LeanMachineLearning

Learning.IsBayesAlgEnvSeq.ae_IsAlgEnvSeqπŸ”—

Lemma

No docstring.

πŸ”—theorem
Learning.IsBayesAlgEnvSeq.ae_IsAlgEnvSeq.{u_1, u_2, u_3, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} {Ξ© : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
  [MeasurableSpace Ξ©] {Q : MeasureTheory.Measure 𝓔}
  {ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨} {alg : Algorithm 𝓐 𝓨}
  {E : Ξ© β†’ 𝓔} {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  {P : MeasureTheory.Measure Ξ©} [MeasureTheory.IsFiniteMeasure P]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐] [StandardBorelSpace 𝓨]
  [Nonempty 𝓨] [ProbabilityTheory.IsMarkovKernel ΞΊ]
  (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P) :
  βˆ€α΅ (e : 𝓔) βˆ‚Q,
    IsAlgEnvSeq IT.action IT.feedback alg
      (stationaryEnv (ProbabilityTheory.Kernel.sectR ΞΊ e))
      (𝓛[trajectory A Y | E; P] e)
Learning.IsBayesAlgEnvSeq.ae_IsAlgEnvSeq.{u_1,
    u_2, u_3, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} {Ξ© : Type u_4}
  [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] [MeasurableSpace Ξ©]
  {Q : MeasureTheory.Measure 𝓔}
  {ΞΊ : ProbabilityTheory.Kernel (𝓔 Γ— 𝓐) 𝓨}
  {alg : Algorithm 𝓐 𝓨} {E : Ξ© β†’ 𝓔}
  {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsFiniteMeasure P]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  [ProbabilityTheory.IsMarkovKernel ΞΊ]
  (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P) :
  βˆ€α΅ (e : 𝓔) βˆ‚Q,
    IsAlgEnvSeq IT.action IT.feedback alg
      (stationaryEnv
        (ProbabilityTheory.Kernel.sectR ΞΊ
          e))
      (𝓛[trajectory A Y | E; P] e)

Code

lemma ae_IsAlgEnvSeq [IsMarkovKernel ΞΊ] (h : IsBayesAlgEnvSeq Q ΞΊ alg E A Y P) :
    βˆ€α΅ e βˆ‚Q, IsAlgEnvSeq IT.action IT.feedback alg (stationaryEnv (ΞΊ.sectR e))
      (condDistrib (trajectory A Y) E P e)
Proof
by
  filter_upwards [hasLaw_IT_action_zero h, hasCondDistrib_IT_feedback_zero h,
    ae_all_iff.2 (hasCondDistrib_IT_action h), ae_all_iff.2 (hasCondDistrib_IT_feedback h)]
    with _ ha0 hr0 hA hR
  exact ⟨IT.measurable_action, IT.measurable_feedback, ha0, hr0, hA, hR⟩

Actions: Source Β· 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 Β· open it in the Lean web editor.

Dependency graph

Audit surface: 10 project declarations, 63 external constants

βœ“ 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.