`Learning.IsAlgEnvSeq.hasLaw_history_withDensity`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.hasLaw_history_withDensity.{u_1, u_2, u_3, u_4}
  {𝓐 : Type u_1} {𝓨 : Type u_2} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  {Ω : Type u_3} [MeasurableSpace Ω] {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨} {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P]
  {Ω₀ : Type u_4} [MeasurableSpace Ω₀] {alg₀ : Algorithm 𝓐 𝓨}
  {A₀ : ℕ → Ω₀ → 𝓐} {Y₀ : ℕ → Ω₀ → 𝓨} {P₀ : MeasureTheory.Measure Ω₀}
  [MeasureTheory.IsProbabilityMeasure P₀]
  [MeasurableSpace.CountablyGenerated 𝓐] (h : IsAlgEnvSeq A Y alg env P)
  (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀)
  (hc : Algorithm.AbsolutelyContinuous alg alg₀) (n : ℕ) :
  ProbabilityTheory.HasLaw (history A Y n)
    (MeasureTheory.Measure.withDensity
      (MeasureTheory.Measure.map (history A₀ Y₀ n) P₀)
      (Algorithm.density alg alg₀ n))
    P
Learning.IsAlgEnvSeq.hasLaw_history_withDensity.{u_1,
    u_2, u_3, u_4}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  {Ω : Type u_3} [MeasurableSpace Ω]
  {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure P]
  {Ω₀ : Type u_4} [MeasurableSpace Ω₀]
  {alg₀ : Algorithm 𝓐 𝓨} {A₀ : ℕ → Ω₀ → 𝓐}
  {Y₀ : ℕ → Ω₀ → 𝓨}
  {P₀ : MeasureTheory.Measure Ω₀}
  [MeasureTheory.IsProbabilityMeasure P₀]
  [MeasurableSpace.CountablyGenerated 𝓐]
  (h : IsAlgEnvSeq A Y alg env P)
  (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀)
  (hc :
    Algorithm.AbsolutelyContinuous alg
      alg₀)
  (n : ℕ) :
  ProbabilityTheory.HasLaw (history A Y n)
    (MeasureTheory.Measure.withDensity
      (MeasureTheory.Measure.map
        (history A₀ Y₀ n) P₀)
      (Algorithm.density alg alg₀ n))
    P

Code

lemma hasLaw_history_withDensity (h : IsAlgEnvSeq A Y alg env P)
    (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀) (hc : alg ≪ₐ alg₀) (n : ℕ) : HasLaw (history A Y n)
      ((P₀.map (history A₀ Y₀ n)).withDensity (alg.density alg₀ n)) P where
  aemeasurable

Proof

(h.measurable_history n).aemeasurable
  map_eq := by
    induction n with
    | zero =>
      rw [h.hasLaw_history_zero.map_eq, h₀.hasLaw_history_zero.map_eq, h.hasLaw_step_zero.map_eq,
        h₀.hasLaw_step_zero.map_eq]
      rw [← Measure.withDensity_rnDeriv_eq _ _ hc.p0,
        Measure.compProd_withDensity_left (by fun_prop)]
      exact map_equiv_withDensity (by fun_prop)
    | succ n ih =>
      let ρ h' (ar : 𝓐 × 𝓨) := Kernel.rnDeriv (alg.policy n) (alg₀.policy n) h' ar.1
      have hs : stepKernel alg env n = (stepKernel alg₀ env n).withDensity ρ := by
        rw [stepKernel, ← Kernel.withDensity_rnDeriv_eq' (hc.policy n)]
        exact Kernel.compProd_withDensity_left (Kernel.measurable_rnDeriv _ _)
      have : IsMarkovKernel ((stepKernel alg₀ env n).withDensity ρ) := by
        rw [← hs]
        infer_instance
      simp_rw [history_succ]
      rw [← Measure.map_map (by fun_prop), ← Measure.map_map (by fun_prop)]
      rotate_left
      · exact (h₀.measurable_history n).prodMk (h₀.measurable_step (n + 1))
      · exact (h.measurable_history n).prodMk (h.measurable_step (n + 1))
      rw [(h.hasCondDistrib_step n).map_eq, (h₀.hasCondDistrib_step n).map_eq, ih, hs,
        Measure.compProd_withDensity_withDensity (by fun_prop) (by fun_prop)]
      exact map_equiv_withDensity (by fun_prop)

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th 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: 7 project declarations, 91 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.hasLaw_history_withDensity🔗

`Learning.IsAlgEnvSeq.hasLaw_history_withDensity`🔗