h m) T_929b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_930","id":89929,"instance":930,"goal":"of_app_930 : of (app q_1 q_2) T_930 <- of q_1 (arrow T_930a T_930) <- of q_2 T_930a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_931","id":89930,"instance":931,"goal":"step_beta_931 : step (app (lam a) q) (a q).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_932","id":89931,"instance":932,"goal":"step_app1_932 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_933","id":89932,"instance":933,"goal":"preservation_933 : of m T_933 -> step m m_p -> of m_p T_933.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_934","id":89933,"instance":934,"goal":"progress_934 : of z T_934 -> value z or (exists z_p. step z z_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_935","id":89934,"instance":935,"goal":"eq_refl_935 : eq h h.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_936","id":89935,"instance":936,"goal":"subst_936 : subst a e (a e).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_937","id":89936,"instance":937,"goal":"lam_937 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_938","id":89937,"instance":938,"goal":"app_938 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_939","id":89938,"instance":939,"goal":"arrow_939 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_940","id":89939,"instance":940,"goal":"of_var_940 : of x T_940 <- of-env x T_940.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_941","id":89940,"instance":941,"goal":"of_lam_941 : of (lam p) (arrow T_941a T_941b) <- ({x} of x T_941a -> of (p x) T_941b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_942","id":89941,"instance":942,"goal":"of_app_942 : of (app x_1 x_2) T_942 <- of x_1 (arrow T_942a T_942) <- of x_2 T_942a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_943","id":89942,"instance":943,"goal":"step_beta_943 : step (app (lam c) f) (c f).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_944","id":89943,"instance":944,"goal":"step_app1_944 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_945","id":89944,"instance":945,"goal":"preservation_945 : of e T_945 -> step e e_p -> of e_p T_945.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_946","id":89945,"instance":946,"goal":"progress_946 : of f T_946 -> value f or (exists f_p. step f f_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_947","id":89946,"instance":947,"goal":"eq_refl_947 : eq m m.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_948","id":89947,"instance":948,"goal":"subst_948 : subst w x (w x).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_949","id":89948,"instance":949,"goal":"lam_949 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_950","id":89949,"instance":950,"goal":"app_950 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_951","id":89950,"instance":951,"goal":"arrow_951 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_952","id":89951,"instance":952,"goal":"of_var_952 : of p T_952 <- of-env p T_952.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_953","id":89952,"instance":953,"goal":"of_lam_953 : of (lam p) (arrow T_953a T_953b) <- ({b} of b T_953a -> of (p b) T_953b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_954","id":89953,"instance":954,"goal":"of_app_954 : of (app p_1 p_2) T_954 <- of p_1 (arrow T_954a T_954) <- of p_2 T_954a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_955","id":89954,"instance":955,"goal":"step_beta_955 : step (app (lam y) w) (y w).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_956","id":89955,"instance":956,"goal":"step_app1_956 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_957","id":89956,"instance":957,"goal":"preservation_957 : of u T_957 -> step u u_p -> of u_p T_957.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_958","id":89957,"instance":958,"goal":"progress_958 : of h T_958 -> value h or (exists h_p. step h h_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_959","id":89958,"instance":959,"goal":"eq_refl_959 : eq h h.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_960","id":89959,"instance":960,"goal":"subst_960 : subst d c (d c).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_961","id":89960,"instance":961,"goal":"lam_961 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_962","id":89961,"instance":962,"goal":"app_962 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_963","id":89962,"instance":963,"goal":"arrow_963 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_964","id":89963,"instance":964,"goal":"of_var_964 : of f T_964 <- of-env f T_964.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_965","id":89964,"instance":965,"goal":"of_lam_965 : of (lam z) (arrow T_965a T_965b) <- ({g} of g T_965a -> of (z g) T_965b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_966","id":89965,"instance":966,"goal":"of_app_966 : of (app u_1 u_2) T_966 <- of u_1 (arrow T_966a T_966) <- of u_2 T_966a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_967","id":89966,"instance":967,"goal":"step_beta_967 : step (app (lam c) v) (c v).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_968","id":89967,"instance":968,"goal":"step_app1_968 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_969","id":89968,"instance":969,"goal":"preservation_969 : of c T_969 -> step c c_p -> of c_p T_969.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_970","id":89969,"instance":970,"goal":"progress_970 : of a T_970 -> value a or (exists a_p. step a a_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_971","id":89970,"instance":971,"goal":"eq_refl_971 : eq v v.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_972","id":89971,"instance":972,"goal":"subst_972 : subst q p (q p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_973","id":89972,"instance":973,"goal":"lam_973 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_974","id":89973,"instance":974,"goal":"app_974 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_975","id":89974,"instance":975,"goal":"arrow_975 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_976","id":89975,"instance":976,"goal":"of_var_976 : of n T_976 <- of-env n T_976.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_977","id":89976,"instance":977,"goal":"of_lam_977 : of (lam p) (arrow T_977a T_977b) <- ({y} of y T_977a -> of (p y) T_977b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_978","id":89977,"instance":978,"goal":"of_app_978 : of (app h_1 h_2) T_978 <- of h_1 (arrow T_978a T_978) <- of h_2 T_978a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_979","id":89978,"instance":979,"goal":"step_beta_979 : step (app (lam q) v) (q v).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_980","id":89979,"instance":980,"goal":"step_app1_980 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_981","id":89980,"instance":981,"goal":"preservation_981 : of u T_981 -> step u u_p -> of u_p T_981.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_982","id":89981,"instance":982,"goal":"progress_982 : of u T_982 -> value u or (exists u_p. step u u_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_983","id":89982,"instance":983,"goal":"eq_refl_983 : eq g g.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_984","id":89983,"instance":984,"goal":"subst_984 : subst p h (p h).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_985","id":89984,"instance":985,"goal":"lam_985 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_986","id":89985,"instance":986,"goal":"app_986 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_987","id":89986,"instance":987,"goal":"arrow_987 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_988","id":89987,"instance":988,"goal":"of_var_988 : of q T_988 <- of-env q T_988.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_989","id":89988,"instance":989,"goal":"of_lam_989 : of (lam w) (arrow T_989a T_989b) <- ({b} of b T_989a -> of (w b) T_989b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_990","id":89989,"instance":990,"goal":"of_app_990 : of (app x_1 x_2) T_990 <- of x_1 (arrow T_990a T_990) <- of x_2 T_990a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_991","id":89990,"instance":991,"goal":"step_beta_991 : step (app (lam e) b) (e b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_992","id":89991,"instance":992,"goal":"step_app1_992 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_993","id":89992,"instance":993,"goal":"preservation_993 : of y T_993 -> step y y_p -> of y_p T_993.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_994","id":89993,"instance":994,"goal":"progress_994 : of b T_994 -> value b or (exists b_p. step b b_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_995","id":89994,"instance":995,"goal":"eq_refl_995 : eq c c.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_996","id":89995,"instance":996,"goal":"subst_996 : subst x q (x q).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_997","id":89996,"instance":997,"goal":"lam_997 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_998","id":89997,"instance":998,"goal":"app_998 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_999","id":89998,"instance":999,"goal":"arrow_999 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1000","id":89999,"instance":1000,"goal":"of_var_1000 : of q T_1000 <- of-env q T_1000.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1001","id":90000,"instance":1001,"goal":"of_lam_1001 : of (lam u) (arrow T_1001a T_1001b) <- ({z} of z T_1001a -> of (u z) T_1001b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1002","id":90001,"instance":1002,"goal":"of_app_1002 : of (app e_1 e_2) T_1002 <- of e_1 (arrow T_1002a T_1002) <- of e_2 T_1002a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1003","id":90002,"instance":1003,"goal":"step_beta_1003 : step (app (lam u) g) (u g).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1004","id":90003,"instance":1004,"goal":"step_app1_1004 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1005","id":90004,"instance":1005,"goal":"preservation_1005 : of u T_1005 -> step u u_p -> of u_p T_1005.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1006","id":90005,"instance":1006,"goal":"progress_1006 : of f T_1006 -> value f or (exists f_p. step f f_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1007","id":90006,"instance":1007,"goal":"eq_refl_1007 : eq u u.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1008","id":90007,"instance":1008,"goal":"subst_1008 : subst d h (d h).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1009","id":90008,"instance":1009,"goal":"lam_1009 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1010","id":90009,"instance":1010,"goal":"app_1010 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1011","id":90010,"instance":1011,"goal":"arrow_1011 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1012","id":90011,"instance":1012,"goal":"of_var_1012 : of z T_1012 <- of-env z T_1012.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1013","id":90012,"instance":1013,"goal":"of_lam_1013 : of (lam a) (arrow T_1013a T_1013b) <- ({d} of d T_1013a -> of (a d) T_1013b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1014","id":90013,"instance":1014,"goal":"of_app_1014 : of (app q_1 q_2) T_1014 <- of q_1 (arrow T_1014a T_1014) <- of q_2 T_1014a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1015","id":90014,"instance":1015,"goal":"step_beta_1015 : step (app (lam w) a) (w a).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1016","id":90015,"instance":1016,"goal":"step_app1_1016 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1017","id":90016,"instance":1017,"goal":"preservation_1017 : of w T_1017 -> step w w_p -> of w_p T_1017.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1018","id":90017,"instance":1018,"goal":"progress_1018 : of c T_1018 -> value c or (exists c_p. step c c_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1019","id":90018,"instance":1019,"goal":"eq_refl_1019 : eq f f.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1020","id":90019,"instance":1020,"goal":"subst_1020 : subst v h (v h).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1021","id":90020,"instance":1021,"goal":"lam_1021 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1022","id":90021,"instance":1022,"goal":"app_1022 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1023","id":90022,"instance":1023,"goal":"arrow_1023 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1024","id":90023,"instance":1024,"goal":"of_var_1024 : of u T_1024 <- of-env u T_1024.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1025","id":90024,"instance":1025,"goal":"of_lam_1025 : of (lam f) (arrow T_1025a T_1025b) <- ({x} of x T_1025a -> of (f x) T_1025b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1026","id":90025,"instance":1026,"goal":"of_app_1026 : of (app c_1 c_2) T_1026 <- of c_1 (arrow T_1026a T_1026) <- of c_2 T_1026a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1027","id":90026,"instance":1027,"goal":"step_beta_1027 : step (app (lam q) p) (q p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1028","id":90027,"instance":1028,"goal":"step_app1_1028 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1029","id":90028,"instance":1029,"goal":"preservation_1029 : of c T_1029 -> step c c_p -> of c_p T_1029.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1030","id":90029,"instance":1030,"goal":"progress_1030 : of c T_1030 -> value c or (exists c_p. step c c_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1031","id":90030,"instance":1031,"goal":"eq_refl_1031 : eq e e.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1032","id":90031,"instance":1032,"goal":"subst_1032 : subst n z (n z).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1033","id":90032,"instance":1033,"goal":"lam_1033 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1034","id":90033,"instance":1034,"goal":"app_1034 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1035","id":90034,"instance":1035,"goal":"arrow_1035 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1036","id":90035,"instance":1036,"goal":"of_var_1036 : of m T_1036 <- of-env m T_1036.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1037","id":90036,"instance":1037,"goal":"of_lam_1037 : of (lam y) (arrow T_1037a T_1037b) <- ({q} of q T_1037a -> of (y q) T_1037b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1038","id":90037,"instance":1038,"goal":"of_app_1038 : of (app z_1 z_2) T_1038 <- of z_1 (arrow T_1038a T_1038) <- of z_2 T_1038a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1039","id":90038,"instance":1039,"goal":"step_beta_1039 : step (app (lam b) v) (b v).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1040","id":90039,"instance":1040,"goal":"step_app1_1040 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1041","id":90040,"instance":1041,"goal":"preservation_1041 : of e T_1041 -> step e e_p -> of e_p T_1041.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1042","id":90041,"instance":1042,"goal":"progress_1042 : of x T_1042 -> value x or (exists x_p. step x x_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1043","id":90042,"instance":1043,"goal":"eq_refl_1043 : eq f f.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1044","id":90043,"instance":1044,"goal":"subst_1044 : subst z d (z d).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1045","id":90044,"instance":1045,"goal":"lam_1045 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1046","id":90045,"instance":1046,"goal":"app_1046 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1047","id":90046,"instance":1047,"goal":"arrow_1047 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1048","id":90047,"instance":1048,"goal":"of_var_1048 : of c T_1048 <- of-env c T_1048.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1049","id":90048,"instance":1049,"goal":"of_lam_1049 : of (lam m) (arrow T_1049a T_1049b) <- ({v} of v T_1049a -> of (m v) T_1049b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1050","id":90049,"instance":1050,"goal":"of_app_1050 : of (app e_1 e_2) T_1050 <- of e_1 (arrow T_1050a T_1050) <- of e_2 T_1050a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1051","id":90050,"instance":1051,"goal":"step_beta_1051 : step (app (lam h) d) (h d).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1052","id":90051,"instance":1052,"goal":"step_app1_1052 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1053","id":90052,"instance":1053,"goal":"preservation_1053 : of x T_1053 -> step x x_p -> of x_p T_1053.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1054","id":90053,"instance":1054,"goal":"progress_1054 : of n T_1054 -> value n or (exists n_p. step n n_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1055","id":90054,"instance":1055,"goal":"eq_refl_1055 : eq a a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1056","id":90055,"instance":1056,"goal":"subst_1056 : subst b a (b a).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1057","id":90056,"instance":1057,"goal":"lam_1057 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1058","id":90057,"instance":1058,"goal":"app_1058 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1059","id":90058,"instance":1059,"goal":"arrow_1059 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1060","id":90059,"instance":1060,"goal":"of_var_1060 : of e T_1060 <- of-env e T_1060.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1061","id":90060,"instance":1061,"goal":"of_lam_1061 : of (lam p) (arrow T_1061a T_1061b) <- ({h} of h T_1061a -> of (p h) T_1061b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1062","id":90061,"instance":1062,"goal":"of_app_1062 : of (app a_1 a_2) T_1062 <- of a_1 (arrow T_1062a T_1062) <- of a_2 T_1062a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1063","id":90062,"instance":1063,"goal":"step_beta_1063 : step (app (lam y) n) (y n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1064","id":90063,"instance":1064,"goal":"step_app1_1064 : step (app w_1 w_2) (app w_1p w_2) <- step w_1 w_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1065","id":90064,"instance":1065,"goal":"preservation_1065 : of b T_1065 -> step b b_p -> of b_p T_1065.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1066","id":90065,"instance":1066,"goal":"progress_1066 : of a T_1066 -> value a or (exists a_p. step a a_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1067","id":90066,"instance":1067,"goal":"eq_refl_1067 : eq y y.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1068","id":90067,"instance":1068,"goal":"subst_1068 : subst y g (y g).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1069","id":90068,"instance":1069,"goal":"lam_1069 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1070","id":90069,"instance":1070,"goal":"app_1070 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1071","id":90070,"instance":1071,"goal":"arrow_1071 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1072","id":90071,"instance":1072,"goal":"of_var_1072 : of y T_1072 <- of-env y T_1072.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1073","id":90072,"instance":1073,"goal":"of_lam_1073 : of (lam m) (arrow T_1073a T_1073b) <- ({u} of u T_1073a -> of (m u) T_1073b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1074","id":90073,"instance":1074,"goal":"of_app_1074 : of (app b_1 b_2) T_1074 <- of b_1 (arrow T_1074a T_1074) <- of b_2 T_1074a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1075","id":90074,"instance":1075,"goal":"step_beta_1075 : step (app (lam a) w) (a w).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1076","id":90075,"instance":1076,"goal":"step_app1_1076 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1077","id":90076,"instance":1077,"goal":"preservation_1077 : of h T_1077 -> step h h_p -> of h_p T_1077.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1078","id":90077,"instance":1078,"goal":"progress_1078 : of g T_1078 -> value g or (exists g_p. step g g_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1079","id":90078,"instance":1079,"goal":"eq_refl_1079 : eq v v.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1080","id":90079,"instance":1080,"goal":"subst_1080 : subst p u (p u).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1081","id":90080,"instance":1081,"goal":"lam_1081 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1082","id":90081,"instance":1082,"goal":"app_1082 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1083","id":90082,"instance":1083,"goal":"arrow_1083 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1084","id":90083,"instance":1084,"goal":"of_var_1084 : of a T_1084 <- of-env a T_1084.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1085","id":90084,"instance":1085,"goal":"of_lam_1085 : of (lam z) (arrow T_1085a T_1085b) <- ({f} of f T_1085a -> of (z f) T_1085b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1086","id":90085,"instance":1086,"goal":"of_app_1086 : of (app w_1 w_2) T_1086 <- of w_1 (arrow T_1086a T_1086) <- of w_2 T_1086a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1087","id":90086,"instance":1087,"goal":"step_beta_1087 : step (app (lam c) b) (c b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1088","id":90087,"instance":1088,"goal":"step_app1_1088 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1089","id":90088,"instance":1089,"goal":"preservation_1089 : of y T_1089 -> step y y_p -> of y_p T_1089.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1090","id":90089,"instance":1090,"goal":"progress_1090 : of v T_1090 -> value v or (exists v_p. step v v_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1091","id":90090,"instance":1091,"goal":"eq_refl_1091 : eq m m.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1092","id":90091,"instance":1092,"goal":"subst_1092 : subst p y (p y).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1093","id":90092,"instance":1093,"goal":"lam_1093 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1094","id":90093,"instance":1094,"goal":"app_1094 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1095","id":90094,"instance":1095,"goal":"arrow_1095 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1096","id":90095,"instance":1096,"goal":"of_var_1096 : of q T_1096 <- of-env q T_1096.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1097","id":90096,"instance":1097,"goal":"of_lam_1097 : of (lam f) (arrow T_1097a T_1097b) <- ({b} of b T_1097a -> of (f b) T_1097b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1098","id":90097,"instance":1098,"goal":"of_app_1098 : of (app n_1 n_2) T_1098 <- of n_1 (arrow T_1098a T_1098) <- of n_2 T_1098a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1099","id":90098,"instance":1099,"goal":"step_beta_1099 : step (app (lam g) e) (g e).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1100","id":90099,"instance":1100,"goal":"step_app1_1100 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1101","id":90100,"instance":1101,"goal":"preservation_1101 : of z T_1101 -> step z z_p -> of z_p T_1101.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1102","id":90101,"instance":1102,"goal":"progress_1102 : of u T_1102 -> value u or (exists u_p. step u u_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1103","id":90102,"instance":1103,"goal":"eq_refl_1103 : eq q q.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1104","id":90103,"instance":1104,"goal":"subst_1104 : subst g u (g u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1105","id":90104,"instance":1105,"goal":"lam_1105 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1106","id":90105,"instance":1106,"goal":"app_1106 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1107","id":90106,"instance":1107,"goal":"arrow_1107 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1108","id":90107,"instance":1108,"goal":"of_var_1108 : of n T_1108 <- of-env n T_1108.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1109","id":90108,"instance":1109,"goal":"of_lam_1109 : of (lam n) (arrow T_1109a T_1109b) <- ({a} of a T_1109a -> of (n a) T_1109b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1110","id":90109,"instance":1110,"goal":"of_app_1110 : of (app z_1 z_2) T_1110 <- of z_1 (arrow T_1110a T_1110) <- of z_2 T_1110a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1111","id":90110,"instance":1111,"goal":"step_beta_1111 : step (app (lam v) a) (v a).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1112","id":90111,"instance":1112,"goal":"step_app1_1112 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1113","id":90112,"instance":1113,"goal":"preservation_1113 : of y T_1113 -> step y y_p -> of y_p T_1113.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1114","id":90113,"instance":1114,"goal":"progress_1114 : of a T_1114 -> value a or (exists a_p. step a a_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1115","id":90114,"instance":1115,"goal":"eq_refl_1115 : eq b b.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1116","id":90115,"instance":1116,"goal":"subst_1116 : subst c q (c q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1117","id":90116,"instance":1117,"goal":"lam_1117 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1118","id":90117,"instance":1118,"goal":"app_1118 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1119","id":90118,"instance":1119,"goal":"arrow_1119 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1120","id":90119,"instance":1120,"goal":"of_var_1120 : of z T_1120 <- of-env z T_1120.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1121","id":90120,"instance":1121,"goal":"of_lam_1121 : of (lam f) (arrow T_1121a T_1121b) <- ({h} of h T_1121a -> of (f h) T_1121b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1122","id":90121,"instance":1122,"goal":"of_app_1122 : of (app c_1 c_2) T_1122 <- of c_1 (arrow T_1122a T_1122) <- of c_2 T_1122a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1123","id":90122,"instance":1123,"goal":"step_beta_1123 : step (app (lam y) f) (y f).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1124","id":90123,"instance":1124,"goal":"step_app1_1124 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1125","id":90124,"instance":1125,"goal":"preservation_1125 : of n T_1125 -> step n n_p -> of n_p T_1125.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1126","id":90125,"instance":1126,"goal":"progress_1126 : of p T_1126 -> value p or (exists p_p. step p p_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1127","id":90126,"instance":1127,"goal":"eq_refl_1127 : eq e e.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1128","id":90127,"instance":1128,"goal":"subst_1128 : subst m p (m p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1129","id":90128,"instance":1129,"goal":"lam_1129 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1130","id":90129,"instance":1130,"goal":"app_1130 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1131","id":90130,"instance":1131,"goal":"arrow_1131 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1132","id":90131,"instance":1132,"goal":"of_var_1132 : of g T_1132 <- of-env g T_1132.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1133","id":90132,"instance":1133,"goal":"of_lam_1133 : of (lam v) (arrow T_1133a T_1133b) <- ({z} of z T_1133a -> of (v z) T_1133b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1134","id":90133,"instance":1134,"goal":"of_app_1134 : of (app q_1 q_2) T_1134 <- of q_1 (arrow T_1134a T_1134) <- of q_2 T_1134a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1135","id":90134,"instance":1135,"goal":"step_beta_1135 : step (app (lam q) g) (q g).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1136","id":90135,"instance":1136,"goal":"step_app1_1136 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1137","id":90136,"instance":1137,"goal":"preservation_1137 : of d T_1137 -> step d d_p -> of d_p T_1137.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1138","id":90137,"instance":1138,"goal":"progress_1138 : of x T_1138 -> value x or (exists x_p. step x x_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1139","id":90138,"instance":1139,"goal":"eq_refl_1139 : eq y y.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1140","id":90139,"instance":1140,"goal":"subst_1140 : subst m y (m y).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1141","id":90140,"instance":1141,"goal":"lam_1141 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1142","id":90141,"instance":1142,"goal":"app_1142 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1143","id":90142,"instance":1143,"goal":"arrow_1143 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1144","id":90143,"instance":1144,"goal":"of_var_1144 : of x T_1144 <- of-env x T_1144.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1145","id":90144,"instance":1145,"goal":"of_lam_1145 : of (lam c) (arrow T_1145a T_1145b) <- ({g} of g T_1145a -> of (c g) T_1145b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1146","id":90145,"instance":1146,"goal":"of_app_1146 : of (app v_1 v_2) T_1146 <- of v_1 (arrow T_1146a T_1146) <- of v_2 T_1146a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1147","id":90146,"instance":1147,"goal":"step_beta_1147 : step (app (lam e) q) (e q).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1148","id":90147,"instance":1148,"goal":"step_app1_1148 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1149","id":90148,"instance":1149,"goal":"preservation_1149 : of x T_1149 -> step x x_p -> of x_p T_1149.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1150","id":90149,"instance":1150,"goal":"progress_1150 : of w T_1150 -> value w or (exists w_p. step w w_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1151","id":90150,"instance":1151,"goal":"eq_refl_1151 : eq v v.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1152","id":90151,"instance":1152,"goal":"subst_1152 : subst b z (b z).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1153","id":90152,"instance":1153,"goal":"lam_1153 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1154","id":90153,"instance":1154,"goal":"app_1154 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1155","id":90154,"instance":1155,"goal":"arrow_1155 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1156","id":90155,"instance":1156,"goal":"of_var_1156 : of y T_1156 <- of-env y T_1156.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1157","id":90156,"instance":1157,"goal":"of_lam_1157 : of (lam g) (arrow T_1157a T_1157b) <- ({n} of n T_1157a -> of (g n) T_1157b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1158","id":90157,"instance":1158,"goal":"of_app_1158 : of (app a_1 a_2) T_1158 <- of a_1 (arrow T_1158a T_1158) <- of a_2 T_1158a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1159","id":90158,"instance":1159,"goal":"step_beta_1159 : step (app (lam c) n) (c n).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1160","id":90159,"instance":1160,"goal":"step_app1_1160 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1161","id":90160,"instance":1161,"goal":"preservation_1161 : of n T_1161 -> step n n_p -> of n_p T_1161.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1162","id":90161,"instance":1162,"goal":"progress_1162 : of d T_1162 -> value d or (exists d_p. step d d_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1163","id":90162,"instance":1163,"goal":"eq_refl_1163 : eq y y.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1164","id":90163,"instance":1164,"goal":"subst_1164 : subst e v (e v).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1165","id":90164,"instance":1165,"goal":"lam_1165 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1166","id":90165,"instance":1166,"goal":"app_1166 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1167","id":90166,"instance":1167,"goal":"arrow_1167 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1168","id":90167,"instance":1168,"goal":"of_var_1168 : of c T_1168 <- of-env c T_1168.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1169","id":90168,"instance":1169,"goal":"of_lam_1169 : of (lam q) (arrow T_1169a T_1169b) <- ({e} of e T_1169a -> of (q e) T_1169b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1170","id":90169,"instance":1170,"goal":"of_app_1170 : of (app f_1 f_2) T_1170 <- of f_1 (arrow T_1170a T_1170) <- of f_2 T_1170a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1171","id":90170,"instance":1171,"goal":"step_beta_1171 : step (app (lam c) w) (c w).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1172","id":90171,"instance":1172,"goal":"step_app1_1172 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1173","id":90172,"instance":1173,"goal":"preservation_1173 : of v T_1173 -> step v v_p -> of v_p T_1173.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1174","id":90173,"instance":1174,"goal":"progress_1174 : of y T_1174 -> value y or (exists y_p. step y y_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1175","id":90174,"instance":1175,"goal":"eq_refl_1175 : eq m m.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1176","id":90175,"instance":1176,"goal":"subst_1176 : subst q b (q b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1177","id":90176,"instance":1177,"goal":"lam_1177 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1178","id":90177,"instance":1178,"goal":"app_1178 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1179","id":90178,"instance":1179,"goal":"arrow_1179 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1180","id":90179,"instance":1180,"goal":"of_var_1180 : of q T_1180 <- of-env q T_1180.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1181","id":90180,"instance":1181,"goal":"of_lam_1181 : of (lam d) (arrow T_1181a T_1181b) <- ({u} of u T_1181a -> of (d u) T_1181b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1182","id":90181,"instance":1182,"goal":"of_app_1182 : of (app m_1 m_2) T_1182 <- of m_1 (arrow T_1182a T_1182) <- of m_2 T_1182a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1183","id":90182,"instance":1183,"goal":"step_beta_1183 : step (app (lam f) c) (f c).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1184","id":90183,"instance":1184,"goal":"step_app1_1184 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1185","id":90184,"instance":1185,"goal":"preservation_1185 : of v T_1185 -> step v v_p -> of v_p T_1185.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1186","id":90185,"instance":1186,"goal":"progress_1186 : of e T_1186 -> value e or (exists e_p. step e e_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1187","id":90186,"instance":1187,"goal":"eq_refl_1187 : eq y y.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1188","id":90187,"instance":1188,"goal":"subst_1188 : subst d w (d w).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1189","id":90188,"instance":1189,"goal":"lam_1189 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1190","id":90189,"instance":1190,"goal":"app_1190 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1191","id":90190,"instance":1191,"goal":"arrow_1191 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1192","id":90191,"instance":1192,"goal":"of_var_1192 : of p T_1192 <- of-env p T_1192.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1193","id":90192,"instance":1193,"goal":"of_lam_1193 : of (lam b) (arrow T_1193a T_1193b) <- ({u} of u T_1193a -> of (b u) T_1193b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1194","id":90193,"instance":1194,"goal":"of_app_1194 : of (app b_1 b_2) T_1194 <- of b_1 (arrow T_1194a T_1194) <- of b_2 T_1194a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1195","id":90194,"instance":1195,"goal":"step_beta_1195 : step (app (lam d) y) (d y).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1196","id":90195,"instance":1196,"goal":"step_app1_1196 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1197","id":90196,"instance":1197,"goal":"preservation_1197 : of b T_1197 -> step b b_p -> of b_p T_1197.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1198","id":90197,"instance":1198,"goal":"progress_1198 : of q T_1198 -> value q or (exists q_p. step q q_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1199","id":90198,"instance":1199,"goal":"eq_refl_1199 : eq n n.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1200","id":90199,"instance":1200,"goal":"subst_1200 : subst h p (h p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1201","id":90200,"instance":1201,"goal":"lam_1201 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1202","id":90201,"instance":1202,"goal":"app_1202 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1203","id":90202,"instance":1203,"goal":"arrow_1203 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1204","id":90203,"instance":1204,"goal":"of_var_1204 : of y T_1204 <- of-env y T_1204.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1205","id":90204,"instance":1205,"goal":"of_lam_1205 : of (lam n) (arrow T_1205a T_1205b) <- ({z} of z T_1205a -> of (n z) T_1205b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1206","id":90205,"instance":1206,"goal":"of_app_1206 : of (app g_1 g_2) T_1206 <- of g_1 (arrow T_1206a T_1206) <- of g_2 T_1206a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1207","id":90206,"instance":1207,"goal":"step_beta_1207 : step (app (lam g) e) (g e).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1208","id":90207,"instance":1208,"goal":"step_app1_1208 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1209","id":90208,"instance":1209,"goal":"preservation_1209 : of d T_1209 -> step d d_p -> of d_p T_1209.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1210","id":90209,"instance":1210,"goal":"progress_1210 : of v T_1210 -> value v or (exists v_p. step v v_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1211","id":90210,"instance":1211,"goal":"eq_refl_1211 : eq f f.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1212","id":90211,"instance":1212,"goal":"subst_1212 : subst z d (z d).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1213","id":90212,"instance":1213,"goal":"lam_1213 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1214","id":90213,"instance":1214,"goal":"app_1214 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1215","id":90214,"instance":1215,"goal":"arrow_1215 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1216","id":90215,"instance":1216,"goal":"of_var_1216 : of q T_1216 <- of-env q T_1216.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1217","id":90216,"instance":1217,"goal":"of_lam_1217 : of (lam c) (arrow T_1217a T_1217b) <- ({d} of d T_1217a -> of (c d) T_1217b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1218","id":90217,"instance":1218,"goal":"of_app_1218 : of (app m_1 m_2) T_1218 <- of m_1 (arrow T_1218a T_1218) <- of m_2 T_1218a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1219","id":90218,"instance":1219,"goal":"step_beta_1219 : step (app (lam n) w) (n w).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1220","id":90219,"instance":1220,"goal":"step_app1_1220 : step (app w_1 w_2) (app w_1p w_2) <- step w_1 w_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1221","id":90220,"instance":1221,"goal":"preservation_1221 : of e T_1221 -> step e e_p -> of e_p T_1221.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1222","id":90221,"instance":1222,"goal":"progress_1222 : of m T_1222 -> value m or (exists m_p. step m m_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1223","id":90222,"instance":1223,"goal":"eq_refl_1223 : eq e e.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1224","id":90223,"instance":1224,"goal":"subst_1224 : subst f e (f e).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1225","id":90224,"instance":1225,"goal":"lam_1225 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1226","id":90225,"instance":1226,"goal":"app_1226 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1227","id":90226,"instance":1227,"goal":"arrow_1227 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1228","id":90227,"instance":1228,"goal":"of_var_1228 : of v T_1228 <- of-env v T_1228.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1229","id":90228,"instance":1229,"goal":"of_lam_1229 : of (lam p) (arrow T_1229a T_1229b) <- ({x} of x T_1229a -> of (p x) T_1229b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1230","id":90229,"instance":1230,"goal":"of_app_1230 : of (app u_1 u_2) T_1230 <- of u_1 (arrow T_1230a T_1230) <- of u_2 T_1230a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1231","id":90230,"instance":1231,"goal":"step_beta_1231 : step (app (lam p) z) (p z).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1232","id":90231,"instance":1232,"goal":"step_app1_1232 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1233","id":90232,"instance":1233,"goal":"preservation_1233 : of w T_1233 -> step w w_p -> of w_p T_1233.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1234","id":90233,"instance":1234,"goal":"progress_1234 : of c T_1234 -> value c or (exists c_p. step c c_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1235","id":90234,"instance":1235,"goal":"eq_refl_1235 : eq v v.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1236","id":90235,"instance":1236,"goal":"subst_1236 : subst n u (n u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1237","id":90236,"instance":1237,"goal":"lam_1237 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1238","id":90237,"instance":1238,"goal":"app_1238 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1239","id":90238,"instance":1239,"goal":"arrow_1239 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1240","id":90239,"instance":1240,"goal":"of_var_1240 : of q T_1240 <- of-env q T_1240.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1241","id":90240,"instance":1241,"goal":"of_lam_1241 : of (lam x) (arrow T_1241a T_1241b) <- ({h} of h T_1241a -> of (x h) T_1241b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1242","id":90241,"instance":1242,"goal":"of_app_1242 : of (app p_1 p_2) T_1242 <- of p_1 (arrow T_1242a T_1242) <- of p_2 T_1242a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1243","id":90242,"instance":1243,"goal":"step_beta_1243 : step (app (lam q) a) (q a).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1244","id":90243,"instance":1244,"goal":"step_app1_1244 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1245","id":90244,"instance":1245,"goal":"preservation_1245 : of d T_1245 -> step d d_p -> of d_p T_1245.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1246","id":90245,"instance":1246,"goal":"progress_1246 : of x T_1246 -> value x or (exists x_p. step x x_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1247","id":90246,"instance":1247,"goal":"eq_refl_1247 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1248","id":90247,"instance":1248,"goal":"subst_1248 : subst w g (w g).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1249","id":90248,"instance":1249,"goal":"lam_1249 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1250","id":90249,"instance":1250,"goal":"app_1250 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1251","id":90250,"instance":1251,"goal":"arrow_1251 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1252","id":90251,"instance":1252,"goal":"of_var_1252 : of z T_1252 <- of-env z T_1252.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1253","id":90252,"instance":1253,"goal":"of_lam_1253 : of (lam z) (arrow T_1253a T_1253b) <- ({y} of y T_1253a -> of (z y) T_1253b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1254","id":90253,"instance":1254,"goal":"of_app_1254 : of (app a_1 a_2) T_1254 <- of a_1 (arrow T_1254a T_1254) <- of a_2 T_1254a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1255","id":90254,"instance":1255,"goal":"step_beta_1255 : step (app (lam q) x) (q x).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1256","id":90255,"instance":1256,"goal":"step_app1_1256 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1257","id":90256,"instance":1257,"goal":"preservation_1257 : of b T_1257 -> step b b_p -> of b_p T_1257.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1258","id":90257,"instance":1258,"goal":"progress_1258 : of e T_1258 -> value e or (exists e_p. step e e_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1259","id":90258,"instance":1259,"goal":"eq_refl_1259 : eq q q.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1260","id":90259,"instance":1260,"goal":"subst_1260 : subst n d (n d).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1261","id":90260,"instance":1261,"goal":"lam_1261 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1262","id":90261,"instance":1262,"goal":"app_1262 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1263","id":90262,"instance":1263,"goal":"arrow_1263 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1264","id":90263,"instance":1264,"goal":"of_var_1264 : of c T_1264 <- of-env c T_1264.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1265","id":90264,"instance":1265,"goal":"of_lam_1265 : of (lam z) (arrow T_1265a T_1265b) <- ({m} of m T_1265a -> of (z m) T_1265b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1266","id":90265,"instance":1266,"goal":"of_app_1266 : of (app c_1 c_2) T_1266 <- of c_1 (arrow T_1266a T_1266) <- of c_2 T_1266a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1267","id":90266,"instance":1267,"goal":"step_beta_1267 : step (app (lam g) e) (g e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1268","id":90267,"instance":1268,"goal":"step_app1_1268 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1269","id":90268,"instance":1269,"goal":"preservation_1269 : of c T_1269 -> step c c_p -> of c_p T_1269.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1270","id":90269,"instance":1270,"goal":"progress_1270 : of d T_1270 -> value d or (exists d_p. step d d_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1271","id":90270,"instance":1271,"goal":"eq_refl_1271 : eq v v.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1272","id":90271,"instance":1272,"goal":"subst_1272 : subst z q (z q).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1273","id":90272,"instance":1273,"goal":"lam_1273 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1274","id":90273,"instance":1274,"goal":"app_1274 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1275","id":90274,"instance":1275,"goal":"arrow_1275 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1276","id":90275,"instance":1276,"goal":"of_var_1276 : of x T_1276 <- of-env x T_1276.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1277","id":90276,"instance":1277,"goal":"of_lam_1277 : of (lam d) (arrow T_1277a T_1277b) <- ({z} of z T_1277a -> of (d z) T_1277b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1278","id":90277,"instance":1278,"goal":"of_app_1278 : of (app n_1 n_2) T_1278 <- of n_1 (arrow T_1278a T_1278) <- of n_2 T_1278a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1279","id":90278,"instance":1279,"goal":"step_beta_1279 : step (app (lam w) z) (w z).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1280","id":90279,"instance":1280,"goal":"step_app1_1280 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1281","id":90280,"instance":1281,"goal":"preservation_1281 : of b T_1281 -> step b b_p -> of b_p T_1281.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1282","id":90281,"instance":1282,"goal":"progress_1282 : of e T_1282 -> value e or (exists e_p. step e e_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1283","id":90282,"instance":1283,"goal":"eq_refl_1283 : eq a a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1284","id":90283,"instance":1284,"goal":"subst_1284 : subst y m (y m).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1285","id":90284,"instance":1285,"goal":"lam_1285 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1286","id":90285,"instance":1286,"goal":"app_1286 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1287","id":90286,"instance":1287,"goal":"arrow_1287 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1288","id":90287,"instance":1288,"goal":"of_var_1288 : of f T_1288 <- of-env f T_1288.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1289","id":90288,"instance":1289,"goal":"of_lam_1289 : of (lam g) (arrow T_1289a T_1289b) <- ({w} of w T_1289a -> of (g w) T_1289b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1290","id":90289,"instance":1290,"goal":"of_app_1290 : of (app y_1 y_2) T_1290 <- of y_1 (arrow T_1290a T_1290) <- of y_2 T_1290a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1291","id":90290,"instance":1291,"goal":"step_beta_1291 : step (app (lam f) n) (f n).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1292","id":90291,"instance":1292,"goal":"step_app1_1292 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1293","id":90292,"instance":1293,"goal":"preservation_1293 : of e T_1293 -> step e e_p -> of e_p T_1293.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1294","id":90293,"instance":1294,"goal":"progress_1294 : of d T_1294 -> value d or (exists d_p. step d d_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1295","id":90294,"instance":1295,"goal":"eq_refl_1295 : eq n n.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1296","id":90295,"instance":1296,"goal":"subst_1296 : subst h d (h d).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1297","id":90296,"instance":1297,"goal":"lam_1297 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1298","id":90297,"instance":1298,"goal":"app_1298 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1299","id":90298,"instance":1299,"goal":"arrow_1299 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1300","id":90299,"instance":1300,"goal":"of_var_1300 : of v T_1300 <- of-env v T_1300.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1301","id":90300,"instance":1301,"goal":"of_lam_1301 : of (lam d) (arrow T_1301a T_1301b) <- ({w} of w T_1301a -> of (d w) T_1301b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1302","id":90301,"instance":1302,"goal":"of_app_1302 : of (app x_1 x_2) T_1302 <- of x_1 (arrow T_1302a T_1302) <- of x_2 T_1302a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1303","id":90302,"instance":1303,"goal":"step_beta_1303 : step (app (lam v) h) (v h).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1304","id":90303,"instance":1304,"goal":"step_app1_1304 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1305","id":90304,"instance":1305,"goal":"preservation_1305 : of h T_1305 -> step h h_p -> of h_p T_1305.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1306","id":90305,"instance":1306,"goal":"progress_1306 : of d T_1306 -> value d or (exists d_p. step d d_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1307","id":90306,"instance":1307,"goal":"eq_refl_1307 : eq g g.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1308","id":90307,"instance":1308,"goal":"subst_1308 : subst n u (n u).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1309","id":90308,"instance":1309,"goal":"lam_1309 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1310","id":90309,"instance":1310,"goal":"app_1310 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1311","id":90310,"instance":1311,"goal":"arrow_1311 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1312","id":90311,"instance":1312,"goal":"of_var_1312 : of n T_1312 <- of-env n T_1312.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1313","id":90312,"instance":1313,"goal":"of_lam_1313 : of (lam d) (arrow T_1313a T_1313b) <- ({c} of c T_1313a -> of (d c) T_1313b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1314","id":90313,"instance":1314,"goal":"of_app_1314 : of (app w_1 w_2) T_1314 <- of w_1 (arrow T_1314a T_1314) <- of w_2 T_1314a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1315","id":90314,"instance":1315,"goal":"step_beta_1315 : step (app (lam a) x) (a x).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1316","id":90315,"instance":1316,"goal":"step_app1_1316 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1317","id":90316,"instance":1317,"goal":"preservation_1317 : of q T_1317 -> step q q_p -> of q_p T_1317.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1318","id":90317,"instance":1318,"goal":"progress_1318 : of q T_1318 -> value q or (exists q_p. step q q_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1319","id":90318,"instance":1319,"goal":"eq_refl_1319 : eq y y.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1320","id":90319,"instance":1320,"goal":"subst_1320 : subst c f (c f).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1321","id":90320,"instance":1321,"goal":"lam_1321 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1322","id":90321,"instance":1322,"goal":"app_1322 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1323","id":90322,"instance":1323,"goal":"arrow_1323 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1324","id":90323,"instance":1324,"goal":"of_var_1324 : of g T_1324 <- of-env g T_1324.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1325","id":90324,"instance":1325,"goal":"of_lam_1325 : of (lam x) (arrow T_1325a T_1325b) <- ({m} of m T_1325a -> of (x m) T_1325b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1326","id":90325,"instance":1326,"goal":"of_app_1326 : of (app v_1 v_2) T_1326 <- of v_1 (arrow T_1326a T_1326) <- of v_2 T_1326a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1327","id":90326,"instance":1327,"goal":"step_beta_1327 : step (app (lam m) z) (m z).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1328","id":90327,"instance":1328,"goal":"step_app1_1328 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1329","id":90328,"instance":1329,"goal":"preservation_1329 : of x T_1329 -> step x x_p -> of x_p T_1329.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1330","id":90329,"instance":1330,"goal":"progress_1330 : of y T_1330 -> value y or (exists y_p. step y y_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1331","id":90330,"instance":1331,"goal":"eq_refl_1331 : eq p p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1332","id":90331,"instance":1332,"goal":"subst_1332 : subst h u (h u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1333","id":90332,"instance":1333,"goal":"lam_1333 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1334","id":90333,"instance":1334,"goal":"app_1334 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1335","id":90334,"instance":1335,"goal":"arrow_1335 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1336","id":90335,"instance":1336,"goal":"of_var_1336 : of a T_1336 <- of-env a T_1336.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1337","id":90336,"instance":1337,"goal":"of_lam_1337 : of (lam f) (arrow T_1337a T_1337b) <- ({u} of u T_1337a -> of (f u) T_1337b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1338","id":90337,"instance":1338,"goal":"of_app_1338 : of (app m_1 m_2) T_1338 <- of m_1 (arrow T_1338a T_1338) <- of m_2 T_1338a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1339","id":90338,"instance":1339,"goal":"step_beta_1339 : step (app (lam u) w) (u w).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1340","id":90339,"instance":1340,"goal":"step_app1_1340 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1341","id":90340,"instance":1341,"goal":"preservation_1341 : of d T_1341 -> step d d_p -> of d_p T_1341.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1342","id":90341,"instance":1342,"goal":"progress_1342 : of p T_1342 -> value p or (exists p_p. step p p_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1343","id":90342,"instance":1343,"goal":"eq_refl_1343 : eq y y.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1344","id":90343,"instance":1344,"goal":"subst_1344 : subst b e (b e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1345","id":90344,"instance":1345,"goal":"lam_1345 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1346","id":90345,"instance":1346,"goal":"app_1346 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1347","id":90346,"instance":1347,"goal":"arrow_1347 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1348","id":90347,"instance":1348,"goal":"of_var_1348 : of h T_1348 <- of-env h T_1348.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1349","id":90348,"instance":1349,"goal":"of_lam_1349 : of (lam p) (arrow T_1349a T_1349b) <- ({n} of n T_1349a -> of (p n) T_1349b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1350","id":90349,"instance":1350,"goal":"of_app_1350 : of (app c_1 c_2) T_1350 <- of c_1 (arrow T_1350a T_1350) <- of c_2 T_1350a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1351","id":90350,"instance":1351,"goal":"step_beta_1351 : step (app (lam f) x) (f x).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1352","id":90351,"instance":1352,"goal":"step_app1_1352 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1353","id":90352,"instance":1353,"goal":"preservation_1353 : of b T_1353 -> step b b_p -> of b_p T_1353.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1354","id":90353,"instance":1354,"goal":"progress_1354 : of u T_1354 -> value u or (exists u_p. step u u_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1355","id":90354,"instance":1355,"goal":"eq_refl_1355 : eq g g.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1356","id":90355,"instance":1356,"goal":"subst_1356 : subst c z (c z).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1357","id":90356,"instance":1357,"goal":"lam_1357 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1358","id":90357,"instance":1358,"goal":"app_1358 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1359","id":90358,"instance":1359,"goal":"arrow_1359 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1360","id":90359,"instance":1360,"goal":"of_var_1360 : of m T_1360 <- of-env m T_1360.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1361","id":90360,"instance":1361,"goal":"of_lam_1361 : of (lam x) (arrow T_1361a T_1361b) <- ({b} of b T_1361a -> of (x b) T_1361b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1362","id":90361,"instance":1362,"goal":"of_app_1362 : of (app p_1 p_2) T_1362 <- of p_1 (arrow T_1362a T_1362) <- of p_2 T_1362a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1363","id":90362,"instance":1363,"goal":"step_beta_1363 : step (app (lam q) h) (q h).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1364","id":90363,"instance":1364,"goal":"step_app1_1364 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1365","id":90364,"instance":1365,"goal":"preservation_1365 : of n T_1365 -> step n n_p -> of n_p T_1365.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1366","id":90365,"instance":1366,"goal":"progress_1366 : of a T_1366 -> value a or (exists a_p. step a a_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1367","id":90366,"instance":1367,"goal":"eq_refl_1367 : eq g g.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1368","id":90367,"instance":1368,"goal":"subst_1368 : subst a m (a m).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1369","id":90368,"instance":1369,"goal":"lam_1369 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1370","id":90369,"instance":1370,"goal":"app_1370 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1371","id":90370,"instance":1371,"goal":"arrow_1371 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1372","id":90371,"instance":1372,"goal":"of_var_1372 : of q T_1372 <- of-env q T_1372.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1373","id":90372,"instance":1373,"goal":"of_lam_1373 : of (lam b) (arrow T_1373a T_1373b) <- ({f} of f T_1373a -> of (b f) T_1373b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1374","id":90373,"instance":1374,"goal":"of_app_1374 : of (app v_1 v_2) T_1374 <- of v_1 (arrow T_1374a T_1374) <- of v_2 T_1374a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1375","id":90374,"instance":1375,"goal":"step_beta_1375 : step (app (lam f) y) (f y).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1376","id":90375,"instance":1376,"goal":"step_app1_1376 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1377","id":90376,"instance":1377,"goal":"preservation_1377 : of f T_1377 -> step f f_p -> of f_p T_1377.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1378","id":90377,"instance":1378,"goal":"progress_1378 : of w T_1378 -> value w or (exists w_p. step w w_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1379","id":90378,"instance":1379,"goal":"eq_refl_1379 : eq q q.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1380","id":90379,"instance":1380,"goal":"subst_1380 : subst a z (a z).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1381","id":90380,"instance":1381,"goal":"lam_1381 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1382","id":90381,"instance":1382,"goal":"app_1382 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1383","id":90382,"instance":1383,"goal":"arrow_1383 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1384","id":90383,"instance":1384,"goal":"of_var_1384 : of f T_1384 <- of-env f T_1384.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1385","id":90384,"instance":1385,"goal":"of_lam_1385 : of (lam z) (arrow T_1385a T_1385b) <- ({n} of n T_1385a -> of (z n) T_1385b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1386","id":90385,"instance":1386,"goal":"of_app_1386 : of (app v_1 v_2) T_1386 <- of v_1 (arrow T_1386a T_1386) <- of v_2 T_1386a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1387","id":90386,"instance":1387,"goal":"step_beta_1387 : step (app (lam n) x) (n x).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1388","id":90387,"instance":1388,"goal":"step_app1_1388 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1389","id":90388,"instance":1389,"goal":"preservation_1389 : of z T_1389 -> step z z_p -> of z_p T_1389.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1390","id":90389,"instance":1390,"goal":"progress_1390 : of n T_1390 -> value n or (exists n_p. step n n_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1391","id":90390,"instance":1391,"goal":"eq_refl_1391 : eq n n.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1392","id":90391,"instance":1392,"goal":"subst_1392 : subst c h (c h).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1393","id":90392,"instance":1393,"goal":"lam_1393 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1394","id":90393,"instance":1394,"goal":"app_1394 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1395","id":90394,"instance":1395,"goal":"arrow_1395 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1396","id":90395,"instance":1396,"goal":"of_var_1396 : of z T_1396 <- of-env z T_1396.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1397","id":90396,"instance":1397,"goal":"of_lam_1397 : of (lam y) (arrow T_1397a T_1397b) <- ({n} of n T_1397a -> of (y n) T_1397b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1398","id":90397,"instance":1398,"goal":"of_app_1398 : of (app v_1 v_2) T_1398 <- of v_1 (arrow T_1398a T_1398) <- of v_2 T_1398a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1399","id":90398,"instance":1399,"goal":"step_beta_1399 : step (app (lam c) p) (c p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1400","id":90399,"instance":1400,"goal":"step_app1_1400 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1401","id":90400,"instance":1401,"goal":"preservation_1401 : of q T_1401 -> step q q_p -> of q_p T_1401.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1402","id":90401,"instance":1402,"goal":"progress_1402 : of m T_1402 -> value m or (exists m_p. step m m_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1403","id":90402,"instance":1403,"goal":"eq_refl_1403 : eq v v.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1404","id":90403,"instance":1404,"goal":"subst_1404 : subst y g (y g).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1405","id":90404,"instance":1405,"goal":"lam_1405 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1406","id":90405,"instance":1406,"goal":"app_1406 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1407","id":90406,"instance":1407,"goal":"arrow_1407 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1408","id":90407,"instance":1408,"goal":"of_var_1408 : of v T_1408 <- of-env v T_1408.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1409","id":90408,"instance":1409,"goal":"of_lam_1409 : of (lam g) (arrow T_1409a T_1409b) <- ({z} of z T_1409a -> of (g z) T_1409b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1410","id":90409,"instance":1410,"goal":"of_app_1410 : of (app m_1 m_2) T_1410 <- of m_1 (arrow T_1410a T_1410) <- of m_2 T_1410a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1411","id":90410,"instance":1411,"goal":"step_beta_1411 : step (app (lam b) h) (b h).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1412","id":90411,"instance":1412,"goal":"step_app1_1412 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1413","id":90412,"instance":1413,"goal":"preservation_1413 : of p T_1413 -> step p p_p -> of p_p T_1413.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1414","id":90413,"instance":1414,"goal":"progress_1414 : of d T_1414 -> value d or (exists d_p. step d d_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1415","id":90414,"instance":1415,"goal":"eq_refl_1415 : eq y y.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1416","id":90415,"instance":1416,"goal":"subst_1416 : subst x u (x u).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1417","id":90416,"instance":1417,"goal":"lam_1417 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1418","id":90417,"instance":1418,"goal":"app_1418 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1419","id":90418,"instance":1419,"goal":"arrow_1419 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1420","id":90419,"instance":1420,"goal":"of_var_1420 : of w T_1420 <- of-env w T_1420.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1421","id":90420,"instance":1421,"goal":"of_lam_1421 : of (lam e) (arrow T_1421a T_1421b) <- ({g} of g T_1421a -> of (e g) T_1421b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1422","id":90421,"instance":1422,"goal":"of_app_1422 : of (app x_1 x_2) T_1422 <- of x_1 (arrow T_1422a T_1422) <- of x_2 T_1422a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1423","id":90422,"instance":1423,"goal":"step_beta_1423 : step (app (lam x) n) (x n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1424","id":90423,"instance":1424,"goal":"step_app1_1424 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1425","id":90424,"instance":1425,"goal":"preservation_1425 : of v T_1425 -> step v v_p -> of v_p T_1425.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1426","id":90425,"instance":1426,"goal":"progress_1426 : of m T_1426 -> value m or (exists m_p. step m m_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1427","id":90426,"instance":1427,"goal":"eq_refl_1427 : eq c c.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1428","id":90427,"instance":1428,"goal":"subst_1428 : subst g n (g n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1429","id":90428,"instance":1429,"goal":"lam_1429 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1430","id":90429,"instance":1430,"goal":"app_1430 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1431","id":90430,"instance":1431,"goal":"arrow_1431 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1432","id":90431,"instance":1432,"goal":"of_var_1432 : of w T_1432 <- of-env w T_1432.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1433","id":90432,"instance":1433,"goal":"of_lam_1433 : of (lam p) (arrow T_1433a T_1433b) <- ({g} of g T_1433a -> of (p g) T_1433b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1434","id":90433,"instance":1434,"goal":"of_app_1434 : of (app a_1 a_2) T_1434 <- of a_1 (arrow T_1434a T_1434) <- of a_2 T_1434a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1435","id":90434,"instance":1435,"goal":"step_beta_1435 : step (app (lam b) u) (b u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1436","id":90435,"instance":1436,"goal":"step_app1_1436 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1437","id":90436,"instance":1437,"goal":"preservation_1437 : of v T_1437 -> step v v_p -> of v_p T_1437.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1438","id":90437,"instance":1438,"goal":"progress_1438 : of z T_1438 -> value z or (exists z_p. step z z_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1439","id":90438,"instance":1439,"goal":"eq_refl_1439 : eq b b.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1440","id":90439,"instance":1440,"goal":"subst_1440 : subst g m (g m).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1441","id":90440,"instance":1441,"goal":"lam_1441 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1442","id":90441,"instance":1442,"goal":"app_1442 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1443","id":90442,"instance":1443,"goal":"arrow_1443 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1444","id":90443,"instance":1444,"goal":"of_var_1444 : of a T_1444 <- of-env a T_1444.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1445","id":90444,"instance":1445,"goal":"of_lam_1445 : of (lam c) (arrow T_1445a T_1445b) <- ({x} of x T_1445a -> of (c x) T_1445b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1446","id":90445,"instance":1446,"goal":"of_app_1446 : of (app u_1 u_2) T_1446 <- of u_1 (arrow T_1446a T_1446) <- of u_2 T_1446a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1447","id":90446,"instance":1447,"goal":"step_beta_1447 : step (app (lam w) q) (w q).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1448","id":90447,"instance":1448,"goal":"step_app1_1448 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1449","id":90448,"instance":1449,"goal":"preservation_1449 : of w T_1449 -> step w w_p -> of w_p T_1449.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1450","id":90449,"instance":1450,"goal":"progress_1450 : of p T_1450 -> value p or (exists p_p. step p p_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1451","id":90450,"instance":1451,"goal":"eq_refl_1451 : eq y y.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1452","id":90451,"instance":1452,"goal":"subst_1452 : subst x m (x m).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1453","id":90452,"instance":1453,"goal":"lam_1453 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1454","id":90453,"instance":1454,"goal":"app_1454 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1455","id":90454,"instance":1455,"goal":"arrow_1455 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1456","id":90455,"instance":1456,"goal":"of_var_1456 : of f T_1456 <- of-env f T_1456.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1457","id":90456,"instance":1457,"goal":"of_lam_1457 : of (lam g) (arrow T_1457a T_1457b) <- ({y} of y T_1457a -> of (g y) T_1457b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1458","id":90457,"instance":1458,"goal":"of_app_1458 : of (app u_1 u_2) T_1458 <- of u_1 (arrow T_1458a T_1458) <- of u_2 T_1458a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1459","id":90458,"instance":1459,"goal":"step_beta_1459 : step (app (lam y) u) (y u).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1460","id":90459,"instance":1460,"goal":"step_app1_1460 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1461","id":90460,"instance":1461,"goal":"preservation_1461 : of u T_1461 -> step u u_p -> of u_p T_1461.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1462","id":90461,"instance":1462,"goal":"progress_1462 : of c T_1462 -> value c or (exists c_p. step c c_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1463","id":90462,"instance":1463,"goal":"eq_refl_1463 : eq f f.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1464","id":90463,"instance":1464,"goal":"subst_1464 : subst x f (x f).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1465","id":90464,"instance":1465,"goal":"lam_1465 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1466","id":90465,"instance":1466,"goal":"app_1466 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1467","id":90466,"instance":1467,"goal":"arrow_1467 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1468","id":90467,"instance":1468,"goal":"of_var_1468 : of v T_1468 <- of-env v T_1468.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1469","id":90468,"instance":1469,"goal":"of_lam_1469 : of (lam h) (arrow T_1469a T_1469b) <- ({p} of p T_1469a -> of (h p) T_1469b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1470","id":90469,"instance":1470,"goal":"of_app_1470 : of (app h_1 h_2) T_1470 <- of h_1 (arrow T_1470a T_1470) <- of h_2 T_1470a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1471","id":90470,"instance":1471,"goal":"step_beta_1471 : step (app (lam n) a) (n a).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1472","id":90471,"instance":1472,"goal":"step_app1_1472 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1473","id":90472,"instance":1473,"goal":"preservation_1473 : of z T_1473 -> step z z_p -> of z_p T_1473.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1474","id":90473,"instance":1474,"goal":"progress_1474 : of q T_1474 -> value q or (exists q_p. step q q_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1475","id":90474,"instance":1475,"goal":"eq_refl_1475 : eq c c.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1476","id":90475,"instance":1476,"goal":"subst_1476 : subst x e (x e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1477","id":90476,"instance":1477,"goal":"lam_1477 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1478","id":90477,"instance":1478,"goal":"app_1478 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1479","id":90478,"instance":1479,"goal":"arrow_1479 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1480","id":90479,"instance":1480,"goal":"of_var_1480 : of x T_1480 <- of-env x T_1480.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1481","id":90480,"instance":1481,"goal":"of_lam_1481 : of (lam q) (arrow T_1481a T_1481b) <- ({h} of h T_1481a -> of (q h) T_1481b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1482","id":90481,"instance":1482,"goal":"of_app_1482 : of (app f_1 f_2) T_1482 <- of f_1 (arrow T_1482a T_1482) <- of f_2 T_1482a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1483","id":90482,"instance":1483,"goal":"step_beta_1483 : step (app (lam z) q) (z q).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1484","id":90483,"instance":1484,"goal":"step_app1_1484 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1485","id":90484,"instance":1485,"goal":"preservation_1485 : of w T_1485 -> step w w_p -> of w_p T_1485.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1486","id":90485,"instance":1486,"goal":"progress_1486 : of f T_1486 -> value f or (exists f_p. step f f_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1487","id":90486,"instance":1487,"goal":"eq_refl_1487 : eq y y.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1488","id":90487,"instance":1488,"goal":"subst_1488 : subst n f (n f).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1489","id":90488,"instance":1489,"goal":"lam_1489 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1490","id":90489,"instance":1490,"goal":"app_1490 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1491","id":90490,"instance":1491,"goal":"arrow_1491 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1492","id":90491,"instance":1492,"goal":"of_var_1492 : of p T_1492 <- of-env p T_1492.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1493","id":90492,"instance":1493,"goal":"of_lam_1493 : of (lam h) (arrow T_1493a T_1493b) <- ({n} of n T_1493a -> of (h n) T_1493b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1494","id":90493,"instance":1494,"goal":"of_app_1494 : of (app m_1 m_2) T_1494 <- of m_1 (arrow T_1494a T_1494) <- of m_2 T_1494a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1495","id":90494,"instance":1495,"goal":"step_beta_1495 : step (app (lam g) b) (g b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1496","id":90495,"instance":1496,"goal":"step_app1_1496 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1497","id":90496,"instance":1497,"goal":"preservation_1497 : of m T_1497 -> step m m_p -> of m_p T_1497.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1498","id":90497,"instance":1498,"goal":"progress_1498 : of n T_1498 -> value n or (exists n_p. step n n_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1499","id":90498,"instance":1499,"goal":"eq_refl_1499 : eq u u.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1500","id":90499,"instance":1500,"goal":"subst_1500 : subst c q (c q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1501","id":90500,"instance":1501,"goal":"lam_1501 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1502","id":90501,"instance":1502,"goal":"app_1502 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1503","id":90502,"instance":1503,"goal":"arrow_1503 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1504","id":90503,"instance":1504,"goal":"of_var_1504 : of v T_1504 <- of-env v T_1504.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1505","id":90504,"instance":1505,"goal":"of_lam_1505 : of (lam g) (arrow T_1505a T_1505b) <- ({p} of p T_1505a -> of (g p) T_1505b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1506","id":90505,"instance":1506,"goal":"of_app_1506 : of (app b_1 b_2) T_1506 <- of b_1 (arrow T_1506a T_1506) <- of b_2 T_1506a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1507","id":90506,"instance":1507,"goal":"step_beta_1507 : step (app (lam e) u) (e u).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1508","id":90507,"instance":1508,"goal":"step_app1_1508 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1509","id":90508,"instance":1509,"goal":"preservation_1509 : of g T_1509 -> step g g_p -> of g_p T_1509.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1510","id":90509,"instance":1510,"goal":"progress_1510 : of z T_1510 -> value z or (exists z_p. step z z_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1511","id":90510,"instance":1511,"goal":"eq_refl_1511 : eq m m.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1512","id":90511,"instance":1512,"goal":"subst_1512 : subst d e (d e).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1513","id":90512,"instance":1513,"goal":"lam_1513 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1514","id":90513,"instance":1514,"goal":"app_1514 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1515","id":90514,"instance":1515,"goal":"arrow_1515 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1516","id":90515,"instance":1516,"goal":"of_var_1516 : of m T_1516 <- of-env m T_1516.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1517","id":90516,"instance":1517,"goal":"of_lam_1517 : of (lam v) (arrow T_1517a T_1517b) <- ({m} of m T_1517a -> of (v m) T_1517b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1518","id":90517,"instance":1518,"goal":"of_app_1518 : of (app a_1 a_2) T_1518 <- of a_1 (arrow T_1518a T_1518) <- of a_2 T_1518a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1519","id":90518,"instance":1519,"goal":"step_beta_1519 : step (app (lam n) y) (n y).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1520","id":90519,"instance":1520,"goal":"step_app1_1520 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1521","id":90520,"instance":1521,"goal":"preservation_1521 : of b T_1521 -> step b b_p -> of b_p T_1521.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1522","id":90521,"instance":1522,"goal":"progress_1522 : of m T_1522 -> value m or (exists m_p. step m m_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1523","id":90522,"instance":1523,"goal":"eq_refl_1523 : eq g g.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1524","id":90523,"instance":1524,"goal":"subst_1524 : subst p u (p u).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1525","id":90524,"instance":1525,"goal":"lam_1525 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1526","id":90525,"instance":1526,"goal":"app_1526 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1527","id":90526,"instance":1527,"goal":"arrow_1527 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1528","id":90527,"instance":1528,"goal":"of_var_1528 : of q T_1528 <- of-env q T_1528.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1529","id":90528,"instance":1529,"goal":"of_lam_1529 : of (lam x) (arrow T_1529a T_1529b) <- ({m} of m T_1529a -> of (x m) T_1529b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1530","id":90529,"instance":1530,"goal":"of_app_1530 : of (app y_1 y_2) T_1530 <- of y_1 (arrow T_1530a T_1530) <- of y_2 T_1530a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1531","id":90530,"instance":1531,"goal":"step_beta_1531 : step (app (lam b) w) (b w).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1532","id":90531,"instance":1532,"goal":"step_app1_1532 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1533","id":90532,"instance":1533,"goal":"preservation_1533 : of v T_1533 -> step v v_p -> of v_p T_1533.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1534","id":90533,"instance":1534,"goal":"progress_1534 : of z T_1534 -> value z or (exists z_p. step z z_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1535","id":90534,"instance":1535,"goal":"eq_refl_1535 : eq w w.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1536","id":90535,"instance":1536,"goal":"subst_1536 : subst q a (q a).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1537","id":90536,"instance":1537,"goal":"lam_1537 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1538","id":90537,"instance":1538,"goal":"app_1538 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1539","id":90538,"instance":1539,"goal":"arrow_1539 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1540","id":90539,"instance":1540,"goal":"of_var_1540 : of e T_1540 <- of-env e T_1540.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1541","id":90540,"instance":1541,"goal":"of_lam_1541 : of (lam q) (arrow T_1541a T_1541b) <- ({h} of h T_1541a -> of (q h) T_1541b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1542","id":90541,"instance":1542,"goal":"of_app_1542 : of (app p_1 p_2) T_1542 <- of p_1 (arrow T_1542a T_1542) <- of p_2 T_1542a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1543","id":90542,"instance":1543,"goal":"step_beta_1543 : step (app (lam n) z) (n z).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1544","id":90543,"instance":1544,"goal":"step_app1_1544 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1545","id":90544,"instance":1545,"goal":"preservation_1545 : of h T_1545 -> step h h_p -> of h_p T_1545.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1546","id":90545,"instance":1546,"goal":"progress_1546 : of m T_1546 -> value m or (exists m_p. step m m_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1547","id":90546,"instance":1547,"goal":"eq_refl_1547 : eq p p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1548","id":90547,"instance":1548,"goal":"subst_1548 : subst w h (w h).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1549","id":90548,"instance":1549,"goal":"lam_1549 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1550","id":90549,"instance":1550,"goal":"app_1550 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1551","id":90550,"instance":1551,"goal":"arrow_1551 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1552","id":90551,"instance":1552,"goal":"of_var_1552 : of d T_1552 <- of-env d T_1552.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1553","id":90552,"instance":1553,"goal":"of_lam_1553 : of (lam p) (arrow T_1553a T_1553b) <- ({c} of c T_1553a -> of (p c) T_1553b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1554","id":90553,"instance":1554,"goal":"of_app_1554 : of (app d_1 d_2) T_1554 <- of d_1 (arrow T_1554a T_1554) <- of d_2 T_1554a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1555","id":90554,"instance":1555,"goal":"step_beta_1555 : step (app (lam b) f) (b f).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1556","id":90555,"instance":1556,"goal":"step_app1_1556 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1557","id":90556,"instance":1557,"goal":"preservation_1557 : of a T_1557 -> step a a_p -> of a_p T_1557.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1558","id":90557,"instance":1558,"goal":"progress_1558 : of q T_1558 -> value q or (exists q_p. step q q_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1559","id":90558,"instance":1559,"goal":"eq_refl_1559 : eq c c.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1560","id":90559,"instance":1560,"goal":"subst_1560 : subst q n (q n).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1561","id":90560,"instance":1561,"goal":"lam_1561 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1562","id":90561,"instance":1562,"goal":"app_1562 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1563","id":90562,"instance":1563,"goal":"arrow_1563 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1564","id":90563,"instance":1564,"goal":"of_var_1564 : of e T_1564 <- of-env e T_1564.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1565","id":90564,"instance":1565,"goal":"of_lam_1565 : of (lam v) (arrow T_1565a T_1565b) <- ({h} of h T_1565a -> of (v h) T_1565b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1566","id":90565,"instance":1566,"goal":"of_app_1566 : of (app d_1 d_2) T_1566 <- of d_1 (arrow T_1566a T_1566) <- of d_2 T_1566a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1567","id":90566,"instance":1567,"goal":"step_beta_1567 : step (app (lam n) g) (n g).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1568","id":90567,"instance":1568,"goal":"step_app1_1568 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1569","id":90568,"instance":1569,"goal":"preservation_1569 : of u T_1569 -> step u u_p -> of u_p T_1569.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1570","id":90569,"instance":1570,"goal":"progress_1570 : of d T_1570 -> value d or (exists d_p. step d d_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1571","id":90570,"instance":1571,"goal":"eq_refl_1571 : eq p p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1572","id":90571,"instance":1572,"goal":"subst_1572 : subst m x (m x).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1573","id":90572,"instance":1573,"goal":"lam_1573 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1574","id":90573,"instance":1574,"goal":"app_1574 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1575","id":90574,"instance":1575,"goal":"arrow_1575 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1576","id":90575,"instance":1576,"goal":"of_var_1576 : of f T_1576 <- of-env f T_1576.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1577","id":90576,"instance":1577,"goal":"of_lam_1577 : of (lam q) (arrow T_1577a T_1577b) <- ({c} of c T_1577a -> of (q c) T_1577b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1578","id":90577,"instance":1578,"goal":"of_app_1578 : of (app d_1 d_2) T_1578 <- of d_1 (arrow T_1578a T_1578) <- of d_2 T_1578a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1579","id":90578,"instance":1579,"goal":"step_beta_1579 : step (app (lam x) d) (x d).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1580","id":90579,"instance":1580,"goal":"step_app1_1580 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1581","id":90580,"instance":1581,"goal":"preservation_1581 : of m T_1581 -> step m m_p -> of m_p T_1581.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1582","id":90581,"instance":1582,"goal":"progress_1582 : of h T_1582 -> value h or (exists h_p. step h h_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1583","id":90582,"instance":1583,"goal":"eq_refl_1583 : eq c c.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1584","id":90583,"instance":1584,"goal":"subst_1584 : subst m y (m y).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1585","id":90584,"instance":1585,"goal":"lam_1585 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1586","id":90585,"instance":1586,"goal":"app_1586 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1587","id":90586,"instance":1587,"goal":"arrow_1587 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1588","id":90587,"instance":1588,"goal":"of_var_1588 : of u T_1588 <- of-env u T_1588.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1589","id":90588,"instance":1589,"goal":"of_lam_1589 : of (lam y) (arrow T_1589a T_1589b) <- ({d} of d T_1589a -> of (y d) T_1589b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1590","id":90589,"instance":1590,"goal":"of_app_1590 : of (app u_1 u_2) T_1590 <- of u_1 (arrow T_1590a T_1590) <- of u_2 T_1590a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1591","id":90590,"instance":1591,"goal":"step_beta_1591 : step (app (lam x) q) (x q).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1592","id":90591,"instance":1592,"goal":"step_app1_1592 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1593","id":90592,"instance":1593,"goal":"preservation_1593 : of n T_1593 -> step n n_p -> of n_p T_1593.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1594","id":90593,"instance":1594,"goal":"progress_1594 : of g T_1594 -> value g or (exists g_p. step g g_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1595","id":90594,"instance":1595,"goal":"eq_refl_1595 : eq d d.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1596","id":90595,"instance":1596,"goal":"subst_1596 : subst x h (x h).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1597","id":90596,"instance":1597,"goal":"lam_1597 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1598","id":90597,"instance":1598,"goal":"app_1598 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1599","id":90598,"instance":1599,"goal":"arrow_1599 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1600","id":90599,"instance":1600,"goal":"of_var_1600 : of n T_1600 <- of-env n T_1600.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1601","id":90600,"instance":1601,"goal":"of_lam_1601 : of (lam f) (arrow T_1601a T_1601b) <- ({g} of g T_1601a -> of (f g) T_1601b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1602","id":90601,"instance":1602,"goal":"of_app_1602 : of (app h_1 h_2) T_1602 <- of h_1 (arrow T_1602a T_1602) <- of h_2 T_1602a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1603","id":90602,"instance":1603,"goal":"step_beta_1603 : step (app (lam x) g) (x g).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1604","id":90603,"instance":1604,"goal":"step_app1_1604 : step (app w_1 w_2) (app w_1p w_2) <- step w_1 w_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1605","id":90604,"instance":1605,"goal":"preservation_1605 : of u T_1605 -> step u u_p -> of u_p T_1605.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1606","id":90605,"instance":1606,"goal":"progress_1606 : of f T_1606 -> value f or (exists f_p. step f f_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1607","id":90606,"instance":1607,"goal":"eq_refl_1607 : eq q q.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1608","id":90607,"instance":1608,"goal":"subst_1608 : subst x q (x q).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1609","id":90608,"instance":1609,"goal":"lam_1609 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1610","id":90609,"instance":1610,"goal":"app_1610 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1611","id":90610,"instance":1611,"goal":"arrow_1611 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1612","id":90611,"instance":1612,"goal":"of_var_1612 : of v T_1612 <- of-env v T_1612.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1613","id":90612,"instance":1613,"goal":"of_lam_1613 : of (lam e) (arrow T_1613a T_1613b) <- ({u} of u T_1613a -> of (e u) T_1613b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1614","id":90613,"instance":1614,"goal":"of_app_1614 : of (app b_1 b_2) T_1614 <- of b_1 (arrow T_1614a T_1614) <- of b_2 T_1614a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1615","id":90614,"instance":1615,"goal":"step_beta_1615 : step (app (lam z) x) (z x).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1616","id":90615,"instance":1616,"goal":"step_app1_1616 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1617","id":90616,"instance":1617,"goal":"preservation_1617 : of u T_1617 -> step u u_p -> of u_p T_1617.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1618","id":90617,"instance":1618,"goal":"progress_1618 : of h T_1618 -> value h or (exists h_p. step h h_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1619","id":90618,"instance":1619,"goal":"eq_refl_1619 : eq u u.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1620","id":90619,"instance":1620,"goal":"subst_1620 : subst f m (f m).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1621","id":90620,"instance":1621,"goal":"lam_1621 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1622","id":90621,"instance":1622,"goal":"app_1622 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1623","id":90622,"instance":1623,"goal":"arrow_1623 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1624","id":90623,"instance":1624,"goal":"of_var_1624 : of c T_1624 <- of-env c T_1624.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1625","id":90624,"instance":1625,"goal":"of_lam_1625 : of (lam c) (arrow T_1625a T_1625b) <- ({z} of z T_1625a -> of (c z) T_1625b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1626","id":90625,"instance":1626,"goal":"of_app_1626 : of (app w_1 w_2) T_1626 <- of w_1 (arrow T_1626a T_1626) <- of w_2 T_1626a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1627","id":90626,"instance":1627,"goal":"step_beta_1627 : step (app (lam m) b) (m b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1628","id":90627,"instance":1628,"goal":"step_app1_1628 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1629","id":90628,"instance":1629,"goal":"preservation_1629 : of w T_1629 -> step w w_p -> of w_p T_1629.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1630","id":90629,"instance":1630,"goal":"progress_1630 : of g T_1630 -> value g or (exists g_p. step g g_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1631","id":90630,"instance":1631,"goal":"eq_refl_1631 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1632","id":90631,"instance":1632,"goal":"subst_1632 : subst m u (m u).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1633","id":90632,"instance":1633,"goal":"lam_1633 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1634","id":90633,"instance":1634,"goal":"app_1634 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1635","id":90634,"instance":1635,"goal":"arrow_1635 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1636","id":90635,"instance":1636,"goal":"of_var_1636 : of z T_1636 <- of-env z T_1636.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1637","id":90636,"instance":1637,"goal":"of_lam_1637 : of (lam v) (arrow T_1637a T_1637b) <- ({h} of h T_1637a -> of (v h) T_1637b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1638","id":90637,"instance":1638,"goal":"of_app_1638 : of (app g_1 g_2) T_1638 <- of g_1 (arrow T_1638a T_1638) <- of g_2 T_1638a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1639","id":90638,"instance":1639,"goal":"step_beta_1639 : step (app (lam y) u) (y u).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1640","id":90639,"instance":1640,"goal":"step_app1_1640 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1641","id":90640,"instance":1641,"goal":"preservation_1641 : of y T_1641 -> step y y_p -> of y_p T_1641.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1642","id":90641,"instance":1642,"goal":"progress_1642 : of y T_1642 -> value y or (exists y_p. step y y_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1643","id":90642,"instance":1643,"goal":"eq_refl_1643 : eq d d.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1644","id":90643,"instance":1644,"goal":"subst_1644 : subst f c (f c).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1645","id":90644,"instance":1645,"goal":"lam_1645 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1646","id":90645,"instance":1646,"goal":"app_1646 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1647","id":90646,"instance":1647,"goal":"arrow_1647 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1648","id":90647,"instance":1648,"goal":"of_var_1648 : of q T_1648 <- of-env q T_1648.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1649","id":90648,"instance":1649,"goal":"of_lam_1649 : of (lam v) (arrow T_1649a T_1649b) <- ({y} of y T_1649a -> of (v y) T_1649b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1650","id":90649,"instance":1650,"goal":"of_app_1650 : of (app b_1 b_2) T_1650 <- of b_1 (arrow T_1650a T_1650) <- of b_2 T_1650a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1651","id":90650,"instance":1651,"goal":"step_beta_1651 : step (app (lam v) f) (v f).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1652","id":90651,"instance":1652,"goal":"step_app1_1652 : step (app h_1 h_2) (app h_1p h_2) <- step h_1 h_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1653","id":90652,"instance":1653,"goal":"preservation_1653 : of u T_1653 -> step u u_p -> of u_p T_1653.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1654","id":90653,"instance":1654,"goal":"progress_1654 : of x T_1654 -> value x or (exists x_p. step x x_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1655","id":90654,"instance":1655,"goal":"eq_refl_1655 : eq c c.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1656","id":90655,"instance":1656,"goal":"subst_1656 : subst b y (b y).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1657","id":90656,"instance":1657,"goal":"lam_1657 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1658","id":90657,"instance":1658,"goal":"app_1658 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1659","id":90658,"instance":1659,"goal":"arrow_1659 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1660","id":90659,"instance":1660,"goal":"of_var_1660 : of h T_1660 <- of-env h T_1660.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1661","id":90660,"instance":1661,"goal":"of_lam_1661 : of (lam b) (arrow T_1661a T_1661b) <- ({w} of w T_1661a -> of (b w) T_1661b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1662","id":90661,"instance":1662,"goal":"of_app_1662 : of (app z_1 z_2) T_1662 <- of z_1 (arrow T_1662a T_1662) <- of z_2 T_1662a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1663","id":90662,"instance":1663,"goal":"step_beta_1663 : step (app (lam c) f) (c f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1664","id":90663,"instance":1664,"goal":"step_app1_1664 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1665","id":90664,"instance":1665,"goal":"preservation_1665 : of e T_1665 -> step e e_p -> of e_p T_1665.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1666","id":90665,"instance":1666,"goal":"progress_1666 : of p T_1666 -> value p or (exists p_p. step p p_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1667","id":90666,"instance":1667,"goal":"eq_refl_1667 : eq w w.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1668","id":90667,"instance":1668,"goal":"subst_1668 : subst n b (n b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1669","id":90668,"instance":1669,"goal":"lam_1669 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1670","id":90669,"instance":1670,"goal":"app_1670 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1671","id":90670,"instance":1671,"goal":"arrow_1671 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1672","id":90671,"instance":1672,"goal":"of_var_1672 : of a T_1672 <- of-env a T_1672.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1673","id":90672,"instance":1673,"goal":"of_lam_1673 : of (lam p) (arrow T_1673a T_1673b) <- ({w} of w T_1673a -> of (p w) T_1673b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1674","id":90673,"instance":1674,"goal":"of_app_1674 : of (app z_1 z_2) T_1674 <- of z_1 (arrow T_1674a T_1674) <- of z_2 T_1674a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1675","id":90674,"instance":1675,"goal":"step_beta_1675 : step (app (lam m) d) (m d).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1676","id":90675,"instance":1676,"goal":"step_app1_1676 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1677","id":90676,"instance":1677,"goal":"preservation_1677 : of y T_1677 -> step y y_p -> of y_p T_1677.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1678","id":90677,"instance":1678,"goal":"progress_1678 : of w T_1678 -> value w or (exists w_p. step w w_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1679","id":90678,"instance":1679,"goal":"eq_refl_1679 : eq x x.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1680","id":90679,"instance":1680,"goal":"subst_1680 : subst h p (h p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1681","id":90680,"instance":1681,"goal":"lam_1681 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1682","id":90681,"instance":1682,"goal":"app_1682 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1683","id":90682,"instance":1683,"goal":"arrow_1683 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1684","id":90683,"instance":1684,"goal":"of_var_1684 : of w T_1684 <- of-env w T_1684.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1685","id":90684,"instance":1685,"goal":"of_lam_1685 : of (lam d) (arrow T_1685a T_1685b) <- ({g} of g T_1685a -> of (d g) T_1685b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1686","id":90685,"instance":1686,"goal":"of_app_1686 : of (app u_1 u_2) T_1686 <- of u_1 (arrow T_1686a T_1686) <- of u_2 T_1686a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1687","id":90686,"instance":1687,"goal":"step_beta_1687 : step (app (lam e) a) (e a).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1688","id":90687,"instance":1688,"goal":"step_app1_1688 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1689","id":90688,"instance":1689,"goal":"preservation_1689 : of g T_1689 -> step g g_p -> of g_p T_1689.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1690","id":90689,"instance":1690,"goal":"progress_1690 : of c T_1690 -> value c or (exists c_p. step c c_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1691","id":90690,"instance":1691,"goal":"eq_refl_1691 : eq b b.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1692","id":90691,"instance":1692,"goal":"subst_1692 : subst m n (m n).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1693","id":90692,"instance":1693,"goal":"lam_1693 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1694","id":90693,"instance":1694,"goal":"app_1694 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1695","id":90694,"instance":1695,"goal":"arrow_1695 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1696","id":90695,"instance":1696,"goal":"of_var_1696 : of w T_1696 <- of-env w T_1696.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1697","id":90696,"instance":1697,"goal":"of_lam_1697 : of (lam n) (arrow T_1697a T_1697b) <- ({f} of f T_1697a -> of (n f) T_1697b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1698","id":90697,"instance":1698,"goal":"of_app_1698 : of (app z_1 z_2) T_1698 <- of z_1 (arrow T_1698a T_1698) <- of z_2 T_1698a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1699","id":90698,"instance":1699,"goal":"step_beta_1699 : step (app (lam z) c) (z c).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1700","id":90699,"instance":1700,"goal":"step_app1_1700 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1701","id":90700,"instance":1701,"goal":"preservation_1701 : of a T_1701 -> step a a_p -> of a_p T_1701.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1702","id":90701,"instance":1702,"goal":"progress_1702 : of g T_1702 -> value g or (exists g_p. step g g_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1703","id":90702,"instance":1703,"goal":"eq_refl_1703 : eq m m.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1704","id":90703,"instance":1704,"goal":"subst_1704 : subst g u (g u).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1705","id":90704,"instance":1705,"goal":"lam_1705 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1706","id":90705,"instance":1706,"goal":"app_1706 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1707","id":90706,"instance":1707,"goal":"arrow_1707 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1708","id":90707,"instance":1708,"goal":"of_var_1708 : of z T_1708 <- of-env z T_1708.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1709","id":90708,"instance":1709,"goal":"of_lam_1709 : of (lam n) (arrow T_1709a T_1709b) <- ({w} of w T_1709a -> of (n w) T_1709b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1710","id":90709,"instance":1710,"goal":"of_app_1710 : of (app w_1 w_2) T_1710 <- of w_1 (arrow T_1710a T_1710) <- of w_2 T_1710a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1711","id":90710,"instance":1711,"goal":"step_beta_1711 : step (app (lam c) g) (c g).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1712","id":90711,"instance":1712,"goal":"step_app1_1712 : step (app u_1 u_2) (app u_1p u_2) <- step u_1 u_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1713","id":90712,"instance":1713,"goal":"preservation_1713 : of u T_1713 -> step u u_p -> of u_p T_1713.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1714","id":90713,"instance":1714,"goal":"progress_1714 : of h T_1714 -> value h or (exists h_p. step h h_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1715","id":90714,"instance":1715,"goal":"eq_refl_1715 : eq q q.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1716","id":90715,"instance":1716,"goal":"subst_1716 : subst d f (d f).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1717","id":90716,"instance":1717,"goal":"lam_1717 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1718","id":90717,"instance":1718,"goal":"app_1718 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1719","id":90718,"instance":1719,"goal":"arrow_1719 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1720","id":90719,"instance":1720,"goal":"of_var_1720 : of c T_1720 <- of-env c T_1720.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1721","id":90720,"instance":1721,"goal":"of_lam_1721 : of (lam v) (arrow T_1721a T_1721b) <- ({f} of f T_1721a -> of (v f) T_1721b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1722","id":90721,"instance":1722,"goal":"of_app_1722 : of (app y_1 y_2) T_1722 <- of y_1 (arrow T_1722a T_1722) <- of y_2 T_1722a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1723","id":90722,"instance":1723,"goal":"step_beta_1723 : step (app (lam w) d) (w d).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1724","id":90723,"instance":1724,"goal":"step_app1_1724 : step (app w_1 w_2) (app w_1p w_2) <- step w_1 w_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1725","id":90724,"instance":1725,"goal":"preservation_1725 : of p T_1725 -> step p p_p -> of p_p T_1725.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1726","id":90725,"instance":1726,"goal":"progress_1726 : of x T_1726 -> value x or (exists x_p. step x x_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1727","id":90726,"instance":1727,"goal":"eq_refl_1727 : eq a a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1728","id":90727,"instance":1728,"goal":"subst_1728 : subst z v (z v).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1729","id":90728,"instance":1729,"goal":"lam_1729 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1730","id":90729,"instance":1730,"goal":"app_1730 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1731","id":90730,"instance":1731,"goal":"arrow_1731 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1732","id":90731,"instance":1732,"goal":"of_var_1732 : of d T_1732 <- of-env d T_1732.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1733","id":90732,"instance":1733,"goal":"of_lam_1733 : of (lam x) (arrow T_1733a T_1733b) <- ({c} of c T_1733a -> of (x c) T_1733b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1734","id":90733,"instance":1734,"goal":"of_app_1734 : of (app p_1 p_2) T_1734 <- of p_1 (arrow T_1734a T_1734) <- of p_2 T_1734a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1735","id":90734,"instance":1735,"goal":"step_beta_1735 : step (app (lam c) b) (c b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1736","id":90735,"instance":1736,"goal":"step_app1_1736 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1737","id":90736,"instance":1737,"goal":"preservation_1737 : of d T_1737 -> step d d_p -> of d_p T_1737.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1738","id":90737,"instance":1738,"goal":"progress_1738 : of h T_1738 -> value h or (exists h_p. step h h_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1739","id":90738,"instance":1739,"goal":"eq_refl_1739 : eq u u.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1740","id":90739,"instance":1740,"goal":"subst_1740 : subst c g (c g).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1741","id":90740,"instance":1741,"goal":"lam_1741 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1742","id":90741,"instance":1742,"goal":"app_1742 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1743","id":90742,"instance":1743,"goal":"arrow_1743 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1744","id":90743,"instance":1744,"goal":"of_var_1744 : of g T_1744 <- of-env g T_1744.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1745","id":90744,"instance":1745,"goal":"of_lam_1745 : of (lam c) (arrow T_1745a T_1745b) <- ({y} of y T_1745a -> of (c y) T_1745b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1746","id":90745,"instance":1746,"goal":"of_app_1746 : of (app x_1 x_2) T_1746 <- of x_1 (arrow T_1746a T_1746) <- of x_2 T_1746a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1747","id":90746,"instance":1747,"goal":"step_beta_1747 : step (app (lam u) d) (u d).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1748","id":90747,"instance":1748,"goal":"step_app1_1748 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1749","id":90748,"instance":1749,"goal":"preservation_1749 : of w T_1749 -> step w w_p -> of w_p T_1749.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1750","id":90749,"instance":1750,"goal":"progress_1750 : of x T_1750 -> value x or (exists x_p. step x x_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1751","id":90750,"instance":1751,"goal":"eq_refl_1751 : eq x x.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1752","id":90751,"instance":1752,"goal":"subst_1752 : subst z b (z b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1753","id":90752,"instance":1753,"goal":"lam_1753 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1754","id":90753,"instance":1754,"goal":"app_1754 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1755","id":90754,"instance":1755,"goal":"arrow_1755 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1756","id":90755,"instance":1756,"goal":"of_var_1756 : of b T_1756 <- of-env b T_1756.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1757","id":90756,"instance":1757,"goal":"of_lam_1757 : of (lam y) (arrow T_1757a T_1757b) <- ({a} of a T_1757a -> of (y a) T_1757b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1758","id":90757,"instance":1758,"goal":"of_app_1758 : of (app z_1 z_2) T_1758 <- of z_1 (arrow T_1758a T_1758) <- of z_2 T_1758a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1759","id":90758,"instance":1759,"goal":"step_beta_1759 : step (app (lam a) y) (a y).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1760","id":90759,"instance":1760,"goal":"step_app1_1760 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1761","id":90760,"instance":1761,"goal":"preservation_1761 : of m T_1761 -> step m m_p -> of m_p T_1761.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1762","id":90761,"instance":1762,"goal":"progress_1762 : of z T_1762 -> value z or (exists z_p. step z z_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1763","id":90762,"instance":1763,"goal":"eq_refl_1763 : eq p p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1764","id":90763,"instance":1764,"goal":"subst_1764 : subst w n (w n).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1765","id":90764,"instance":1765,"goal":"lam_1765 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1766","id":90765,"instance":1766,"goal":"app_1766 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1767","id":90766,"instance":1767,"goal":"arrow_1767 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1768","id":90767,"instance":1768,"goal":"of_var_1768 : of a T_1768 <- of-env a T_1768.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1769","id":90768,"instance":1769,"goal":"of_lam_1769 : of (lam g) (arrow T_1769a T_1769b) <- ({y} of y T_1769a -> of (g y) T_1769b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1770","id":90769,"instance":1770,"goal":"of_app_1770 : of (app e_1 e_2) T_1770 <- of e_1 (arrow T_1770a T_1770) <- of e_2 T_1770a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1771","id":90770,"instance":1771,"goal":"step_beta_1771 : step (app (lam p) a) (p a).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1772","id":90771,"instance":1772,"goal":"step_app1_1772 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1773","id":90772,"instance":1773,"goal":"preservation_1773 : of x T_1773 -> step x x_p -> of x_p T_1773.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1774","id":90773,"instance":1774,"goal":"progress_1774 : of x T_1774 -> value x or (exists x_p. step x x_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1775","id":90774,"instance":1775,"goal":"eq_refl_1775 : eq a a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1776","id":90775,"instance":1776,"goal":"subst_1776 : subst x m (x m).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1777","id":90776,"instance":1777,"goal":"lam_1777 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1778","id":90777,"instance":1778,"goal":"app_1778 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1779","id":90778,"instance":1779,"goal":"arrow_1779 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1780","id":90779,"instance":1780,"goal":"of_var_1780 : of c T_1780 <- of-env c T_1780.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1781","id":90780,"instance":1781,"goal":"of_lam_1781 : of (lam b) (arrow T_1781a T_1781b) <- ({p} of p T_1781a -> of (b p) T_1781b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1782","id":90781,"instance":1782,"goal":"of_app_1782 : of (app q_1 q_2) T_1782 <- of q_1 (arrow T_1782a T_1782) <- of q_2 T_1782a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1783","id":90782,"instance":1783,"goal":"step_beta_1783 : step (app (lam b) h) (b h).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1784","id":90783,"instance":1784,"goal":"step_app1_1784 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1785","id":90784,"instance":1785,"goal":"preservation_1785 : of d T_1785 -> step d d_p -> of d_p T_1785.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1786","id":90785,"instance":1786,"goal":"progress_1786 : of b T_1786 -> value b or (exists b_p. step b b_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1787","id":90786,"instance":1787,"goal":"eq_refl_1787 : eq m m.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1788","id":90787,"instance":1788,"goal":"subst_1788 : subst h g (h g).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1789","id":90788,"instance":1789,"goal":"lam_1789 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1790","id":90789,"instance":1790,"goal":"app_1790 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1791","id":90790,"instance":1791,"goal":"arrow_1791 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1792","id":90791,"instance":1792,"goal":"of_var_1792 : of g T_1792 <- of-env g T_1792.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1793","id":90792,"instance":1793,"goal":"of_lam_1793 : of (lam z) (arrow T_1793a T_1793b) <- ({c} of c T_1793a -> of (z c) T_1793b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1794","id":90793,"instance":1794,"goal":"of_app_1794 : of (app w_1 w_2) T_1794 <- of w_1 (arrow T_1794a T_1794) <- of w_2 T_1794a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1795","id":90794,"instance":1795,"goal":"step_beta_1795 : step (app (lam n) x) (n x).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1796","id":90795,"instance":1796,"goal":"step_app1_1796 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1797","id":90796,"instance":1797,"goal":"preservation_1797 : of m T_1797 -> step m m_p -> of m_p T_1797.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1798","id":90797,"instance":1798,"goal":"progress_1798 : of m T_1798 -> value m or (exists m_p. step m m_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1799","id":90798,"instance":1799,"goal":"eq_refl_1799 : eq b b.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1800","id":90799,"instance":1800,"goal":"subst_1800 : subst c w (c w).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1801","id":90800,"instance":1801,"goal":"lam_1801 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1802","id":90801,"instance":1802,"goal":"app_1802 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1803","id":90802,"instance":1803,"goal":"arrow_1803 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1804","id":90803,"instance":1804,"goal":"of_var_1804 : of f T_1804 <- of-env f T_1804.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1805","id":90804,"instance":1805,"goal":"of_lam_1805 : of (lam c) (arrow T_1805a T_1805b) <- ({m} of m T_1805a -> of (c m) T_1805b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1806","id":90805,"instance":1806,"goal":"of_app_1806 : of (app h_1 h_2) T_1806 <- of h_1 (arrow T_1806a T_1806) <- of h_2 T_1806a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1807","id":90806,"instance":1807,"goal":"step_beta_1807 : step (app (lam h) u) (h u).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1808","id":90807,"instance":1808,"goal":"step_app1_1808 : step (app y_1 y_2) (app y_1p y_2) <- step y_1 y_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1809","id":90808,"instance":1809,"goal":"preservation_1809 : of x T_1809 -> step x x_p -> of x_p T_1809.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1810","id":90809,"instance":1810,"goal":"progress_1810 : of p T_1810 -> value p or (exists p_p. step p p_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1811","id":90810,"instance":1811,"goal":"eq_refl_1811 : eq g g.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1812","id":90811,"instance":1812,"goal":"subst_1812 : subst y z (y z).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1813","id":90812,"instance":1813,"goal":"lam_1813 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1814","id":90813,"instance":1814,"goal":"app_1814 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1815","id":90814,"instance":1815,"goal":"arrow_1815 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1816","id":90815,"instance":1816,"goal":"of_var_1816 : of v T_1816 <- of-env v T_1816.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1817","id":90816,"instance":1817,"goal":"of_lam_1817 : of (lam p) (arrow T_1817a T_1817b) <- ({v} of v T_1817a -> of (p v) T_1817b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1818","id":90817,"instance":1818,"goal":"of_app_1818 : of (app w_1 w_2) T_1818 <- of w_1 (arrow T_1818a T_1818) <- of w_2 T_1818a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1819","id":90818,"instance":1819,"goal":"step_beta_1819 : step (app (lam g) x) (g x).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1820","id":90819,"instance":1820,"goal":"step_app1_1820 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1821","id":90820,"instance":1821,"goal":"preservation_1821 : of m T_1821 -> step m m_p -> of m_p T_1821.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1822","id":90821,"instance":1822,"goal":"progress_1822 : of m T_1822 -> value m or (exists m_p. step m m_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1823","id":90822,"instance":1823,"goal":"eq_refl_1823 : eq a a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1824","id":90823,"instance":1824,"goal":"subst_1824 : subst f u (f u).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1825","id":90824,"instance":1825,"goal":"lam_1825 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1826","id":90825,"instance":1826,"goal":"app_1826 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1827","id":90826,"instance":1827,"goal":"arrow_1827 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1828","id":90827,"instance":1828,"goal":"of_var_1828 : of x T_1828 <- of-env x T_1828.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1829","id":90828,"instance":1829,"goal":"of_lam_1829 : of (lam v) (arrow T_1829a T_1829b) <- ({m} of m T_1829a -> of (v m) T_1829b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1830","id":90829,"instance":1830,"goal":"of_app_1830 : of (app v_1 v_2) T_1830 <- of v_1 (arrow T_1830a T_1830) <- of v_2 T_1830a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1831","id":90830,"instance":1831,"goal":"step_beta_1831 : step (app (lam v) f) (v f).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1832","id":90831,"instance":1832,"goal":"step_app1_1832 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1833","id":90832,"instance":1833,"goal":"preservation_1833 : of c T_1833 -> step c c_p -> of c_p T_1833.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1834","id":90833,"instance":1834,"goal":"progress_1834 : of b T_1834 -> value b or (exists b_p. step b b_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1835","id":90834,"instance":1835,"goal":"eq_refl_1835 : eq h h.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1836","id":90835,"instance":1836,"goal":"subst_1836 : subst n c (n c).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1837","id":90836,"instance":1837,"goal":"lam_1837 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1838","id":90837,"instance":1838,"goal":"app_1838 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1839","id":90838,"instance":1839,"goal":"arrow_1839 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1840","id":90839,"instance":1840,"goal":"of_var_1840 : of d T_1840 <- of-env d T_1840.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1841","id":90840,"instance":1841,"goal":"of_lam_1841 : of (lam c) (arrow T_1841a T_1841b) <- ({f} of f T_1841a -> of (c f) T_1841b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1842","id":90841,"instance":1842,"goal":"of_app_1842 : of (app u_1 u_2) T_1842 <- of u_1 (arrow T_1842a T_1842) <- of u_2 T_1842a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1843","id":90842,"instance":1843,"goal":"step_beta_1843 : step (app (lam z) h) (z h).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1844","id":90843,"instance":1844,"goal":"step_app1_1844 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1845","id":90844,"instance":1845,"goal":"preservation_1845 : of h T_1845 -> step h h_p -> of h_p T_1845.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1846","id":90845,"instance":1846,"goal":"progress_1846 : of e T_1846 -> value e or (exists e_p. step e e_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1847","id":90846,"instance":1847,"goal":"eq_refl_1847 : eq d d.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1848","id":90847,"instance":1848,"goal":"subst_1848 : subst w y (w y).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1849","id":90848,"instance":1849,"goal":"lam_1849 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1850","id":90849,"instance":1850,"goal":"app_1850 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1851","id":90850,"instance":1851,"goal":"arrow_1851 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1852","id":90851,"instance":1852,"goal":"of_var_1852 : of c T_1852 <- of-env c T_1852.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1853","id":90852,"instance":1853,"goal":"of_lam_1853 : of (lam v) (arrow T_1853a T_1853b) <- ({u} of u T_1853a -> of (v u) T_1853b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1854","id":90853,"instance":1854,"goal":"of_app_1854 : of (app z_1 z_2) T_1854 <- of z_1 (arrow T_1854a T_1854) <- of z_2 T_1854a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1855","id":90854,"instance":1855,"goal":"step_beta_1855 : step (app (lam g) u) (g u).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1856","id":90855,"instance":1856,"goal":"step_app1_1856 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1857","id":90856,"instance":1857,"goal":"preservation_1857 : of z T_1857 -> step z z_p -> of z_p T_1857.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1858","id":90857,"instance":1858,"goal":"progress_1858 : of u T_1858 -> value u or (exists u_p. step u u_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1859","id":90858,"instance":1859,"goal":"eq_refl_1859 : eq a a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1860","id":90859,"instance":1860,"goal":"subst_1860 : subst f u (f u).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1861","id":90860,"instance":1861,"goal":"lam_1861 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1862","id":90861,"instance":1862,"goal":"app_1862 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1863","id":90862,"instance":1863,"goal":"arrow_1863 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1864","id":90863,"instance":1864,"goal":"of_var_1864 : of y T_1864 <- of-env y T_1864.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1865","id":90864,"instance":1865,"goal":"of_lam_1865 : of (lam h) (arrow T_1865a T_1865b) <- ({c} of c T_1865a -> of (h c) T_1865b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1866","id":90865,"instance":1866,"goal":"of_app_1866 : of (app x_1 x_2) T_1866 <- of x_1 (arrow T_1866a T_1866) <- of x_2 T_1866a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1867","id":90866,"instance":1867,"goal":"step_beta_1867 : step (app (lam c) p) (c p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1868","id":90867,"instance":1868,"goal":"step_app1_1868 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1869","id":90868,"instance":1869,"goal":"preservation_1869 : of z T_1869 -> step z z_p -> of z_p T_1869.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1870","id":90869,"instance":1870,"goal":"progress_1870 : of n T_1870 -> value n or (exists n_p. step n n_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1871","id":90870,"instance":1871,"goal":"eq_refl_1871 : eq w w.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1872","id":90871,"instance":1872,"goal":"subst_1872 : subst x v (x v).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1873","id":90872,"instance":1873,"goal":"lam_1873 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1874","id":90873,"instance":1874,"goal":"app_1874 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1875","id":90874,"instance":1875,"goal":"arrow_1875 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1876","id":90875,"instance":1876,"goal":"of_var_1876 : of b T_1876 <- of-env b T_1876.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1877","id":90876,"instance":1877,"goal":"of_lam_1877 : of (lam h) (arrow T_1877a T_1877b) <- ({f} of f T_1877a -> of (h f) T_1877b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1878","id":90877,"instance":1878,"goal":"of_app_1878 : of (app x_1 x_2) T_1878 <- of x_1 (arrow T_1878a T_1878) <- of x_2 T_1878a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1879","id":90878,"instance":1879,"goal":"step_beta_1879 : step (app (lam h) z) (h z).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1880","id":90879,"instance":1880,"goal":"step_app1_1880 : step (app f_1 f_2) (app f_1p f_2) <- step f_1 f_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1881","id":90880,"instance":1881,"goal":"preservation_1881 : of g T_1881 -> step g g_p -> of g_p T_1881.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1882","id":90881,"instance":1882,"goal":"progress_1882 : of y T_1882 -> value y or (exists y_p. step y y_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1883","id":90882,"instance":1883,"goal":"eq_refl_1883 : eq g g.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1884","id":90883,"instance":1884,"goal":"subst_1884 : subst e c (e c).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1885","id":90884,"instance":1885,"goal":"lam_1885 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1886","id":90885,"instance":1886,"goal":"app_1886 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1887","id":90886,"instance":1887,"goal":"arrow_1887 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1888","id":90887,"instance":1888,"goal":"of_var_1888 : of f T_1888 <- of-env f T_1888.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1889","id":90888,"instance":1889,"goal":"of_lam_1889 : of (lam n) (arrow T_1889a T_1889b) <- ({u} of u T_1889a -> of (n u) T_1889b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1890","id":90889,"instance":1890,"goal":"of_app_1890 : of (app w_1 w_2) T_1890 <- of w_1 (arrow T_1890a T_1890) <- of w_2 T_1890a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1891","id":90890,"instance":1891,"goal":"step_beta_1891 : step (app (lam f) p) (f p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1892","id":90891,"instance":1892,"goal":"step_app1_1892 : step (app a_1 a_2) (app a_1p a_2) <- step a_1 a_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1893","id":90892,"instance":1893,"goal":"preservation_1893 : of q T_1893 -> step q q_p -> of q_p T_1893.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1894","id":90893,"instance":1894,"goal":"progress_1894 : of q T_1894 -> value q or (exists q_p. step q q_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1895","id":90894,"instance":1895,"goal":"eq_refl_1895 : eq a a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1896","id":90895,"instance":1896,"goal":"subst_1896 : subst y e (y e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1897","id":90896,"instance":1897,"goal":"lam_1897 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1898","id":90897,"instance":1898,"goal":"app_1898 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1899","id":90898,"instance":1899,"goal":"arrow_1899 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1900","id":90899,"instance":1900,"goal":"of_var_1900 : of b T_1900 <- of-env b T_1900.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1901","id":90900,"instance":1901,"goal":"of_lam_1901 : of (lam v) (arrow T_1901a T_1901b) <- ({m} of m T_1901a -> of (v m) T_1901b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1902","id":90901,"instance":1902,"goal":"of_app_1902 : of (app u_1 u_2) T_1902 <- of u_1 (arrow T_1902a T_1902) <- of u_2 T_1902a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1903","id":90902,"instance":1903,"goal":"step_beta_1903 : step (app (lam z) g) (z g).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1904","id":90903,"instance":1904,"goal":"step_app1_1904 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1905","id":90904,"instance":1905,"goal":"preservation_1905 : of z T_1905 -> step z z_p -> of z_p T_1905.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1906","id":90905,"instance":1906,"goal":"progress_1906 : of d T_1906 -> value d or (exists d_p. step d d_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1907","id":90906,"instance":1907,"goal":"eq_refl_1907 : eq n n.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1908","id":90907,"instance":1908,"goal":"subst_1908 : subst z q (z q).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1909","id":90908,"instance":1909,"goal":"lam_1909 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1910","id":90909,"instance":1910,"goal":"app_1910 : tm -> tm -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1911","id":90910,"instance":1911,"goal":"arrow_1911 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1912","id":90911,"instance":1912,"goal":"of_var_1912 : of c T_1912 <- of-env c T_1912.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1913","id":90912,"instance":1913,"goal":"of_lam_1913 : of (lam h) (arrow T_1913a T_1913b) <- ({m} of m T_1913a -> of (h m) T_1913b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1914","id":90913,"instance":1914,"goal":"of_app_1914 : of (app m_1 m_2) T_1914 <- of m_1 (arrow T_1914a T_1914) <- of m_2 T_1914a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1915","id":90914,"instance":1915,"goal":"step_beta_1915 : step (app (lam y) v) (y v).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1916","id":90915,"instance":1916,"goal":"step_app1_1916 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1917","id":90916,"instance":1917,"goal":"preservation_1917 : of y T_1917 -> step y y_p -> of y_p T_1917.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1918","id":90917,"instance":1918,"goal":"progress_1918 : of p T_1918 -> value p or (exists p_p. step p p_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1919","id":90918,"instance":1919,"goal":"eq_refl_1919 : eq c c.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1920","id":90919,"instance":1920,"goal":"subst_1920 : subst b v (b v).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1921","id":90920,"instance":1921,"goal":"lam_1921 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1922","id":90921,"instance":1922,"goal":"app_1922 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1923","id":90922,"instance":1923,"goal":"arrow_1923 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1924","id":90923,"instance":1924,"goal":"of_var_1924 : of m T_1924 <- of-env m T_1924.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1925","id":90924,"instance":1925,"goal":"of_lam_1925 : of (lam n) (arrow T_1925a T_1925b) <- ({z} of z T_1925a -> of (n z) T_1925b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1926","id":90925,"instance":1926,"goal":"of_app_1926 : of (app v_1 v_2) T_1926 <- of v_1 (arrow T_1926a T_1926) <- of v_2 T_1926a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1927","id":90926,"instance":1927,"goal":"step_beta_1927 : step (app (lam n) u) (n u).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1928","id":90927,"instance":1928,"goal":"step_app1_1928 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1929","id":90928,"instance":1929,"goal":"preservation_1929 : of z T_1929 -> step z z_p -> of z_p T_1929.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1930","id":90929,"instance":1930,"goal":"progress_1930 : of d T_1930 -> value d or (exists d_p. step d d_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1931","id":90930,"instance":1931,"goal":"eq_refl_1931 : eq u u.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1932","id":90931,"instance":1932,"goal":"subst_1932 : subst w v (w v).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1933","id":90932,"instance":1933,"goal":"lam_1933 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1934","id":90933,"instance":1934,"goal":"app_1934 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1935","id":90934,"instance":1935,"goal":"arrow_1935 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1936","id":90935,"instance":1936,"goal":"of_var_1936 : of u T_1936 <- of-env u T_1936.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1937","id":90936,"instance":1937,"goal":"of_lam_1937 : of (lam z) (arrow T_1937a T_1937b) <- ({u} of u T_1937a -> of (z u) T_1937b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1938","id":90937,"instance":1938,"goal":"of_app_1938 : of (app m_1 m_2) T_1938 <- of m_1 (arrow T_1938a T_1938) <- of m_2 T_1938a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1939","id":90938,"instance":1939,"goal":"step_beta_1939 : step (app (lam e) g) (e g).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1940","id":90939,"instance":1940,"goal":"step_app1_1940 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1941","id":90940,"instance":1941,"goal":"preservation_1941 : of m T_1941 -> step m m_p -> of m_p T_1941.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1942","id":90941,"instance":1942,"goal":"progress_1942 : of d T_1942 -> value d or (exists d_p. step d d_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1943","id":90942,"instance":1943,"goal":"eq_refl_1943 : eq f f.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1944","id":90943,"instance":1944,"goal":"subst_1944 : subst f y (f y).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1945","id":90944,"instance":1945,"goal":"lam_1945 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1946","id":90945,"instance":1946,"goal":"app_1946 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1947","id":90946,"instance":1947,"goal":"arrow_1947 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1948","id":90947,"instance":1948,"goal":"of_var_1948 : of n T_1948 <- of-env n T_1948.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1949","id":90948,"instance":1949,"goal":"of_lam_1949 : of (lam z) (arrow T_1949a T_1949b) <- ({v} of v T_1949a -> of (z v) T_1949b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1950","id":90949,"instance":1950,"goal":"of_app_1950 : of (app b_1 b_2) T_1950 <- of b_1 (arrow T_1950a T_1950) <- of b_2 T_1950a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1951","id":90950,"instance":1951,"goal":"step_beta_1951 : step (app (lam z) y) (z y).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1952","id":90951,"instance":1952,"goal":"step_app1_1952 : step (app p_1 p_2) (app p_1p p_2) <- step p_1 p_1p.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1953","id":90952,"instance":1953,"goal":"preservation_1953 : of g T_1953 -> step g g_p -> of g_p T_1953.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1954","id":90953,"instance":1954,"goal":"progress_1954 : of e T_1954 -> value e or (exists e_p. step e e_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1955","id":90954,"instance":1955,"goal":"eq_refl_1955 : eq u u.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1956","id":90955,"instance":1956,"goal":"subst_1956 : subst b z (b z).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1957","id":90956,"instance":1957,"goal":"lam_1957 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1958","id":90957,"instance":1958,"goal":"app_1958 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1959","id":90958,"instance":1959,"goal":"arrow_1959 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1960","id":90959,"instance":1960,"goal":"of_var_1960 : of q T_1960 <- of-env q T_1960.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1961","id":90960,"instance":1961,"goal":"of_lam_1961 : of (lam q) (arrow T_1961a T_1961b) <- ({p} of p T_1961a -> of (q p) T_1961b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1962","id":90961,"instance":1962,"goal":"of_app_1962 : of (app a_1 a_2) T_1962 <- of a_1 (arrow T_1962a T_1962) <- of a_2 T_1962a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1963","id":90962,"instance":1963,"goal":"step_beta_1963 : step (app (lam m) v) (m v).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1964","id":90963,"instance":1964,"goal":"step_app1_1964 : step (app e_1 e_2) (app e_1p e_2) <- step e_1 e_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1965","id":90964,"instance":1965,"goal":"preservation_1965 : of a T_1965 -> step a a_p -> of a_p T_1965.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1966","id":90965,"instance":1966,"goal":"progress_1966 : of a T_1966 -> value a or (exists a_p. step a a_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1967","id":90966,"instance":1967,"goal":"eq_refl_1967 : eq h h.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1968","id":90967,"instance":1968,"goal":"subst_1968 : subst p e (p e).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1969","id":90968,"instance":1969,"goal":"lam_1969 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1970","id":90969,"instance":1970,"goal":"app_1970 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1971","id":90970,"instance":1971,"goal":"arrow_1971 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1972","id":90971,"instance":1972,"goal":"of_var_1972 : of b T_1972 <- of-env b T_1972.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1973","id":90972,"instance":1973,"goal":"of_lam_1973 : of (lam e) (arrow T_1973a T_1973b) <- ({b} of b T_1973a -> of (e b) T_1973b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1974","id":90973,"instance":1974,"goal":"of_app_1974 : of (app c_1 c_2) T_1974 <- of c_1 (arrow T_1974a T_1974) <- of c_2 T_1974a.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1975","id":90974,"instance":1975,"goal":"step_beta_1975 : step (app (lam e) h) (e h).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1976","id":90975,"instance":1976,"goal":"step_app1_1976 : step (app g_1 g_2) (app g_1p g_2) <- step g_1 g_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1977","id":90976,"instance":1977,"goal":"preservation_1977 : of f T_1977 -> step f f_p -> of f_p T_1977.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1978","id":90977,"instance":1978,"goal":"progress_1978 : of v T_1978 -> value v or (exists v_p. step v v_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1979","id":90978,"instance":1979,"goal":"eq_refl_1979 : eq m m.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1980","id":90979,"instance":1980,"goal":"subst_1980 : subst d c (d c).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1981","id":90980,"instance":1981,"goal":"lam_1981 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1982","id":90981,"instance":1982,"goal":"app_1982 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1983","id":90982,"instance":1983,"goal":"arrow_1983 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1984","id":90983,"instance":1984,"goal":"of_var_1984 : of m T_1984 <- of-env m T_1984.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1985","id":90984,"instance":1985,"goal":"of_lam_1985 : of (lam w) (arrow T_1985a T_1985b) <- ({c} of c T_1985a -> of (w c) T_1985b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1986","id":90985,"instance":1986,"goal":"of_app_1986 : of (app f_1 f_2) T_1986 <- of f_1 (arrow T_1986a T_1986) <- of f_2 T_1986a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1987","id":90986,"instance":1987,"goal":"step_beta_1987 : step (app (lam z) e) (z e).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_1988","id":90987,"instance":1988,"goal":"step_app1_1988 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_1989","id":90988,"instance":1989,"goal":"preservation_1989 : of v T_1989 -> step v v_p -> of v_p T_1989.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_1990","id":90989,"instance":1990,"goal":"progress_1990 : of u T_1990 -> value u or (exists u_p. step u u_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_1991","id":90990,"instance":1991,"goal":"eq_refl_1991 : eq w w.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_1992","id":90991,"instance":1992,"goal":"subst_1992 : subst u c (u c).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_1993","id":90992,"instance":1993,"goal":"lam_1993 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_1994","id":90993,"instance":1994,"goal":"app_1994 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_1995","id":90994,"instance":1995,"goal":"arrow_1995 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_1996","id":90995,"instance":1996,"goal":"of_var_1996 : of d T_1996 <- of-env d T_1996.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_1997","id":90996,"instance":1997,"goal":"of_lam_1997 : of (lam h) (arrow T_1997a T_1997b) <- ({g} of g T_1997a -> of (h g) T_1997b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_1998","id":90997,"instance":1998,"goal":"of_app_1998 : of (app y_1 y_2) T_1998 <- of y_1 (arrow T_1998a T_1998) <- of y_2 T_1998a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_1999","id":90998,"instance":1999,"goal":"step_beta_1999 : step (app (lam y) p) (y p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2000","id":90999,"instance":2000,"goal":"step_app1_2000 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2001","id":91000,"instance":2001,"goal":"preservation_2001 : of e T_2001 -> step e e_p -> of e_p T_2001.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2002","id":91001,"instance":2002,"goal":"progress_2002 : of w T_2002 -> value w or (exists w_p. step w w_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2003","id":91002,"instance":2003,"goal":"eq_refl_2003 : eq h h.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2004","id":91003,"instance":2004,"goal":"subst_2004 : subst n g (n g).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2005","id":91004,"instance":2005,"goal":"lam_2005 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2006","id":91005,"instance":2006,"goal":"app_2006 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2007","id":91006,"instance":2007,"goal":"arrow_2007 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2008","id":91007,"instance":2008,"goal":"of_var_2008 : of m T_2008 <- of-env m T_2008.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2009","id":91008,"instance":2009,"goal":"of_lam_2009 : of (lam u) (arrow T_2009a T_2009b) <- ({q} of q T_2009a -> of (u q) T_2009b).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2010","id":91009,"instance":2010,"goal":"of_app_2010 : of (app p_1 p_2) T_2010 <- of p_1 (arrow T_2010a T_2010) <- of p_2 T_2010a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2011","id":91010,"instance":2011,"goal":"step_beta_2011 : step (app (lam h) v) (h v).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2012","id":91011,"instance":2012,"goal":"step_app1_2012 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2013","id":91012,"instance":2013,"goal":"preservation_2013 : of g T_2013 -> step g g_p -> of g_p T_2013.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2014","id":91013,"instance":2014,"goal":"progress_2014 : of m T_2014 -> value m or (exists m_p. step m m_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2015","id":91014,"instance":2015,"goal":"eq_refl_2015 : eq z z.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2016","id":91015,"instance":2016,"goal":"subst_2016 : subst f x (f x).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2017","id":91016,"instance":2017,"goal":"lam_2017 : (tm -> tm) -> tm.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2018","id":91017,"instance":2018,"goal":"app_2018 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2019","id":91018,"instance":2019,"goal":"arrow_2019 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2020","id":91019,"instance":2020,"goal":"of_var_2020 : of m T_2020 <- of-env m T_2020.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2021","id":91020,"instance":2021,"goal":"of_lam_2021 : of (lam h) (arrow T_2021a T_2021b) <- ({d} of d T_2021a -> of (h d) T_2021b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2022","id":91021,"instance":2022,"goal":"of_app_2022 : of (app m_1 m_2) T_2022 <- of m_1 (arrow T_2022a T_2022) <- of m_2 T_2022a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2023","id":91022,"instance":2023,"goal":"step_beta_2023 : step (app (lam h) e) (h e).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2024","id":91023,"instance":2024,"goal":"step_app1_2024 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2025","id":91024,"instance":2025,"goal":"preservation_2025 : of v T_2025 -> step v v_p -> of v_p T_2025.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2026","id":91025,"instance":2026,"goal":"progress_2026 : of c T_2026 -> value c or (exists c_p. step c c_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2027","id":91026,"instance":2027,"goal":"eq_refl_2027 : eq y y.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2028","id":91027,"instance":2028,"goal":"subst_2028 : subst g f (g f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2029","id":91028,"instance":2029,"goal":"lam_2029 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2030","id":91029,"instance":2030,"goal":"app_2030 : tm -> tm -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2031","id":91030,"instance":2031,"goal":"arrow_2031 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2032","id":91031,"instance":2032,"goal":"of_var_2032 : of n T_2032 <- of-env n T_2032.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2033","id":91032,"instance":2033,"goal":"of_lam_2033 : of (lam u) (arrow T_2033a T_2033b) <- ({b} of b T_2033a -> of (u b) T_2033b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2034","id":91033,"instance":2034,"goal":"of_app_2034 : of (app w_1 w_2) T_2034 <- of w_1 (arrow T_2034a T_2034) <- of w_2 T_2034a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2035","id":91034,"instance":2035,"goal":"step_beta_2035 : step (app (lam a) m) (a m).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2036","id":91035,"instance":2036,"goal":"step_app1_2036 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2037","id":91036,"instance":2037,"goal":"preservation_2037 : of a T_2037 -> step a a_p -> of a_p T_2037.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2038","id":91037,"instance":2038,"goal":"progress_2038 : of d T_2038 -> value d or (exists d_p. step d d_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2039","id":91038,"instance":2039,"goal":"eq_refl_2039 : eq n n.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2040","id":91039,"instance":2040,"goal":"subst_2040 : subst h w (h w).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2041","id":91040,"instance":2041,"goal":"lam_2041 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2042","id":91041,"instance":2042,"goal":"app_2042 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2043","id":91042,"instance":2043,"goal":"arrow_2043 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2044","id":91043,"instance":2044,"goal":"of_var_2044 : of f T_2044 <- of-env f T_2044.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2045","id":91044,"instance":2045,"goal":"of_lam_2045 : of (lam v) (arrow T_2045a T_2045b) <- ({p} of p T_2045a -> of (v p) T_2045b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2046","id":91045,"instance":2046,"goal":"of_app_2046 : of (app f_1 f_2) T_2046 <- of f_1 (arrow T_2046a T_2046) <- of f_2 T_2046a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2047","id":91046,"instance":2047,"goal":"step_beta_2047 : step (app (lam p) f) (p f).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2048","id":91047,"instance":2048,"goal":"step_app1_2048 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2049","id":91048,"instance":2049,"goal":"preservation_2049 : of u T_2049 -> step u u_p -> of u_p T_2049.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2050","id":91049,"instance":2050,"goal":"progress_2050 : of z T_2050 -> value z or (exists z_p. step z z_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2051","id":91050,"instance":2051,"goal":"eq_refl_2051 : eq h h.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2052","id":91051,"instance":2052,"goal":"subst_2052 : subst f p (f p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2053","id":91052,"instance":2053,"goal":"lam_2053 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2054","id":91053,"instance":2054,"goal":"app_2054 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2055","id":91054,"instance":2055,"goal":"arrow_2055 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2056","id":91055,"instance":2056,"goal":"of_var_2056 : of u T_2056 <- of-env u T_2056.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2057","id":91056,"instance":2057,"goal":"of_lam_2057 : of (lam u) (arrow T_2057a T_2057b) <- ({y} of y T_2057a -> of (u y) T_2057b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2058","id":91057,"instance":2058,"goal":"of_app_2058 : of (app a_1 a_2) T_2058 <- of a_1 (arrow T_2058a T_2058) <- of a_2 T_2058a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2059","id":91058,"instance":2059,"goal":"step_beta_2059 : step (app (lam z) f) (z f).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2060","id":91059,"instance":2060,"goal":"step_app1_2060 : step (app u_1 u_2) (app u_1p u_2) <- step u_1 u_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2061","id":91060,"instance":2061,"goal":"preservation_2061 : of m T_2061 -> step m m_p -> of m_p T_2061.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2062","id":91061,"instance":2062,"goal":"progress_2062 : of w T_2062 -> value w or (exists w_p. step w w_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2063","id":91062,"instance":2063,"goal":"eq_refl_2063 : eq h h.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2064","id":91063,"instance":2064,"goal":"subst_2064 : subst n p (n p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2065","id":91064,"instance":2065,"goal":"lam_2065 : (tm -> tm) -> tm.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2066","id":91065,"instance":2066,"goal":"app_2066 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2067","id":91066,"instance":2067,"goal":"arrow_2067 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2068","id":91067,"instance":2068,"goal":"of_var_2068 : of u T_2068 <- of-env u T_2068.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2069","id":91068,"instance":2069,"goal":"of_lam_2069 : of (lam n) (arrow T_2069a T_2069b) <- ({b} of b T_2069a -> of (n b) T_2069b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2070","id":91069,"instance":2070,"goal":"of_app_2070 : of (app y_1 y_2) T_2070 <- of y_1 (arrow T_2070a T_2070) <- of y_2 T_2070a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2071","id":91070,"instance":2071,"goal":"step_beta_2071 : step (app (lam f) v) (f v).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2072","id":91071,"instance":2072,"goal":"step_app1_2072 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2073","id":91072,"instance":2073,"goal":"preservation_2073 : of c T_2073 -> step c c_p -> of c_p T_2073.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2074","id":91073,"instance":2074,"goal":"progress_2074 : of h T_2074 -> value h or (exists h_p. step h h_p).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2075","id":91074,"instance":2075,"goal":"eq_refl_2075 : eq d d.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2076","id":91075,"instance":2076,"goal":"subst_2076 : subst v p (v p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2077","id":91076,"instance":2077,"goal":"lam_2077 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2078","id":91077,"instance":2078,"goal":"app_2078 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2079","id":91078,"instance":2079,"goal":"arrow_2079 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2080","id":91079,"instance":2080,"goal":"of_var_2080 : of f T_2080 <- of-env f T_2080.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2081","id":91080,"instance":2081,"goal":"of_lam_2081 : of (lam c) (arrow T_2081a T_2081b) <- ({a} of a T_2081a -> of (c a) T_2081b).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2082","id":91081,"instance":2082,"goal":"of_app_2082 : of (app e_1 e_2) T_2082 <- of e_1 (arrow T_2082a T_2082) <- of e_2 T_2082a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2083","id":91082,"instance":2083,"goal":"step_beta_2083 : step (app (lam w) v) (w v).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2084","id":91083,"instance":2084,"goal":"step_app1_2084 : step (app z_1 z_2) (app z_1p z_2) <- step z_1 z_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2085","id":91084,"instance":2085,"goal":"preservation_2085 : of f T_2085 -> step f f_p -> of f_p T_2085.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2086","id":91085,"instance":2086,"goal":"progress_2086 : of a T_2086 -> value a or (exists a_p. step a a_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2087","id":91086,"instance":2087,"goal":"eq_refl_2087 : eq q q.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2088","id":91087,"instance":2088,"goal":"subst_2088 : subst v a (v a).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2089","id":91088,"instance":2089,"goal":"lam_2089 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2090","id":91089,"instance":2090,"goal":"app_2090 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2091","id":91090,"instance":2091,"goal":"arrow_2091 : tp -> tp -> tp.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2092","id":91091,"instance":2092,"goal":"of_var_2092 : of n T_2092 <- of-env n T_2092.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2093","id":91092,"instance":2093,"goal":"of_lam_2093 : of (lam z) (arrow T_2093a T_2093b) <- ({f} of f T_2093a -> of (z f) T_2093b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2094","id":91093,"instance":2094,"goal":"of_app_2094 : of (app p_1 p_2) T_2094 <- of p_1 (arrow T_2094a T_2094) <- of p_2 T_2094a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2095","id":91094,"instance":2095,"goal":"step_beta_2095 : step (app (lam f) x) (f x).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2096","id":91095,"instance":2096,"goal":"step_app1_2096 : step (app q_1 q_2) (app q_1p q_2) <- step q_1 q_1p.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2097","id":91096,"instance":2097,"goal":"preservation_2097 : of u T_2097 -> step u u_p -> of u_p T_2097.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2098","id":91097,"instance":2098,"goal":"progress_2098 : of n T_2098 -> value n or (exists n_p. step n n_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2099","id":91098,"instance":2099,"goal":"eq_refl_2099 : eq p p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2100","id":91099,"instance":2100,"goal":"subst_2100 : subst a g (a g).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2101","id":91100,"instance":2101,"goal":"lam_2101 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2102","id":91101,"instance":2102,"goal":"app_2102 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2103","id":91102,"instance":2103,"goal":"arrow_2103 : tp -> tp -> tp.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2104","id":91103,"instance":2104,"goal":"of_var_2104 : of d T_2104 <- of-env d T_2104.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2105","id":91104,"instance":2105,"goal":"of_lam_2105 : of (lam b) (arrow T_2105a T_2105b) <- ({p} of p T_2105a -> of (b p) T_2105b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2106","id":91105,"instance":2106,"goal":"of_app_2106 : of (app q_1 q_2) T_2106 <- of q_1 (arrow T_2106a T_2106) <- of q_2 T_2106a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2107","id":91106,"instance":2107,"goal":"step_beta_2107 : step (app (lam v) b) (v b).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2108","id":91107,"instance":2108,"goal":"step_app1_2108 : step (app v_1 v_2) (app v_1p v_2) <- step v_1 v_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2109","id":91108,"instance":2109,"goal":"preservation_2109 : of g T_2109 -> step g g_p -> of g_p T_2109.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2110","id":91109,"instance":2110,"goal":"progress_2110 : of x T_2110 -> value x or (exists x_p. step x x_p).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2111","id":91110,"instance":2111,"goal":"eq_refl_2111 : eq m m.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2112","id":91111,"instance":2112,"goal":"subst_2112 : subst p q (p q).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2113","id":91112,"instance":2113,"goal":"lam_2113 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2114","id":91113,"instance":2114,"goal":"app_2114 : tm -> tm -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2115","id":91114,"instance":2115,"goal":"arrow_2115 : tp -> tp -> tp.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2116","id":91115,"instance":2116,"goal":"of_var_2116 : of c T_2116 <- of-env c T_2116.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2117","id":91116,"instance":2117,"goal":"of_lam_2117 : of (lam e) (arrow T_2117a T_2117b) <- ({w} of w T_2117a -> of (e w) T_2117b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2118","id":91117,"instance":2118,"goal":"of_app_2118 : of (app h_1 h_2) T_2118 <- of h_1 (arrow T_2118a T_2118) <- of h_2 T_2118a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2119","id":91118,"instance":2119,"goal":"step_beta_2119 : step (app (lam n) q) (n q).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2120","id":91119,"instance":2120,"goal":"step_app1_2120 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2121","id":91120,"instance":2121,"goal":"preservation_2121 : of q T_2121 -> step q q_p -> of q_p T_2121.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2122","id":91121,"instance":2122,"goal":"progress_2122 : of w T_2122 -> value w or (exists w_p. step w w_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2123","id":91122,"instance":2123,"goal":"eq_refl_2123 : eq x x.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2124","id":91123,"instance":2124,"goal":"subst_2124 : subst v e (v e).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2125","id":91124,"instance":2125,"goal":"lam_2125 : (tm -> tm) -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2126","id":91125,"instance":2126,"goal":"app_2126 : tm -> tm -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2127","id":91126,"instance":2127,"goal":"arrow_2127 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2128","id":91127,"instance":2128,"goal":"of_var_2128 : of e T_2128 <- of-env e T_2128.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2129","id":91128,"instance":2129,"goal":"of_lam_2129 : of (lam x) (arrow T_2129a T_2129b) <- ({a} of a T_2129a -> of (x a) T_2129b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2130","id":91129,"instance":2130,"goal":"of_app_2130 : of (app c_1 c_2) T_2130 <- of c_1 (arrow T_2130a T_2130) <- of c_2 T_2130a.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2131","id":91130,"instance":2131,"goal":"step_beta_2131 : step (app (lam h) d) (h d).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2132","id":91131,"instance":2132,"goal":"step_app1_2132 : step (app m_1 m_2) (app m_1p m_2) <- step m_1 m_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2133","id":91132,"instance":2133,"goal":"preservation_2133 : of v T_2133 -> step v v_p -> of v_p T_2133.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2134","id":91133,"instance":2134,"goal":"progress_2134 : of v T_2134 -> value v or (exists v_p. step v v_p).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2135","id":91134,"instance":2135,"goal":"eq_refl_2135 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2136","id":91135,"instance":2136,"goal":"subst_2136 : subst a v (a v).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2137","id":91136,"instance":2137,"goal":"lam_2137 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2138","id":91137,"instance":2138,"goal":"app_2138 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2139","id":91138,"instance":2139,"goal":"arrow_2139 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2140","id":91139,"instance":2140,"goal":"of_var_2140 : of z T_2140 <- of-env z T_2140.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2141","id":91140,"instance":2141,"goal":"of_lam_2141 : of (lam h) (arrow T_2141a T_2141b) <- ({z} of z T_2141a -> of (h z) T_2141b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2142","id":91141,"instance":2142,"goal":"of_app_2142 : of (app h_1 h_2) T_2142 <- of h_1 (arrow T_2142a T_2142) <- of h_2 T_2142a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2143","id":91142,"instance":2143,"goal":"step_beta_2143 : step (app (lam z) x) (z x).","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2144","id":91143,"instance":2144,"goal":"step_app1_2144 : step (app n_1 n_2) (app n_1p n_2) <- step n_1 n_1p.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2145","id":91144,"instance":2145,"goal":"preservation_2145 : of b T_2145 -> step b b_p -> of b_p T_2145.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2146","id":91145,"instance":2146,"goal":"progress_2146 : of v T_2146 -> value v or (exists v_p. step v v_p).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2147","id":91146,"instance":2147,"goal":"eq_refl_2147 : eq a a.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2148","id":91147,"instance":2148,"goal":"subst_2148 : subst g f (g f).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2149","id":91148,"instance":2149,"goal":"lam_2149 : (tm -> tm) -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2150","id":91149,"instance":2150,"goal":"app_2150 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2151","id":91150,"instance":2151,"goal":"arrow_2151 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2152","id":91151,"instance":2152,"goal":"of_var_2152 : of d T_2152 <- of-env d T_2152.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2153","id":91152,"instance":2153,"goal":"of_lam_2153 : of (lam e) (arrow T_2153a T_2153b) <- ({g} of g T_2153a -> of (e g) T_2153b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2154","id":91153,"instance":2154,"goal":"of_app_2154 : of (app m_1 m_2) T_2154 <- of m_1 (arrow T_2154a T_2154) <- of m_2 T_2154a.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2155","id":91154,"instance":2155,"goal":"step_beta_2155 : step (app (lam y) d) (y d).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2156","id":91155,"instance":2156,"goal":"step_app1_2156 : step (app c_1 c_2) (app c_1p c_2) <- step c_1 c_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2157","id":91156,"instance":2157,"goal":"preservation_2157 : of n T_2157 -> step n n_p -> of n_p T_2157.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2158","id":91157,"instance":2158,"goal":"progress_2158 : of q T_2158 -> value q or (exists q_p. step q q_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2159","id":91158,"instance":2159,"goal":"eq_refl_2159 : eq c c.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2160","id":91159,"instance":2160,"goal":"subst_2160 : subst e g (e g).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2161","id":91160,"instance":2161,"goal":"lam_2161 : (tm -> tm) -> tm.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2162","id":91161,"instance":2162,"goal":"app_2162 : tm -> tm -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2163","id":91162,"instance":2163,"goal":"arrow_2163 : tp -> tp -> tp.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2164","id":91163,"instance":2164,"goal":"of_var_2164 : of p T_2164 <- of-env p T_2164.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2165","id":91164,"instance":2165,"goal":"of_lam_2165 : of (lam e) (arrow T_2165a T_2165b) <- ({f} of f T_2165a -> of (e f) T_2165b).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2166","id":91165,"instance":2166,"goal":"of_app_2166 : of (app e_1 e_2) T_2166 <- of e_1 (arrow T_2166a T_2166) <- of e_2 T_2166a.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2167","id":91166,"instance":2167,"goal":"step_beta_2167 : step (app (lam m) x) (m x).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2168","id":91167,"instance":2168,"goal":"step_app1_2168 : step (app d_1 d_2) (app d_1p d_2) <- step d_1 d_1p.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2169","id":91168,"instance":2169,"goal":"preservation_2169 : of u T_2169 -> step u u_p -> of u_p T_2169.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2170","id":91169,"instance":2170,"goal":"progress_2170 : of n T_2170 -> value n or (exists n_p. step n n_p).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2171","id":91170,"instance":2171,"goal":"eq_refl_2171 : eq e e.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2172","id":91171,"instance":2172,"goal":"subst_2172 : subst n p (n p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2173","id":91172,"instance":2173,"goal":"lam_2173 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2174","id":91173,"instance":2174,"goal":"app_2174 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2175","id":91174,"instance":2175,"goal":"arrow_2175 : tp -> tp -> tp.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2176","id":91175,"instance":2176,"goal":"of_var_2176 : of u T_2176 <- of-env u T_2176.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2177","id":91176,"instance":2177,"goal":"of_lam_2177 : of (lam b) (arrow T_2177a T_2177b) <- ({x} of x T_2177a -> of (b x) T_2177b).","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2178","id":91177,"instance":2178,"goal":"of_app_2178 : of (app h_1 h_2) T_2178 <- of h_1 (arrow T_2178a T_2178) <- of h_2 T_2178a.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2179","id":91178,"instance":2179,"goal":"step_beta_2179 : step (app (lam q) b) (q b).","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2180","id":91179,"instance":2180,"goal":"step_app1_2180 : step (app x_1 x_2) (app x_1p x_2) <- step x_1 x_1p.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2181","id":91180,"instance":2181,"goal":"preservation_2181 : of p T_2181 -> step p p_p -> of p_p T_2181.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2182","id":91181,"instance":2182,"goal":"progress_2182 : of w T_2182 -> value w or (exists w_p. step w w_p).","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2183","id":91182,"instance":2183,"goal":"eq_refl_2183 : eq n n.","lf_family":"tp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2184","id":91183,"instance":2184,"goal":"subst_2184 : subst q y (q y).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2185","id":91184,"instance":2185,"goal":"lam_2185 : (tm -> tm) -> tm.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2186","id":91185,"instance":2186,"goal":"app_2186 : tm -> tm -> tm.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2187","id":91186,"instance":2187,"goal":"arrow_2187 : tp -> tp -> tp.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2188","id":91187,"instance":2188,"goal":"of_var_2188 : of d T_2188 <- of-env d T_2188.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_lam","theorem":"of_lam_2189","id":91188,"instance":2189,"goal":"of_lam_2189 : of (lam n) (arrow T_2189a T_2189b) <- ({g} of g T_2189a -> of (n g) T_2189b).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_app","theorem":"of_app_2190","id":91189,"instance":2190,"goal":"of_app_2190 : of (app d_1 d_2) T_2190 <- of d_1 (arrow T_2190a T_2190) <- of d_2 T_2190a.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_beta","theorem":"step_beta_2191","id":91190,"instance":2191,"goal":"step_beta_2191 : step (app (lam n) c) (n c).","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"step_app1","theorem":"step_app1_2192","id":91191,"instance":2192,"goal":"step_app1_2192 : step (app b_1 b_2) (app b_1p b_2) <- step b_1 b_1p.","lf_family":"exp","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"pres","theorem":"pres_2193","id":91192,"instance":2193,"goal":"preservation_2193 : of g T_2193 -> step g g_p -> of g_p T_2193.","lf_family":"ctx","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"prog","theorem":"prog_2194","id":91193,"instance":2194,"goal":"progress_2194 : of x T_2194 -> value x or (exists x_p. step x x_p).","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"eq_refl","theorem":"eq_refl_2195","id":91194,"instance":2195,"goal":"eq_refl_2195 : eq d d.","lf_family":"tm","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"subst","theorem":"subst_2196","id":91195,"instance":2196,"goal":"subst_2196 : subst v u (v u).","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_lam","theorem":"tm_lam_2197","id":91196,"instance":2197,"goal":"lam_2197 : (tm -> tm) -> tm.","lf_family":"env","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tm_app","theorem":"tm_app_2198","id":91197,"instance":2198,"goal":"app_2198 : tm -> tm -> tm.","lf_family":"val","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"tp_arr","theorem":"tp_arr_2199","id":91198,"instance":2199,"goal":"arrow_2199 : tp -> tp -> tp.","lf_family":"typ","context":["mode"]} -{"prover":"Twelf","tactic_proof":"%mode.","source":"twelf_synthetic/scaled","variant":"of_var","theorem":"of_var_2200","id":91199,"instance":2200,"goal":"of_var_2200 : of v T_2200 <- of-env v T_2200.","lf_family":"tp","context":["mode"]} +{"id":89000,"prover":"Twelf","theorem":"nat","goal":"nat : type.","context":["nat"],"tactic_proof":"z : nat.\ns : nat -> nat.","source":"twelf_synthetic/natural_numbers"} +{"id":89001,"prover":"Twelf","theorem":"plus","goal":"plus : nat -> nat -> nat -> type.","context":["plus"],"tactic_proof":"plus-z : plus z N N.\nplus-s : plus (s M) N (s P) <- plus M N P.","source":"twelf_synthetic/natural_numbers"} +{"id":89002,"prover":"Twelf","theorem":"plus_comm","goal":"plus-comm : plus M N P -> plus N M P -> type.","context":["plus-comm"],"tactic_proof":"plus-comm-z : plus-comm plus-z D <- plus-zero-right D.\nplus-comm-s : plus-comm (plus-s D1) D2 <- plus-comm D1 D3 <- plus-succ-right D3 D2.","source":"twelf_synthetic/natural_numbers"} +{"id":89003,"prover":"Twelf","theorem":"plus_assoc","goal":"plus-assoc : plus A B AB -> plus AB C ABC -> plus B C BC -> plus A BC ABC -> type.","context":["plus-assoc"],"tactic_proof":"%mode plus-assoc +D1 +D2 -D3 -D4.","source":"twelf_synthetic/natural_numbers"} +{"id":89004,"prover":"Twelf","theorem":"times","goal":"times : nat -> nat -> nat -> type.","context":["times"],"tactic_proof":"times-z : times z N z.\ntimes-s : times (s M) N P <- times M N Q <- plus N Q P.","source":"twelf_synthetic/natural_numbers"} +{"id":89005,"prover":"Twelf","theorem":"leq","goal":"leq : nat -> nat -> type.","context":["leq"],"tactic_proof":"leq-z : leq z N.\nleq-s : leq (s M) (s N) <- leq M N.","source":"twelf_synthetic/natural_numbers"} +{"id":89006,"prover":"Twelf","theorem":"leq_refl","goal":"leq-refl : {N:nat} leq N N -> type.","context":["leq-refl","N"],"tactic_proof":"leq-refl-z : leq-refl z leq-z.\nleq-refl-s : leq-refl (s N) (leq-s D) <- leq-refl N D.","source":"twelf_synthetic/natural_numbers"} +{"id":89007,"prover":"Twelf","theorem":"leq_trans","goal":"leq-trans : leq A B -> leq B C -> leq A C -> type.","context":["leq-trans"],"tactic_proof":"%mode leq-trans +D1 +D2 -D3.","source":"twelf_synthetic/natural_numbers"} +{"id":89008,"prover":"Twelf","theorem":"max","goal":"max : nat -> nat -> nat -> type.","context":["max"],"tactic_proof":"max-zl : max z N N.\nmax-zr : max N z N.\nmax-ss : max (s M) (s N) (s P) <- max M N P.","source":"twelf_synthetic/natural_numbers"} +{"id":89009,"prover":"Twelf","theorem":"even_odd","goal":"even : nat -> type.\nodd : nat -> type.","context":["even","odd"],"tactic_proof":"even-z : even z.\neven-s : even (s N) <- odd N.\nodd-s : odd (s N) <- even N.","source":"twelf_synthetic/natural_numbers"} +{"id":89010,"prover":"Twelf","theorem":"tp","goal":"tp : type.","context":["tp"],"tactic_proof":"arr : tp -> tp -> tp.\nunit : tp.","source":"twelf_synthetic/lambda_calculus"} +{"id":89011,"prover":"Twelf","theorem":"tm","goal":"tm : type.","context":["tm"],"tactic_proof":"app : tm -> tm -> tm.\nlam : tp -> (tm -> tm) -> tm.\nempty : tm.","source":"twelf_synthetic/lambda_calculus"} +{"id":89012,"prover":"Twelf","theorem":"of","goal":"of : tm -> tp -> type.","context":["of"],"tactic_proof":"of-app : of (app E1 E2) T <- of E1 (arr T2 T) <- of E2 T2.\nof-lam : of (lam T1 E) (arr T1 T2) <- ({x:tm} of x T1 -> of (E x) T2).\nof-empty : of empty unit.","source":"twelf_synthetic/lambda_calculus"} +{"id":89013,"prover":"Twelf","theorem":"value","goal":"value : tm -> type.","context":["value"],"tactic_proof":"value-lam : value (lam T E).\nvalue-empty : value empty.","source":"twelf_synthetic/lambda_calculus"} +{"id":89014,"prover":"Twelf","theorem":"step","goal":"step : tm -> tm -> type.","context":["step"],"tactic_proof":"step-app1 : step (app E1 E2) (app E1' E2) <- step E1 E1'.\nstep-app2 : step (app V E2) (app V E2') <- value V <- step E2 E2'.\nstep-beta : step (app (lam T E) V) (E V) <- value V.","source":"twelf_synthetic/lambda_calculus"} +{"id":89015,"prover":"Twelf","theorem":"preservation","goal":"preservation : of E T -> step E E' -> of E' T -> type.","context":["preservation"],"tactic_proof":"%mode preservation +Dof +Dstep -Dof'.\npreservation-app1 : preservation (of-app D1 D2) (step-app1 S) (of-app D1' D2) <- preservation D1 S D1'.\npreservation-beta : preservation (of-app (of-lam D1) D2) (step-beta V) (D1 _ D2).","source":"twelf_synthetic/lambda_calculus"} +{"id":89016,"prover":"Twelf","theorem":"progress","goal":"progress : of E T -> (value E + step E E') -> type.","context":["progress"],"tactic_proof":"%mode progress +Dof -Dresult.","source":"twelf_synthetic/lambda_calculus"} +{"id":89017,"prover":"Twelf","theorem":"subst","goal":"subst : (tm -> tm) -> tm -> tm -> type.","context":["subst"],"tactic_proof":"subst-var : subst ([x] x) V V.\nsubst-const : subst ([x] C) V C.\nsubst-app : subst ([x] app (E1 x) (E2 x)) V (app E1' E2') <- subst E1 V E1' <- subst E2 V E2'.\nsubst-lam : subst ([x] lam T ([y] E x y)) V (lam T E') <- ({y:tm} subst ([x] E x y) V (E' y)).","source":"twelf_synthetic/lambda_calculus"} +{"id":89018,"prover":"Twelf","theorem":"list","goal":"list : type.","context":["list"],"tactic_proof":"nil : list.\ncons : nat -> list -> list.","source":"twelf_synthetic/lists"} +{"id":89019,"prover":"Twelf","theorem":"append","goal":"append : list -> list -> list -> type.","context":["append"],"tactic_proof":"append-nil : append nil L L.\nappend-cons : append (cons H T) L (cons H R) <- append T L R.","source":"twelf_synthetic/lists"} +{"id":89020,"prover":"Twelf","theorem":"append_assoc","goal":"append-assoc : append L1 L2 L12 -> append L12 L3 L123 -> append L2 L3 L23 -> append L1 L23 L123 -> type.","context":["append-assoc"],"tactic_proof":"%mode append-assoc +D1 +D2 -D3 -D4.","source":"twelf_synthetic/lists"} +{"id":89021,"prover":"Twelf","theorem":"reverse","goal":"reverse : list -> list -> type.","context":["reverse"],"tactic_proof":"reverse-nil : reverse nil nil.\nreverse-cons : reverse (cons H T) R <- reverse T T' <- append T' (cons H nil) R.","source":"twelf_synthetic/lists"} +{"id":89022,"prover":"Twelf","theorem":"length","goal":"length : list -> nat -> type.","context":["length"],"tactic_proof":"length-nil : length nil z.\nlength-cons : length (cons H T) (s N) <- length T N.","source":"twelf_synthetic/lists"} +{"id":89023,"prover":"Twelf","theorem":"member","goal":"member : nat -> list -> type.","context":["member"],"tactic_proof":"member-head : member X (cons X L).\nmember-tail : member X (cons Y L) <- member X L.","source":"twelf_synthetic/lists"} +{"id":89024,"prover":"Twelf","theorem":"sorted","goal":"sorted : list -> type.","context":["sorted"],"tactic_proof":"sorted-nil : sorted nil.\nsorted-one : sorted (cons X nil).\nsorted-cons : sorted (cons X (cons Y L)) <- leq X Y <- sorted (cons Y L).","source":"twelf_synthetic/lists"} +{"id":89025,"prover":"Twelf","theorem":"insert_sorted","goal":"insert : nat -> list -> list -> type.","context":["insert"],"tactic_proof":"insert-nil : insert X nil (cons X nil).\ninsert-leq : insert X (cons Y L) (cons X (cons Y L)) <- leq X Y.\ninsert-gt : insert X (cons Y L) (cons Y L') <- insert X L L'.","source":"twelf_synthetic/lists"} +{"id":89026,"prover":"Twelf","theorem":"prop","goal":"o : type.","context":["o"],"tactic_proof":"imp : o -> o -> o.\nand : o -> o -> o.\nor : o -> o -> o.\nnot : o -> o.\nforall : (i -> o) -> o.\nexists : (i -> o) -> o.","source":"twelf_synthetic/logic_framework"} +{"id":89027,"prover":"Twelf","theorem":"pf","goal":"pf : o -> type.","context":["pf"],"tactic_proof":"imp-i : ({x:pf A} pf B) -> pf (imp A B).\nimp-e : pf (imp A B) -> pf A -> pf B.\nand-i : pf A -> pf B -> pf (and A B).\nand-e1 : pf (and A B) -> pf A.\nand-e2 : pf (and A B) -> pf B.\nor-i1 : pf A -> pf (or A B).\nor-i2 : pf B -> pf (or A B).\nor-e : pf (or A B) -> ({x:pf A} pf C) -> ({x:pf B} pf C) -> pf C.\nnot-i : ({x:pf A} pf false) -> pf (not A).\nnot-e : pf (not A) -> pf A -> pf false.\nforall-i : ({x:i} pf (A x)) -> pf (forall A).\nforall-e : pf (forall A) -> {T:i} pf (A T).\nexists-i : {T:i} pf (A T) -> pf (exists A).\nexists-e : pf (exists A) -> ({x:i} {y:pf (A x)} pf C) -> pf C.","source":"twelf_synthetic/logic_framework"} +{"id":89028,"prover":"Twelf","theorem":"and_comm_pf","goal":"and-comm : pf (imp (and A B) (and B A)).","context":["and-comm"],"tactic_proof":"and-comm = imp-i ([h:pf (and A B)] and-i (and-e2 h) (and-e1 h)).","source":"twelf_synthetic/logic_framework"} +{"id":89029,"prover":"Twelf","theorem":"or_comm_pf","goal":"or-comm : pf (imp (or A B) (or B A)).","context":["or-comm"],"tactic_proof":"or-comm = imp-i ([h:pf (or A B)] or-e h ([a] or-i2 a) ([b] or-i1 b)).","source":"twelf_synthetic/logic_framework"} +{"id":89030,"prover":"Twelf","theorem":"modus_ponens_pf","goal":"mp : pf A -> pf (imp A B) -> pf B.","context":["mp"],"tactic_proof":"mp = [a:pf A] [f:pf (imp A B)] imp-e f a.","source":"twelf_synthetic/logic_framework"} +{"id":89031,"prover":"Twelf","theorem":"double_neg_intro_pf","goal":"dn-intro : pf (imp A (not (not A))).","context":["dn-intro"],"tactic_proof":"dn-intro = imp-i ([a:pf A] not-i ([h:pf (not A)] not-e h a)).","source":"twelf_synthetic/logic_framework"} +{"id":89032,"prover":"Twelf","theorem":"contra_pf","goal":"contra : pf (imp (imp A B) (imp (not B) (not A))).","context":["contra"],"tactic_proof":"contra = imp-i ([f:pf (imp A B)] imp-i ([nb:pf (not B)] not-i ([a:pf A] not-e nb (imp-e f a)))).","source":"twelf_synthetic/logic_framework"} +{"id":89033,"prover":"Twelf","theorem":"tree","goal":"tree : type.","context":["tree"],"tactic_proof":"empty : tree.\nnode : nat -> tree -> tree -> tree.","source":"twelf_synthetic/binary_trees"} +{"id":89034,"prover":"Twelf","theorem":"insert_bst","goal":"insert : nat -> tree -> tree -> type.","context":["insert"],"tactic_proof":"insert-empty : insert X empty (node X empty empty).\ninsert-lt : insert X (node Y L R) (node Y L' R) <- leq X Y <- insert X L L'.\ninsert-gt : insert X (node Y L R) (node Y L R') <- leq Y X <- insert X R R'.","source":"twelf_synthetic/binary_trees"} +{"id":89035,"prover":"Twelf","theorem":"tree_size","goal":"tree-size : tree -> nat -> type.","context":["tree-size"],"tactic_proof":"tree-size-empty : tree-size empty z.\ntree-size-node : tree-size (node X L R) (s N) <- tree-size L NL <- tree-size R NR <- plus NL NR N.","source":"twelf_synthetic/binary_trees"} +{"id":89036,"prover":"Twelf","theorem":"tree_height","goal":"tree-height : tree -> nat -> type.","context":["tree-height"],"tactic_proof":"tree-height-empty : tree-height empty z.\ntree-height-node : tree-height (node X L R) (s H) <- tree-height L HL <- tree-height R HR <- max HL HR H.","source":"twelf_synthetic/binary_trees"} +{"id":89037,"prover":"Twelf","theorem":"tree_member","goal":"tree-member : nat -> tree -> type.","context":["tree-member"],"tactic_proof":"tree-member-node : tree-member X (node X L R).\ntree-member-left : tree-member X (node Y L R) <- tree-member X L.\ntree-member-right : tree-member X (node Y L R) <- tree-member X R.","source":"twelf_synthetic/binary_trees"} +{"id":89038,"prover":"Twelf","theorem":"tree_mirror","goal":"mirror : tree -> tree -> type.","context":["mirror"],"tactic_proof":"mirror-empty : mirror empty empty.\nmirror-node : mirror (node X L R) (node X R' L') <- mirror L L' <- mirror R R'.","source":"twelf_synthetic/binary_trees"} +{"id":89039,"prover":"Twelf","theorem":"mirror_involutive","goal":"mirror-involutive : mirror T T' -> mirror T' T -> type.","context":["mirror-involutive"],"tactic_proof":"%mode mirror-involutive +D1 -D2.","source":"twelf_synthetic/binary_trees"} +{"id":89040,"prover":"Twelf","theorem":"inorder_flatten","goal":"inorder : tree -> list -> type.","context":["inorder"],"tactic_proof":"inorder-empty : inorder empty nil.\ninorder-node : inorder (node X L R) M <- inorder L ML <- inorder R MR <- append ML (cons X MR) M.","source":"twelf_synthetic/binary_trees"} +{"id":89041,"prover":"Twelf","theorem":"substitution","goal":"subst' : (tm -> tm) -> tm -> tm -> type.","context":[],"tactic_proof":"subst'-id : subst' ([x] x) V V.\nsubst'-const : subst' ([x] C) V C.","source":"twelf_synthetic/substitution"} +{"id":89042,"prover":"Twelf","theorem":"capture_avoid","goal":"fresh : tm -> (tm -> tm) -> type.","context":["fresh"],"tactic_proof":"fresh-const : fresh V ([x] C).\nfresh-app : fresh V ([x] app (E1 x) (E2 x)) <- fresh V E1 <- fresh V E2.","source":"twelf_synthetic/substitution"} +{"id":89043,"prover":"Twelf","theorem":"alpha_eq","goal":"alpha-eq : tm -> tm -> type.","context":["alpha-eq"],"tactic_proof":"alpha-eq-refl : alpha-eq E E.\nalpha-eq-lam : alpha-eq (lam T ([x] E x)) (lam T ([y] E y)) <- ({x} alpha-eq (E x) (E x)).","source":"twelf_synthetic/substitution"} +{"id":89044,"prover":"Twelf","theorem":"beta_red","goal":"beta-red : tm -> tm -> type.","context":["beta-red"],"tactic_proof":"beta-red-step : beta-red (app (lam T E) V) (E V) <- value V.","source":"twelf_synthetic/substitution"} +{"id":89045,"prover":"Twelf","theorem":"eta_red","goal":"eta-red : tm -> tm -> type.","context":["eta-red"],"tactic_proof":"eta-red-step : eta-red (lam T ([x] app E x)) E.","source":"twelf_synthetic/substitution"} +{"id":89046,"prover":"Twelf","theorem":"subst_lemma","goal":"subst-lemma : subst' E1 V1 E1' -> subst' E2 V2 E2' -> subst' ([x] E1 x) V1 E1'' -> type.","context":["subst-lemma"],"tactic_proof":"%mode subst-lemma +D1 +D2 +D3.","source":"twelf_synthetic/substitution"} +{"id":89047,"prover":"Twelf","theorem":"channel","goal":"chan : type.","context":["chan"],"tactic_proof":"new-chan : chan.","source":"twelf_synthetic/session_types"} +{"id":89048,"prover":"Twelf","theorem":"msg_type","goal":"msg : type.","context":["msg"],"tactic_proof":"unit-msg : msg.\nsum-msg : msg -> msg -> msg.","source":"twelf_synthetic/session_types"} +{"id":89049,"prover":"Twelf","theorem":"protocol","goal":"proto : type.","context":["proto"],"tactic_proof":"send-proto : msg -> proto -> proto.\nrecv-proto : msg -> proto -> proto.\nend-proto : proto.","source":"twelf_synthetic/session_types"} +{"id":89050,"prover":"Twelf","theorem":"duality","goal":"dual : proto -> proto -> type.","context":["dual"],"tactic_proof":"dual-end : dual end-proto end-proto.\ndual-send : dual (send-proto M P) (recv-proto M Q) <- dual P Q.\ndual-recv : dual (recv-proto M P) (send-proto M Q) <- dual P Q.","source":"twelf_synthetic/session_types"} +{"id":89051,"prover":"Twelf","theorem":"session_safe","goal":"session-safe : chan -> proto -> type.","context":["session-safe"],"tactic_proof":"%mode session-safe +C +P.","source":"twelf_synthetic/session_types"} +{"id":89052,"prover":"Twelf","theorem":"linear_use","goal":"linear : chan -> type.","context":["linear"],"tactic_proof":"linear-new : linear C <- new-chan.","source":"twelf_synthetic/session_types"} +{"id":89053,"prover":"Twelf","theorem":"linear_type","goal":"ltp : type.","context":["ltp"],"tactic_proof":"lolli : ltp -> ltp -> ltp.\ntensor : ltp -> ltp -> ltp.\nwith-type : ltp -> ltp -> ltp.\none-type : ltp.","source":"twelf_synthetic/linear_lf"} +{"id":89054,"prover":"Twelf","theorem":"linear_tm","goal":"ltm : type.","context":["ltm"],"tactic_proof":"lapp : ltm -> ltm -> ltm.\nllam : ltp -> (ltm -> ltm) -> ltm.\npair-ltm : ltm -> ltm -> ltm.\nunit-ltm : ltm.","source":"twelf_synthetic/linear_lf"} +{"id":89055,"prover":"Twelf","theorem":"linear_of","goal":"lof : ltm -> ltp -> type.","context":["lof"],"tactic_proof":"lof-lam : lof (llam T1 E) (lolli T1 T2) <- ({x:ltm} lof x T1 -> lof (E x) T2).\nlof-pair : lof (pair-ltm E1 E2) (tensor T1 T2) <- lof E1 T1 <- lof E2 T2.\nlof-unit : lof unit-ltm one-type.","source":"twelf_synthetic/linear_lf"} +{"id":89056,"prover":"Twelf","theorem":"linear_safety","goal":"linear-safety : lof E T -> step E E' -> lof E' T -> type.","context":["linear-safety"],"tactic_proof":"%mode linear-safety +D1 +D2 -D3.","source":"twelf_synthetic/linear_lf"} +{"id":89057,"prover":"Twelf","theorem":"linear_single","goal":"linear-single : ltm -> type.","context":["linear-single"],"tactic_proof":"linear-single-var : linear-single (var X).\nlinear-single-app : linear-single (lapp E1 E2) <- linear-single E1 <- linear-single E2.","source":"twelf_synthetic/linear_lf"} +{"id":89058,"prover":"Twelf","theorem":"weakening","goal":"weakening : of E T -> of E T -> type.","context":["weakening"],"tactic_proof":"%mode weakening +Dof -Dof'.","source":"twelf_synthetic/metatheory"} +{"id":89059,"prover":"Twelf","theorem":"type_unique","goal":"type-unique : of E T1 -> of E T2 -> eq T1 T2 -> type.","context":["type-unique"],"tactic_proof":"%mode type-unique +D1 +D2 -D3.","source":"twelf_synthetic/metatheory"} +{"id":89060,"prover":"Twelf","theorem":"canonical_forms","goal":"canonical : value V -> of V T -> type.","context":["canonical"],"tactic_proof":"%mode canonical +Dv +Dof.","source":"twelf_synthetic/metatheory"} +{"id":89061,"prover":"Twelf","theorem":"progress_lemma","goal":"progress-lemma : of E T -> (value E + step E E') -> type.","context":["progress-lemma"],"tactic_proof":"%mode progress-lemma +Dof -Dresult.","source":"twelf_synthetic/metatheory"} +{"id":89062,"prover":"Twelf","theorem":"preservation_lemma","goal":"preservation-lemma : of E T -> step E E' -> of E' T -> type.","context":["preservation-lemma"],"tactic_proof":"%mode preservation-lemma +Dof +Dstep -Dof'.","source":"twelf_synthetic/metatheory"} +{"id":89063,"prover":"Twelf","theorem":"normalization","goal":"strong-norm : of E T -> type.","context":["strong-norm"],"tactic_proof":"%mode strong-norm +Dof.","source":"twelf_synthetic/metatheory"} +{"id":89064,"prover":"Twelf","theorem":"confluence","goal":"confluence : reduces E E1 -> reduces E E2 -> (reduces E1 E' /\\ reduces E2 E') -> type.","context":["confluence"],"tactic_proof":"%mode confluence +D1 +D2 -D3.","source":"twelf_synthetic/metatheory"} +{"id":89065,"prover":"Twelf","theorem":"forall_type","goal":"forall-tp : (tp -> tp) -> tp.","context":["forall-tp"],"tactic_proof":"forall-intro : of E (forall-tp A) <- ({X:tp} of E (A X)).\nforall-elim : of (tapp E T) (A T) <- of E (forall-tp A).","source":"twelf_synthetic/polymorphism"} +{"id":89066,"prover":"Twelf","theorem":"exists_type","goal":"exists-tp : (tp -> tp) -> tp.","context":["exists-tp"],"tactic_proof":"pack : of (pack T E) (exists-tp A) <- of E (A T).\nunpack : of (open E1 [X] [x] E2) C <- of E1 (exists-tp A) <- ({X:tp} {x:tm} of x (A X) -> of (E2 X x) C).","source":"twelf_synthetic/polymorphism"} +{"id":89067,"prover":"Twelf","theorem":"type_abstraction","goal":"tabs : tp -> (tp -> tm) -> tm.","context":["tabs"],"tactic_proof":"of-tabs : of (tabs K E) (forall-tp A) <- ({X:tp} of (E X) (A X)).","source":"twelf_synthetic/polymorphism"} +{"id":89068,"prover":"Twelf","theorem":"param_poly","goal":"parametricity : of E (forall-tp A) -> type.","context":["parametricity"],"tactic_proof":"%mode parametricity +D.","source":"twelf_synthetic/polymorphism"} +{"id":89069,"prover":"Twelf","theorem":"subtype","goal":"sub : tp -> tp -> type.","context":["sub"],"tactic_proof":"sub-refl : sub T T.\nsub-trans : sub T1 T3 <- sub T1 T2 <- sub T2 T3.\nsub-arrow : sub (arr S1 T1) (arr S2 T2) <- sub S2 S1 <- sub T1 T2.","source":"twelf_synthetic/polymorphism"} +{"id":89070,"prover":"Twelf","theorem":"module_signature","goal":"module-sig : type.","context":["module-sig"],"tactic_proof":"mk-sig : list -> module-sig.","source":"twelf_synthetic/modules"} +{"id":89071,"prover":"Twelf","theorem":"module_impl","goal":"module-impl : module-sig -> type.","context":["module-impl"],"tactic_proof":"mk-impl : list -> module-impl Sig.","source":"twelf_synthetic/modules"} +{"id":89072,"prover":"Twelf","theorem":"functor","goal":"functor : module-sig -> module-sig -> type.","context":["functor"],"tactic_proof":"apply-functor : functor S1 S2 -> module-impl S1 -> module-impl S2 -> type.","source":"twelf_synthetic/modules"} +{"id":89073,"prover":"Twelf","theorem":"abstract_type","goal":"abs-tp : type.","context":["abs-tp"],"tactic_proof":"pack-abs : tp -> (tp -> tp) -> abs-tp.\nopen-abs : abs-tp -> (tp -> tp -> tp) -> tp -> type.","source":"twelf_synthetic/modules"} +{"id":89074,"prover":"Twelf","theorem":"structure_eq","goal":"struct-eq : module-impl S -> module-impl S -> type.","context":["struct-eq"],"tactic_proof":"%mode struct-eq +M1 +M2.","source":"twelf_synthetic/modules"} +{"id":89075,"prover":"Twelf","theorem":"include_module","goal":"include : module-impl S1 -> module-impl S2 -> module-impl S3 -> type.","context":["include"],"tactic_proof":"%mode include +M1 +M2 -M3.","source":"twelf_synthetic/modules"} +{"id":89076,"prover":"Twelf","theorem":"record","goal":"record : type.","context":["record"],"tactic_proof":"empty-rec : record.\ncons-rec : label -> tm -> record -> record.","source":"twelf_synthetic/records_twelf"} +{"id":89077,"prover":"Twelf","theorem":"label","goal":"label : type.","context":["label"],"tactic_proof":"lab-a : label.\nlab-b : label.\nlab-c : label.","source":"twelf_synthetic/records_twelf"} +{"id":89078,"prover":"Twelf","theorem":"rec_lookup","goal":"lookup : label -> record -> tm -> type.","context":["lookup"],"tactic_proof":"lookup-head : lookup L (cons-rec L V R) V.\nlookup-tail : lookup L (cons-rec L' V R) V' <- lookup L R V'.","source":"twelf_synthetic/records_twelf"} +{"id":89079,"prover":"Twelf","theorem":"rec_update","goal":"update : label -> tm -> record -> record -> type.","context":["update"],"tactic_proof":"update-head : update L V (cons-rec L V' R) (cons-rec L V R).\nupdate-tail : update L V (cons-rec L' V' R) (cons-rec L' V' R') <- update L V R R'.","source":"twelf_synthetic/records_twelf"} +{"id":89080,"prover":"Twelf","theorem":"rec_concat","goal":"rec-concat : record -> record -> record -> type.","context":["rec-concat"],"tactic_proof":"rec-concat-nil : rec-concat empty-rec R R.\nrec-concat-cons : rec-concat (cons-rec L V R1) R2 (cons-rec L V R3) <- rec-concat R1 R2 R3.","source":"twelf_synthetic/records_twelf"} +{"id":89081,"prover":"Twelf","theorem":"trace","goal":"trace : type.","context":["trace"],"tactic_proof":"empty-trace : trace.\nstep-trace : tm -> tm -> trace -> trace.","source":"twelf_synthetic/traces"} +{"id":89082,"prover":"Twelf","theorem":"trace_step","goal":"trace-step : trace -> tm -> tm -> type.","context":["trace-step"],"tactic_proof":"%mode trace-step +T +E -E'.","source":"twelf_synthetic/traces"} +{"id":89083,"prover":"Twelf","theorem":"trace_append","goal":"trace-append : trace -> trace -> trace -> type.","context":["trace-append"],"tactic_proof":"trace-app-nil : trace-append empty-trace T T.\ntrace-app-cons : trace-append (step-trace E1 E2 T1) T2 (step-trace E1 E2 T3) <- trace-append T1 T2 T3.","source":"twelf_synthetic/traces"} +{"id":89084,"prover":"Twelf","theorem":"evaluates","goal":"evaluates : tm -> tm -> type.","context":["evaluates"],"tactic_proof":"eval-refl : evaluates E E.\neval-step : evaluates E E'' <- step E E' <- evaluates E' E''.","source":"twelf_synthetic/traces"} +{"id":89085,"prover":"Twelf","theorem":"multistep","goal":"multistep : tm -> tm -> trace -> type.","context":["multistep"],"tactic_proof":"multistep-refl : multistep E E empty-trace.\nmultistep-step : multistep E E'' (step-trace E E' T) <- step E E' <- multistep E' E'' T.","source":"twelf_synthetic/traces"} +{"id":89086,"prover":"Twelf","theorem":"bigstep","goal":"bigstep : tm -> tm -> type.","context":["bigstep"],"tactic_proof":"bigstep-val : bigstep V V <- value V.\nbigstep-app : bigstep (app E1 E2) V <- bigstep E1 (lam T E) <- bigstep E2 V2 <- bigstep (E V2) V.","source":"twelf_synthetic/bigstep"} +{"id":89087,"prover":"Twelf","theorem":"bigstep_det","goal":"bigstep-det : bigstep E V1 -> bigstep E V2 -> eq V1 V2 -> type.","context":["bigstep-det"],"tactic_proof":"%mode bigstep-det +D1 +D2 -D3.","source":"twelf_synthetic/bigstep"} +{"id":89088,"prover":"Twelf","theorem":"bigstep_value","goal":"bigstep-value : bigstep E V -> value V -> type.","context":["bigstep-value"],"tactic_proof":"%mode bigstep-value +D -Dv.","source":"twelf_synthetic/bigstep"} +{"id":89089,"prover":"Twelf","theorem":"bigstep_of","goal":"bigstep-of : bigstep E V -> of E T -> of V T -> type.","context":["bigstep-of"],"tactic_proof":"%mode bigstep-of +D1 +D2 -D3.","source":"twelf_synthetic/bigstep"} +{"id":89090,"prover":"Twelf","theorem":"bigstep_smallstep","goal":"bigstep-smallstep : bigstep E V -> multistep E V _ -> type.","context":["bigstep-smallstep"],"tactic_proof":"%mode bigstep-smallstep +D1 -D2.","source":"twelf_synthetic/bigstep"} +{"id":89091,"prover":"Twelf","theorem":"hoas_abs","goal":"habs : (tm -> tm) -> tm.","context":["habs"],"tactic_proof":"habs-intro : of (habs E) (arr T1 T2) <- ({x:tm} of x T1 -> of (E x) T2).","source":"twelf_synthetic/hoas"} +{"id":89092,"prover":"Twelf","theorem":"hoas_app","goal":"happ : tm -> tm -> tm.","context":["happ"],"tactic_proof":"happ-intro : of (happ E1 E2) T2 <- of E1 (arr T1 T2) <- of E2 T1.","source":"twelf_synthetic/hoas"} +{"id":89093,"prover":"Twelf","theorem":"hoas_beta","goal":"hstep : tm -> tm -> type.","context":["hstep"],"tactic_proof":"hstep-beta : hstep (happ (habs E) V) (E V) <- value V.","source":"twelf_synthetic/hoas"} +{"id":89094,"prover":"Twelf","theorem":"hoas_subst","goal":"hsubst : (tm -> tm) -> tm -> tm -> type.","context":["hsubst"],"tactic_proof":"%mode hsubst +E +V -E'.","source":"twelf_synthetic/hoas"} +{"id":89095,"prover":"Twelf","theorem":"parametric_hoas","goal":"phabs : (tm -> tm) -> tm.","context":["phabs"],"tactic_proof":"%block phabs-block : block {x:tm}.","source":"twelf_synthetic/hoas"} +{"id":89096,"prover":"Twelf","theorem":"token","goal":"token : type.","context":["token"],"tactic_proof":"tok-ident : string -> token.\ntok-num : nat -> token.\ntok-lparen : token.\ntok-rparen : token.\ntok-plus : token.\ntok-times : token.","source":"twelf_synthetic/parsing"} +{"id":89097,"prover":"Twelf","theorem":"parse_expr","goal":"parse : list -> tm -> list -> type.","context":["parse"],"tactic_proof":"%mode parse +Input -Result -Rest.","source":"twelf_synthetic/parsing"} +{"id":89098,"prover":"Twelf","theorem":"lex","goal":"lex : string -> list -> type.","context":["lex"],"tactic_proof":"%mode lex +Input -Tokens.","source":"twelf_synthetic/parsing"} +{"id":89099,"prover":"Twelf","theorem":"ast","goal":"ast : type.","context":["ast"],"tactic_proof":"ast-num : nat -> ast.\nast-var : string -> ast.\nast-bin : op -> ast -> ast -> ast.","source":"twelf_synthetic/parsing"} +{"id":89100,"prover":"Twelf","theorem":"op","goal":"op : type.","context":["op"],"tactic_proof":"op-add : op.\nop-sub : op.\nop-mul : op.\nop-div : op.","source":"twelf_synthetic/parsing"} +{"id":89101,"prover":"Twelf","theorem":"grammar","goal":"grammar-rule : type.","context":["grammar-rule"],"tactic_proof":"gr-term : symbol -> grammar-rule.\ngr-alt : grammar-rule -> grammar-rule -> grammar-rule.\ngr-seq : grammar-rule -> grammar-rule -> grammar-rule.","source":"twelf_synthetic/parsing"} +{"id":89102,"prover":"Twelf","theorem":"dfa_state","goal":"state : type.","context":["state"],"tactic_proof":"init-state : state.\ntrans-state : state -> char -> state.","source":"twelf_synthetic/automata"} +{"id":89103,"prover":"Twelf","theorem":"dfa_accept","goal":"accepts : dfa -> string -> type.","context":["accepts"],"tactic_proof":"%mode accepts +D +S.","source":"twelf_synthetic/automata"} +{"id":89104,"prover":"Twelf","theorem":"nfa_run","goal":"nfa-run : nfa -> list -> state -> type.","context":["nfa-run"],"tactic_proof":"%mode nfa-run +N +S -Q.","source":"twelf_synthetic/automata"} +{"id":89105,"prover":"Twelf","theorem":"regex","goal":"regex : type.","context":["regex"],"tactic_proof":"re-char : char -> regex.\nre-concat : regex -> regex -> regex.\nre-alt : regex -> regex -> regex.\nre-star : regex -> regex.\nre-epsilon : regex.","source":"twelf_synthetic/automata"} +{"id":89106,"prover":"Twelf","theorem":"regex_match","goal":"re-match : regex -> string -> type.","context":["re-match"],"tactic_proof":"%mode re-match +R +S.","source":"twelf_synthetic/automata"} +{"id":89107,"prover":"Twelf","theorem":"pda","goal":"pda-state : type.","context":["pda-state"],"tactic_proof":"pda-init : pda-state.\npda-push : pda-state -> char -> stack -> pda-state.","source":"twelf_synthetic/automata"} +{"id":89108,"prover":"Twelf","theorem":"io_monad","goal":"io : type -> type.","context":["io"],"tactic_proof":"io-return : A -> io A.\nio-bind : io A -> (A -> io B) -> io B.","source":"twelf_synthetic/effects"} +{"id":89109,"prover":"Twelf","theorem":"state_monad","goal":"state-m : type -> type.","context":["state-m"],"tactic_proof":"state-return : A -> state-m A.\nstate-bind : state-m A -> (A -> state-m B) -> state-m B.\nstate-get : state-m state.\nstate-put : state -> state-m unit.","source":"twelf_synthetic/effects"} +{"id":89110,"prover":"Twelf","theorem":"exception","goal":"exc : type -> type.","context":["exc"],"tactic_proof":"exc-pure : A -> exc A.\nexc-raise : string -> exc A.\nexc-catch : exc A -> (string -> exc A) -> exc A.","source":"twelf_synthetic/effects"} +{"id":89111,"prover":"Twelf","theorem":"continuation","goal":"cont : type -> type -> type.","context":["cont"],"tactic_proof":"cont-pure : A -> cont A R.\ncont-callcc : ((A -> cont B R) -> cont A R) -> cont A R.","source":"twelf_synthetic/effects"} +{"id":89112,"prover":"Twelf","theorem":"free_monad","goal":"free : type -> type.","context":["free"],"tactic_proof":"free-pure : A -> free A.\nfree-op : op-sig -> (result -> free A) -> free A.","source":"twelf_synthetic/effects"} +{"id":89113,"prover":"Twelf","theorem":"add_zero_left","goal":"add-zero-left : {N:nat} plus z N N -> type.","context":["add-zero-left","N"],"tactic_proof":"%mode add-zero-left +N -D.\nadd-zero-left-z : add-zero-left z plus-z.\nadd-zero-left-s : add-zero-left (s N) plus-z <- add-zero-left N _.","source":"twelf_synthetic/numerics_twelf"} +{"id":89114,"prover":"Twelf","theorem":"add_zero_right","goal":"add-zero-right : {N:nat} plus N z N -> type.","context":["add-zero-right","N"],"tactic_proof":"%mode add-zero-right +N -D.\nadd-zero-right-z : add-zero-right z plus-z.\nadd-zero-right-s : add-zero-right (s N) (plus-s D) <- add-zero-right N D.","source":"twelf_synthetic/numerics_twelf"} +{"id":89115,"prover":"Twelf","theorem":"times_zero_right","goal":"times-zero-right : {N:nat} times N z z -> type.","context":["times-zero-right","N"],"tactic_proof":"%mode times-zero-right +N -D.","source":"twelf_synthetic/numerics_twelf"} +{"id":89116,"prover":"Twelf","theorem":"div_algorithm","goal":"div-mod : nat -> nat -> nat -> nat -> type.","context":["div-mod"],"tactic_proof":"div-mod-base : div-mod M N z M <- lt M N.\ndiv-mod-rec : div-mod M N (s Q) R <- leq N M <- sub M N M' <- div-mod M' N Q R.","source":"twelf_synthetic/numerics_twelf"} +{"id":89117,"prover":"Twelf","theorem":"gcd","goal":"gcd : nat -> nat -> nat -> type.","context":["gcd"],"tactic_proof":"gcd-zero : gcd M z M.\ngcd-rec : gcd M N G <- modulo M N R <- gcd N R G.","source":"twelf_synthetic/numerics_twelf"}