`Learning.IsAlgEnvSeq.hasLaw_history_zero`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.hasLaw_history_zero.{u_1, u_2, u_3} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P]
  (h : IsAlgEnvSeq A Y alg env P) :
  ProbabilityTheory.HasLaw (history A Y 0)
    (MeasureTheory.Measure.map
      (⇑(MeasurableEquiv.symm
          (MeasurableEquiv.piUnique fun x => 𝓐 × 𝓨)))
      (MeasureTheory.Measure.map (step A Y 0) P))
    P
Learning.IsAlgEnvSeq.hasLaw_history_zero.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure P]
  (h : IsAlgEnvSeq A Y alg env P) :
  ProbabilityTheory.HasLaw (history A Y 0)
    (MeasureTheory.Measure.map
      (⇑(MeasurableEquiv.symm
          (MeasurableEquiv.piUnique
            fun x => 𝓐 × 𝓨)))
      (MeasureTheory.Measure.map
        (step A Y 0) P))
    P

Code

lemma IsAlgEnvSeq.hasLaw_history_zero (h : IsAlgEnvSeq A Y alg env P) : HasLaw (history A Y 0)
    ((P.map (step A Y 0)).map (MeasurableEquiv.piUnique (fun _ : Iic 0 ↦ 𝓐 × 𝓨)).symm) P where
  aemeasurable

Proof

(h.measurable_history 0).aemeasurable
  map_eq := by
    have he : (MeasurableEquiv.piUnique (fun _ : Iic 0 ↦ 𝓐 × 𝓨)).symm ∘ step A Y 0 =
        history A Y 0 := by
      funext _ ⟨0, _⟩
      rfl
    rw [← he]
    have hA := h.measurable_action
    have hY := h.measurable_feedback
    exact (Measure.map_map (by fun_prop) (by fun_prop)).symm

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, 47 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_zero🔗

`Learning.IsAlgEnvSeq.hasLaw_history_zero`🔗