diff --git a/Analysis/Section_11_2.lean b/Analysis/Section_11_2.lean index 8eeb5550..5b9aebb6 100644 --- a/Analysis/Section_11_2.lean +++ b/Analysis/Section_11_2.lean @@ -266,7 +266,7 @@ theorem PiecewiseConstantOn.integ_const (c: ℝ) (I: BoundedInterval) : integ (fun _ ↦ c) I = c * |I|ₗ := by sorry -/-- Theorem 11.2.16 (f) (Laws of integration) / Exercise 11.2.4 -/ +/-- Theorem 11.2.16 (f') (Laws of integration) / Exercise 11.2.4 -/ theorem PiecewiseConstantOn.integ_const' {f:ℝ → ℝ} {I: BoundedInterval} (h: ConstantOn f I) : integ f I = (constant_value_on f I) * |I|ₗ := by sorry @@ -279,7 +279,7 @@ theorem PiecewiseConstantOn.of_extend {I J: BoundedInterval} (hIJ: I ⊆ J) sorry open Classical in -/-- Theorem 11.2.16 (g) (Laws of integration) / Exercise 11.2.4 -/ +/-- Theorem 11.2.16 (g') (Laws of integration) / Exercise 11.2.4 -/ theorem PiecewiseConstantOn.integ_of_extend {I J: BoundedInterval} (hIJ: I ⊆ J) {f: ℝ → ℝ} (h: PiecewiseConstantOn f I) : integ (fun x ↦ if x ∈ I then f x else 0) J = integ f I := by @@ -290,7 +290,7 @@ theorem PiecewiseConstantOn.of_join {I J K: BoundedInterval} (hIJK: K.joins I J) (f: ℝ → ℝ) : PiecewiseConstantOn f K ↔ PiecewiseConstantOn f I ∧ PiecewiseConstantOn f J := by sorry -/-- Theorem 11.2.16 (h) (Laws of integration) / Exercise 11.2.4 -/ +/-- Theorem 11.2.16 (h') (Laws of integration) / Exercise 11.2.4 -/ theorem PiecewiseConstantOn.integ_of_join {I J K: BoundedInterval} (hIJK: K.joins I J) {f: ℝ → ℝ} (h: PiecewiseConstantOn f K) : integ f K = integ f I + integ f J := by diff --git a/Analysis/Section_11_8.lean b/Analysis/Section_11_8.lean index 3a62bffa..d4b7cd60 100644 --- a/Analysis/Section_11_8.lean +++ b/Analysis/Section_11_8.lean @@ -315,7 +315,7 @@ theorem PiecewiseConstantOn.RS_integ_const (c: ℝ) (I: BoundedInterval) {α:ℝ RS_integ (fun _ ↦ c) I α = c * α[I]ₗ := by sorry -/-- Theorem 11.8.8 (f) (Laws of RS integration) / Exercise 11.8.3 -/ +/-- Theorem 11.8.8 (f') (Laws of RS integration) / Exercise 11.8.3 -/ theorem PiecewiseConstantOn.RS_integ_const' {f:ℝ → ℝ} {I: BoundedInterval} {α:ℝ → ℝ} (hα: Monotone α) (h: ConstantOn f I) : RS_integ f I α = (constant_value_on f I) * α[I]ₗ := by @@ -329,7 +329,7 @@ theorem PiecewiseConstantOn.RS_of_extend {I J: BoundedInterval} (hIJ: I ⊆ J) sorry open Classical in -/-- Theorem 11.8.8 (g) (Laws of RS integration) / Exercise 11.8.3 -/ +/-- Theorem 11.8.8 (g') (Laws of RS integration) / Exercise 11.8.3 -/ theorem PiecewiseConstantOn.RS_integ_of_extend {I J: BoundedInterval} (hIJ: I ⊆ J) {f: ℝ → ℝ} (h: PiecewiseConstantOn f I) {α:ℝ → ℝ} (hα: Monotone α): RS_integ (fun x ↦ if x ∈ I then f x else 0) J α = RS_integ f I α := by diff --git a/Analysis/Section_7_5.lean b/Analysis/Section_7_5.lean index a677b7ea..1336e151 100644 --- a/Analysis/Section_7_5.lean +++ b/Analysis/Section_7_5.lean @@ -190,12 +190,12 @@ theorem Series.ratio_test_neg {s : Series} (hnon: ∀ n ≥ s.m, s.seq n ≠ 0) convert (ratio_ineq s.m _).1.trans (ratio_ineq s.m _).2.1 with n; rfl all_goals convert hnon using 1 with n; simp -/-- Corollary 7.5.3 (Ratio test) / Exercise 7.5.3 -/ +/-- Corollary 7.5.3(i) (Ratio test inconclusive, diverges) / Exercise 7.5.3 -/ theorem Series.ratio_test_inconclusive: ∃ s:Series, (∀ n ≥ s.m, s.seq n ≠ 0) ∧ atTop.Tendsto (fun n ↦ |s.seq (n+1)| / |s.seq n|) (nhds 1) ∧ s.diverges := by sorry -/-- Corollary 7.5.3 (Ratio test) / Exercise 7.5.3 -/ +/-- Corollary 7.5.3(ii) (Ratio test inconclusive, absConverges) / Exercise 7.5.3 -/ theorem Series.ratio_test_inconclusive' : ∃ s:Series, (∀ n ≥ s.m, s.seq n ≠ 0) ∧ atTop.Tendsto (fun n ↦ |s.seq (n+1)| / |s.seq n|) (nhds 1) ∧ s.absConverges := by sorry