Delete unnecessary definitions (#59)

Author: RemyDegenne

LeanBandits/Bandit/Bandit.lean

@@ -8,7 +8,6 @@ import LeanBandits.ForMathlib.IndepFun
 import LeanBandits.ForMathlib.IndepInfinitePi
 import LeanBandits.ForMathlib.KernelRepresentation
 import LeanBandits.ForMathlib.StandardBorel
-import LeanBandits.SequentialLearning.Deterministic
 import LeanBandits.SequentialLearning.FiniteActions
 import LeanBandits.SequentialLearning.StationaryEnv
 
@@ -26,32 +25,6 @@ variable {α R : Type*} {mα : MeasurableSpace α} {mR : MeasurableSpace R}
 
 section MeasureSpace
 
-namespace Bandit
-
-/-- Kernel describing the distribution of the next action-reward pair given the history up to
-time `n`. -/
-noncomputable
-def stepKernel (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
-    Kernel (Iic n → α × R) (α × R) :=
-  Learning.stepKernel alg (stationaryEnv ν) n
-deriving IsMarkovKernel
-
-@[simp]
-lemma fst_stepKernel (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
-    (stepKernel alg ν n).fst = alg.policy n := by
-  rw [stepKernel, Learning.fst_stepKernel]
-
-@[simp]
-lemma snd_stepKernel (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
-    (stepKernel alg ν n).snd = ν ∘ₖ alg.policy n := by
-  rw [stepKernel, Learning.stepKernel, stationaryEnv_feedback, Kernel.snd_compProd_prodMkLeft]
-
-/-- Measure on the sequence of actions pulled and rewards observed generated by the bandit. -/
-noncomputable
-def trajMeasure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] : Measure (ℕ → α × R) :=
-  Learning.trajMeasure alg (stationaryEnv ν)
-deriving IsProbabilityMeasure
-
 /-- Measure of an infinite stream of rewards from each action. -/
 noncomputable
 def streamMeasure (ν : Kernel α R) : Measure (ℕ → α → R) :=
@@ -61,26 +34,6 @@ instance (ν : Kernel α R) [IsMarkovKernel ν] : IsProbabilityMeasure (streamMe
   unfold streamMeasure
   infer_instance
 
-/-- Joint distribution of the sequence of action pulled and rewards, and a stream of independent
-rewards from all actions. -/
-noncomputable
-def measure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
-    Measure ((ℕ → α × R) × (ℕ → α → R)) :=
-  (trajMeasure alg ν).prod (streamMeasure ν)
-deriving IsProbabilityMeasure
-
-@[simp]
-lemma fst_measure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
-    (measure alg ν).fst = trajMeasure alg ν := by
-  rw [measure, Measure.fst_prod]
-
-@[simp]
-lemma snd_measure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν] :
-    (measure alg ν).snd = streamMeasure ν := by
-  rw [measure, Measure.snd_prod]
-
-end Bandit
-
 section StreamMeasure
 
 lemma _root_.hasLaw_eval_infinitePi {ι : Type*} {X : ι → Type*} {mX : ∀ i, MeasurableSpace (X i)}
@@ -90,15 +43,15 @@ lemma _root_.hasLaw_eval_infinitePi {ι : Type*} {X : ι → Type*} {mX : ∀ i,
   map_eq := by exact (measurePreserving_eval_infinitePi μ i).map_eq
 
 lemma hasLaw_eval_streamMeasure (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
-    HasLaw (fun h : ℕ → α → R ↦ h n) (Measure.infinitePi ν) (Bandit.streamMeasure ν) :=
+    HasLaw (fun h : ℕ → α → R ↦ h n) (Measure.infinitePi ν) (streamMeasure ν) :=
   hasLaw_eval_infinitePi (fun _ ↦ Measure.infinitePi ν) n
 
 lemma hasLaw_eval_eval_streamMeasure (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) (a : α) :
-    HasLaw (fun h : ℕ → α → R ↦ h n a) (ν a) (Bandit.streamMeasure ν) :=
+    HasLaw (fun h : ℕ → α → R ↦ h n a) (ν a) (streamMeasure ν) :=
   (hasLaw_eval_infinitePi ν a).comp (hasLaw_eval_streamMeasure ν n)
 
 lemma identDistrib_eval_eval_id_streamMeasure (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) (a : α) :
-    IdentDistrib (fun h : ℕ → α → R ↦ h n a) id (Bandit.streamMeasure ν) (ν a) where
+    IdentDistrib (fun h : ℕ → α → R ↦ h n a) id (streamMeasure ν) (ν a) where
   aemeasurable_fst := Measurable.aemeasurable (by fun_prop)
   aemeasurable_snd := Measurable.aemeasurable (by fun_prop)
   map_eq := by
@@ -118,47 +71,40 @@ lemma Integrable.congr_identDistrib {Ω Ω' : Type*}
 
 lemma integrable_eval_streamMeasure (ν : Kernel α ℝ) [IsMarkovKernel ν] (n : ℕ) (a : α)
     (h_int : Integrable id (ν a)) :
-    Integrable (fun h : ℕ → α → ℝ ↦ h n a) (Bandit.streamMeasure ν) :=
+    Integrable (fun h : ℕ → α → ℝ ↦ h n a) (streamMeasure ν) :=
   Integrable.congr_identDistrib h_int (identDistrib_eval_eval_id_streamMeasure ν n a).symm
 
 lemma integral_eval_streamMeasure (ν : Kernel α ℝ) [IsMarkovKernel ν] (n : ℕ) (a : α) :
-    ∫ h, h n a ∂(Bandit.streamMeasure ν) = (ν a)[id] := by
-  calc ∫ h, h n a ∂(Bandit.streamMeasure ν)
-  _ = ∫ x, x ∂((Bandit.streamMeasure ν).map (fun h ↦ h n a)) := by
+    ∫ h, h n a ∂(streamMeasure ν) = (ν a)[id] := by
+  calc ∫ h, h n a ∂(streamMeasure ν)
+  _ = ∫ x, x ∂((streamMeasure ν).map (fun h ↦ h n a)) := by
     rw [integral_map (Measurable.aemeasurable (by fun_prop)) (by fun_prop)]
   _ = (ν a)[id] := by simp [(hasLaw_eval_eval_streamMeasure ν n a).map_eq]
 
 lemma iIndepFun_eval_streamMeasure' (ν : Kernel α R) [IsMarkovKernel ν] :
-    iIndepFun (fun n ω ↦ ω n) (Bandit.streamMeasure ν) :=
+    iIndepFun (fun n ω ↦ ω n) (streamMeasure ν) :=
   iIndepFun_infinitePi (P := fun (_ : ℕ) ↦ Measure.infinitePi ν) (Ω := fun _ ↦ α → R)
     (X := fun i u ↦ u) (fun i ↦ by fun_prop)
 
 lemma iIndepFun_eval_streamMeasure'' (ν : Kernel α R) [IsMarkovKernel ν] (a : α) :
-    iIndepFun (fun n ω ↦ ω n a) (Bandit.streamMeasure ν) :=
+    iIndepFun (fun n ω ↦ ω n a) (streamMeasure ν) :=
   (iIndepFun_eval_streamMeasure' ν).comp (g := fun i ω ↦ ω a) (by fun_prop)
 
 lemma iIndepFun_eval_streamMeasure (ν : Kernel α R) [IsMarkovKernel ν] :
-    iIndepFun (fun (p : ℕ × α) ω ↦ ω p.1 p.2) (Bandit.streamMeasure ν) :=
+    iIndepFun (fun (p : ℕ × α) ω ↦ ω p.1 p.2) (streamMeasure ν) :=
   iIndepFun_uncurry_infinitePi' (X := fun _ _ ↦ id) (fun _ ↦ ν) (by fun_prop)
 
 lemma indepFun_eval_streamMeasure (ν : Kernel α R) [IsMarkovKernel ν] {n m : ℕ} {a b : α}
     (h : n ≠ m ∨ a ≠ b) :
-    IndepFun (fun ω ↦ ω n a) (fun ω ↦ ω m b) (Bandit.streamMeasure ν) := by
+    IndepFun (fun ω ↦ ω n a) (fun ω ↦ ω m b) (streamMeasure ν) := by
   change IndepFun (fun ω ↦ ω (n, a).1 (n, a).2) (fun ω ↦ ω (m, b).1 (m, b).2)
-    (Bandit.streamMeasure ν)
+    (streamMeasure ν)
   exact (iIndepFun_eval_streamMeasure ν).indepFun (by grind)
 
 lemma indepFun_eval_streamMeasure' (ν : Kernel α R) [IsMarkovKernel ν] {a b : α} (h : a ≠ b) :
-    IndepFun (fun ω n ↦ ω n a) (fun ω n ↦ ω n b) (Bandit.streamMeasure ν) :=
+    IndepFun (fun ω n ↦ ω n a) (fun ω n ↦ ω n b) (streamMeasure ν) :=
   indepFun_proj_infinitePi_infinitePi h
 
-lemma indepFun_eval_snd_measure (alg : Algorithm α R) (ν : Kernel α R) [IsMarkovKernel ν]
-    {a b : α} (h : a ≠ b) :
-    IndepFun (fun ω n ↦ ω.2 n a) (fun ω n ↦ ω.2 n b) (Bandit.measure alg ν) := by
-  refine indepFun_snd_prod ?_ ?_ (indepFun_eval_streamMeasure' ν h) (Bandit.trajMeasure alg ν)
-  · exact Measurable.aemeasurable (by fun_prop)
-  · exact Measurable.aemeasurable (by fun_prop)
-
 end StreamMeasure
 
 namespace ArrayModel
@@ -181,7 +127,7 @@ instance {α R : Type*} [Countable α] [MeasurableSpace R] [StandardBorelSpace R
 /-- Probability measure for the array model of stochastic bandits. -/
 noncomputable
 def arrayMeasure (ν : Kernel α R) : Measure (probSpace α R) :=
-  (Measure.infinitePi fun _ ↦ volume).prod (Bandit.streamMeasure ν)
+  (Measure.infinitePi fun _ ↦ volume).prod (streamMeasure ν)
 
 instance (ν : Kernel α R) [IsMarkovKernel ν] : IsProbabilityMeasure (arrayMeasure ν) :=
   Measure.prod.instIsProbabilityMeasure _ _
@@ -606,7 +552,7 @@ lemma map_snd_apply_arrayMeasure {ν : Kernel α R} [IsMarkovKernel ν] (n : ℕ
     rw [Measure.snd, Measure.map_map (by fun_prop) (by fun_prop)]
     rfl
   _ = ν a := by
-    rw [arrayMeasure, Measure.snd_prod, Bandit.streamMeasure]
+    rw [arrayMeasure, Measure.snd_prod, streamMeasure]
     have : (fun ω ↦ ω n a) = (fun h : α → R ↦ h a) ∘ (fun ω : ℕ → α → R ↦ ω n) := rfl
     rw [this, ← Measure.map_map (by fun_prop) (by fun_prop), Measure.infinitePi_map_eval,
       Measure.infinitePi_map_eval]
@@ -629,7 +575,7 @@ omit [DecidableEq α] [Nonempty α] [StandardBorelSpace α] in
 lemma indepFun_fst_add_one_aux (ν : Kernel α R) [IsMarkovKernel ν] (n : ℕ) :
     (fun ω ↦ ω.1 (n + 1)) ⟂ᵢ[arrayMeasure ν] (fun ω ↦ (fun (i : Iic n) ↦ ω.1 i, ω.2)) := by
   let μ₁ : Measure (ℕ → I) := Measure.infinitePi fun _ ↦ volume
-  let μ₂ : Measure (ℕ → α → R) := Bandit.streamMeasure ν
+  let μ₂ : Measure (ℕ → α → R) := streamMeasure ν
   -- Coordinates of μ₁ are independent
   have h_indep : iIndepFun (fun i (ω : ℕ → I) ↦ ω i) μ₁ :=
     iIndepFun_infinitePi (fun _ ↦ measurable_id)
@@ -1016,10 +962,10 @@ lemma hasCondDistrib_action' (alg : Algorithm α R) (ν : Kernel α R) [IsMarkov
     rw [h_indep']
     congr
     simp only [arrayMeasure]
-    calc ((Measure.infinitePi fun x ↦ ℙ).prod (Bandit.streamMeasure ν)).map (fun ω ↦ ω.1 (n + 1))
+    calc ((Measure.infinitePi fun x ↦ ℙ).prod (streamMeasure ν)).map (fun ω ↦ ω.1 (n + 1))
     _ = (Measure.infinitePi fun x ↦ ℙ).map (Function.eval (n + 1)) := by
       nth_rw 2 [← Measure.fst_prod (μ := Measure.infinitePi fun x ↦ ℙ)
-        (ν := Bandit.streamMeasure ν)]
+        (ν := streamMeasure ν)]
       rw [Measure.fst, Measure.map_map (by fun_prop) (by fun_prop)]
       rfl
     _ = ℙ := by rw [Measure.infinitePi_map_eval]

LeanBandits/Bandit/RewardByCountMeasure.lean

@@ -20,7 +20,7 @@ variable {α Ω : Type*} {mα : MeasurableSpace α} {mΩ : MeasurableSpace Ω} [
   {alg : Algorithm α ℝ} {ν : Kernel α ℝ} [IsMarkovKernel ν]
   {h_inter : IsAlgEnvSeq A R alg (stationaryEnv ν) P}
 
-local notation "𝔓'" => P.prod (Bandit.streamMeasure ν)
+local notation "𝔓'" => P.prod (streamMeasure ν)
 
 omit [DecidableEq α] [StandardBorelSpace α] [Nonempty α] in
 lemma hasLaw_Z (a : α) (m : ℕ) :
@@ -30,10 +30,10 @@ lemma hasLaw_Z (a : α) (m : ℕ) :
     _ = ((𝔓').snd).map (fun ω ↦ ω m a) := by
       rw [Measure.snd, Measure.map_map (by fun_prop) (by fun_prop)]
       rfl
-    _ = (Bandit.streamMeasure ν).map (fun ω ↦ ω m a) := by simp
+    _ = (streamMeasure ν).map (fun ω ↦ ω m a) := by simp
     _ = ((Measure.infinitePi fun _ ↦ Measure.infinitePi ν).map (fun ω ↦ ω m)).map
         (fun ω ↦ ω a) := by
-      rw [Bandit.streamMeasure, Measure.map_map (by fun_prop) (by fun_prop)]
+      rw [streamMeasure, Measure.map_map (by fun_prop) (by fun_prop)]
       rfl
     _ = ν a := by simp_rw [(measurePreserving_eval_infinitePi _ _).map_eq]
 
@@ -44,9 +44,6 @@ notation "𝓛[" Y " | " X " in " s "; " μ "]" => Measure.map Y (μ[|X ⁻¹' s
 /-- Law of `Y` conditioned on the event that `X` equals `x`. -/
 notation "𝓛[" Y " | " X " ← " x "; " μ "]" => Measure.map Y (μ[|X ⁻¹' {x}])
 
-local notation "𝔓t" => Bandit.trajMeasure alg ν
-local notation "𝔓" => Bandit.measure alg ν
-
 omit [DecidableEq α] in
 lemma condDistrib_reward'' [Countable α]
     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (n : ℕ) :
@@ -107,7 +104,7 @@ lemma condIndepFun_reward_stepsUntil_action [StandardBorelSpace Ω] [Countable 
       (fun ω ↦ R n ω.1) ({ω | stepsUntil A a m ω.1 = ↑n}.indicator (fun _ ↦ 1)) 𝔓' := by
   have hA := h.measurable_A
   have hR := h.measurable_R
-  exact condIndepFun_fst_prod (ν := Bandit.streamMeasure ν)
+  exact condIndepFun_fst_prod (ν := streamMeasure ν)
     (measurable_indicator_stepsUntil_eq hA hR a m n) (by fun_prop) (by fun_prop)
     (condIndepFun_reward_stepsUntil_action' h a m n)
 
@@ -229,97 +226,8 @@ lemma identDistrib_rewardByCount_id [StandardBorelSpace Ω] [Countable α]
 
 lemma identDistrib_rewardByCount_eval [StandardBorelSpace Ω] [Countable α]
     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : α) (n m : ℕ) (hn : n ≠ 0) :
-    IdentDistrib (rewardByCount A R a n) (fun ω ↦ ω m a) 𝔓' (Bandit.streamMeasure ν) :=
+    IdentDistrib (rewardByCount A R a n) (fun ω ↦ ω m a) 𝔓' (streamMeasure ν) :=
   (identDistrib_rewardByCount_id h a n hn).trans
     (identDistrib_eval_eval_id_streamMeasure ν m a).symm
 
--- lemma indepFun_rewardByCount_Iic [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : α)
---     (n : ℕ) :
---     (rewardByCount A R a (n + 1)) ⟂ᵢ[𝔓'] fun ω (i : Iic n) ↦ rewardByCount A R a i ω := by
---   sorry
-
--- lemma iIndepFun_rewardByCount' [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : α) :
---     iIndepFun (rewardByCount A R a) 𝔓' := by
---   have hA := h.measurable_A
---   have hR := h.measurable_R
---   rw [iIndepFun_nat_iff_forall_indepFun (by fun_prop)]
---   exact indepFun_rewardByCount_Iic h a
-
--- lemma iIndepFun_rewardByCount [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) :
---     iIndepFun (fun (p : α × ℕ) ↦ rewardByCount A R p.1 (p.2 + 1)) 𝔓' := by
---   sorry
-
--- lemma identDistrib_rewardByCount_stream_all [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) :
---     IdentDistrib (fun ω (p : α × ℕ) ↦ rewardByCount A R p.1 (p.2 + 1) ω)
---       (fun ω p ↦ ω p.2 p.1) 𝔓' (Bandit.streamMeasure ν) := by
---   refine IdentDistrib.pi (fun p ↦ ?_) ?_ ?_
---   · refine identDistrib_rewardByCount_eval h p.1 (p.2 + 1) p.2 (by simp) (ν := ν)
---   · exact iIndepFun_rewardByCount h
---   · sorry
-
--- lemma identDistrib_rewardByCount_stream' [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : α) :
---     IdentDistrib (fun ω n ↦ rewardByCount A R a (n + 1) ω) (fun ω n ↦ ω n a)
---       𝔓' (Bandit.streamMeasure ν) := by
---   refine IdentDistrib.pi (fun n ↦ ?_) ?_ ?_
---   · refine identDistrib_rewardByCount_eval h a (n + 1) n (by simp) (ν := ν)
---   · have h_indep := iIndepFun_rewardByCount' h a
---     exact iIndepFun.precomp (g := fun n ↦ n + 1) (fun i j hij ↦ by grind) h_indep
---   · exact iIndepFun_eval_streamMeasure'' ν a
-
-omit [DecidableEq α] [StandardBorelSpace α] [Nonempty α] in
-lemma identDistrib_eval_streamMeasure_measure (a : α) :
-    IdentDistrib (fun ω n ↦ ω n a) (fun ω n ↦ ω.2 n a)
-      (Bandit.streamMeasure ν) 𝔓 := by
-  refine IdentDistrib.pi (fun n ↦ ?_) ?_ ?_
-  · rw [← Bandit.snd_measure alg ν, Measure.snd,
-      identDistrib_map_left_iff (by fun_prop) (by fun_prop)
-        (Measurable.aemeasurable <| by fun_prop)]
-    exact IdentDistrib.refl (by fun_prop)
-  · exact iIndepFun_eval_streamMeasure'' ν a
-  · change iIndepFun (fun n ↦ ((fun ω ↦ ω n a) ∘ Prod.snd)) 𝔓
-    rw [← iIndepFun_map_iff (by fun_prop) (fun _ ↦ Measurable.aemeasurable (by fun_prop))]
-    rw [← Measure.snd, Bandit.snd_measure]
-    exact iIndepFun_eval_streamMeasure'' ν a
-
--- lemma identDistrib_rewardByCount_stream [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : α) :
---     IdentDistrib (fun ω n ↦ rewardByCount A R a (n + 1) ω) (fun ω n ↦ ω.2 n a) 𝔓' 𝔓 :=
---   (identDistrib_rewardByCount_stream' h a).trans (identDistrib_eval_streamMeasure_measure a)
-
--- lemma indepFun_rewardByCount_of_ne [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) {a b : α} (hab : a ≠ b) :
---     IndepFun (fun ω s ↦ rewardByCount A R a s ω) (fun ω s ↦ rewardByCount A R b s ω) 𝔓' := by
---   sorry
-
--- lemma identDistrib_sum_Icc_rewardByCount [StandardBorelSpace Ω] [Nonempty Ω] [Countable α]
---     (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (m : ℕ) (a : α) :
---     IdentDistrib (fun ω ↦ ∑ s ∈ Icc 1 m, rewardByCount A R a s ω)
---       (fun ω ↦ ∑ s ∈ range m, ω.2 s a) 𝔓' 𝔓 := by
---   have h1 (a : α) :
---       IdentDistrib (fun ω s ↦ rewardByCount A R a (s + 1) ω) (fun ω s ↦ ω.2 s a) 𝔓' 𝔓 :=
---     identDistrib_rewardByCount_stream h a
---   have h_eq (ω : Ω × (ℕ → α → ℝ)) : ∑ s ∈ Icc 1 m, rewardByCount A R a s ω
---       = ∑ s ∈ range m, rewardByCount A R a (s + 1) ω := by
---     let e : Icc 1 m ≃ range m :=
---     { toFun x := ⟨x - 1, by have h := x.2; simp only [mem_Icc] at h; simp; grind⟩
---       invFun x := ⟨x + 1, by
---         have h := x.2
---         simp only [mem_Icc, le_add_iff_nonneg_left, zero_le, true_and, ge_iff_le]
---         simp only [mem_range] at h
---         grind⟩
---       left_inv x := by have h := x.2; simp only [mem_Icc] at h; grind
---       right_inv x := by have h := x.2; grind }
---     rw [← sum_coe_sort (Icc 1 m), ← sum_coe_sort (range m), sum_equiv e]
---     · simp
---     · simp only [univ_eq_attach, mem_attach, forall_const, Subtype.forall, mem_Icc,
---         forall_and_index]
---       grind
---   simp_rw [h_eq]
---   exact IdentDistrib.comp (h1 a) (u := fun p ↦ ∑ s ∈ range m, p s) (by fun_prop)
-
 end Bandits