LeanMachineLearning

Learning.measurableSet_stepsUntil_eq๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
Learning.measurableSet_stepsUntil_eq.{u_1, u_2, u_3} {๐“ : Type u_1}
  {R : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {alg : Algorithm ๐“ R} {env : Environment ๐“ R}
  {P : MeasureTheory.Measure ฮฉ} [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R' : โ„• โ†’ ฮฉ โ†’ R} [MeasurableSingletonClass ๐“]
  (h : IsAlgEnvSeq A R' alg env P) (a : ๐“) (m n : โ„•) :
  MeasurableSet {ฯ‰ | stepsUntil A a m ฯ‰ = โ†‘n}
Learning.measurableSet_stepsUntil_eq.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {R : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {alg : Algorithm ๐“ R}
  {env : Environment ๐“ R}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R' : โ„• โ†’ ฮฉ โ†’ R}
  [MeasurableSingletonClass ๐“]
  (h : IsAlgEnvSeq A R' alg env P) (a : ๐“)
  (m n : โ„•) :
  MeasurableSet
    {ฯ‰ | stepsUntil A a m ฯ‰ = โ†‘n}

Code

lemma measurableSet_stepsUntil_eq [MeasurableSingletonClass ๐“]
    (h : IsAlgEnvSeq A R' alg env P) (a : ๐“) (m n : โ„•) :
    MeasurableSet[MeasurableSpace.comap (fun ฯ‰ : ฮฉ โ†ฆ (history A R' (n-1) ฯ‰, A n ฯ‰))
        inferInstance]
      {ฯ‰ : ฮฉ | stepsUntil A a m ฯ‰ = โ†‘n}
Proof
by
  let mProd := MeasurableSpace.comap
    (fun ฯ‰ : ฮฉ โ†ฆ (history A R' (n-1) ฯ‰, A n ฯ‰)) inferInstance
  suffices Measurable[mProd] ({ฯ‰ | stepsUntil A a m ฯ‰ = โ†‘n}.indicator fun x โ†ฆ 1) by
    rwa [measurable_indicator_const_iff] at this
  exact measurable_comap_indicator_stepsUntil_eq h a m n

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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 ยท open it in the Lean web editor.

Dependency graph

Audit surface: 6 project declarations, 66 external constants

โœ“ 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.