LeanMachineLearning

Learning.isStoppingTime_stepsUntil_filtrationActionπŸ”—

Lemma

stepsUntil a m is a stopping time with respect to the filtration filtrationAction.

πŸ”—theorem
Learning.isStoppingTime_stepsUntil_filtrationAction.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {R : Type u_2} {Ξ© : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} {mΞ© : MeasurableSpace Ξ©} [DecidableEq 𝓐]
  {alg : Algorithm 𝓐 R} {env : Environment 𝓐 R}
  {P : MeasureTheory.Measure Ξ©} [MeasureTheory.IsProbabilityMeasure P]
  {A : β„• β†’ Ξ© β†’ 𝓐} {R' : β„• β†’ Ξ© β†’ R} [MeasurableSingletonClass 𝓐]
  (h : IsAlgEnvSeq A R' alg env P) (a : 𝓐) (m : β„•) :
  MeasureTheory.IsStoppingTime (IsAlgEnvSeq.filtrationAction h)
    (stepsUntil A a m)
Learning.isStoppingTime_stepsUntil_filtrationAction.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {R : Type u_2}
  {Ξ© : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  {mΞ© : MeasurableSpace Ξ©} [DecidableEq 𝓐]
  {alg : Algorithm 𝓐 R}
  {env : Environment 𝓐 R}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : β„• β†’ Ξ© β†’ 𝓐} {R' : β„• β†’ Ξ© β†’ R}
  [MeasurableSingletonClass 𝓐]
  (h : IsAlgEnvSeq A R' alg env P) (a : 𝓐)
  (m : β„•) :
  MeasureTheory.IsStoppingTime
    (IsAlgEnvSeq.filtrationAction h)
    (stepsUntil A a m)

Code

lemma isStoppingTime_stepsUntil_filtrationAction [MeasurableSingletonClass 𝓐]
    (h : IsAlgEnvSeq A R' alg env P) (a : 𝓐) (m : β„•) :
    IsStoppingTime h.filtrationAction (stepsUntil A a m)
Proof
by
  refine isStoppingTime_of_measurableSet_eq fun n ↦ ?_
  by_cases hn : n = 0
  Β· subst hn
    simp only [WithTop.coe_zero]
    exact measurableSet_stepsUntil_eq_zero a m
  Β· rw [h.filtrationAction_eq_comap _ hn]
    exact measurableSet_stepsUntil_eq h a m n

Actions: Source Β· Open Issue

Meaning last changed in v4.34.0-rc2-2-g4830b8c (2026-08-23), the 2th 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: 8 project declarations, 77 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.