`Learning.IsAlgEnvSeq.absolutelyContinuous_map_history`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.absolutelyContinuous_map_history.{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]
  (h : IsAlgEnvSeq A Y alg env P) (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀)
  (hc : Algorithm.AbsolutelyContinuous alg alg₀) (n : ℕ) :
  MeasureTheory.Measure.AbsolutelyContinuous
    (MeasureTheory.Measure.map (history A Y n) P)
    (MeasureTheory.Measure.map (history A₀ Y₀ n) P₀)
Learning.IsAlgEnvSeq.absolutelyContinuous_map_history.{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]
  (h : IsAlgEnvSeq A Y alg env P)
  (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀)
  (hc :
    Algorithm.AbsolutelyContinuous alg
      alg₀)
  (n : ℕ) :
  MeasureTheory.Measure.AbsolutelyContinuous
    (MeasureTheory.Measure.map
      (history A Y n) P)
    (MeasureTheory.Measure.map
      (history A₀ Y₀ n) P₀)

Code

lemma absolutelyContinuous_map_history (h : IsAlgEnvSeq A Y alg env P)
    (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀) (hc : alg ≪ₐ alg₀) (n : ℕ) :
    P.map (history A Y n) ≪ P₀.map (history A₀ Y₀ n)

Proof

by
  induction n with
  | zero =>
    rw [h.hasLaw_history_zero.map_eq, h₀.hasLaw_history_zero.map_eq]
    apply Measure.AbsolutelyContinuous.map _ (by fun_prop)
    rw [h.hasLaw_step_zero.map_eq, h₀.hasLaw_step_zero.map_eq]
    exact Measure.AbsolutelyContinuous.compProd_left hc.p0 _
  | succ n ih =>
    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))
    apply Measure.AbsolutelyContinuous.map _ (by fun_prop)
    rw [(h.hasCondDistrib_step n).map_eq, (h₀.hasCondDistrib_step n).map_eq]
    apply Measure.AbsolutelyContinuous.compProd ih
    filter_upwards with h' using Measure.AbsolutelyContinuous.compProd_left_apply (hc.policy n h') _

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: 5 project declarations, 45 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.absolutelyContinuous_map_history🔗

`Learning.IsAlgEnvSeq.absolutelyContinuous_map_history`🔗