`Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2} {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P'] {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ} [MeasurableSingletonClass 𝓐]
  (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
  (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω n a => (pullCount A a n ω, sumRewards A R a n ω))
    (fun ω' n a => (pullCount A₂ a n ω', sumRewards A₂ R₂ a n ω')) P P'
Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P']
  {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ}
  [MeasurableSingletonClass 𝓐]
  (h1 :
    IsAlgEnvSeq A R alg (stationaryEnv ν)
      P)
  (h2 :
    IsAlgEnvSeq A₂ R₂ alg
      (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω n a =>
      (pullCount A a n ω,
        sumRewards A R a n ω))
    (fun ω' n a =>
      (pullCount A₂ a n ω',
        sumRewards A₂ R₂ a n ω'))
    P P'

Code

lemma _root_.Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards [MeasurableSingletonClass 𝓐]
    (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
    (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
    IdentDistrib (fun ω n a ↦ (pullCount A a n ω, sumRewards A R a n ω))
      (fun ω' n a ↦ (pullCount A₂ a n ω', sumRewards A₂ R₂ a n ω')) P P'

Proof

by
  let f (τ : ℕ → 𝓐 × ℝ) (n : ℕ) (a : 𝓐) : ℕ × ℝ :=
    (∑ i ∈ range n, if (τ i).1 = a then 1 else 0,
     ∑ i ∈ range n, if (τ i).1 = a then (τ i).2 else 0)
  have hc1 : (fun ω n a ↦ (pullCount A a n ω, sumRewards A R a n ω)) =
      f ∘ (trajectory A R) := by
    ext ω n a : 3
    simp_rw [Function.comp, f, pullCount, card_filter, sumRewards, trajectory]
    rfl
  have hc2 : (fun ω' n a ↦ (pullCount A₂ a n ω', sumRewards A₂ R₂ a n ω')) =
      f ∘ (trajectory A₂ R₂) := by
    ext ω' n a : 3
    simp_rw [Function.comp, f, pullCount, card_filter, sumRewards, trajectory]
    rfl
  have hf : Measurable f := by
    simp_rw [f, measurable_pi_iff]
    intro n a
    apply Measurable.prod
    · dsimp only
      exact measurable_sum _
        (fun _ _ ↦ Measurable.ite (by measurability) (by fun_prop) (by fun_prop))
    · dsimp only
      exact measurable_sum _
        (fun _ _ ↦ Measurable.ite (by measurability) (by fun_prop) (by fun_prop))
  rw [hc1, hc2]
  exact (h1.identDistrib_trajectory h2).comp hf

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 5th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 8 project declarations, 63 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards🔗

`Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards`🔗