LeanMachineLearning

Learning.randomSampling.hasLaw_feeback๐Ÿ”—

Lemma

Each reward follows the distribution ฮผ.map f.

๐Ÿ”—theorem
Learning.randomSampling.hasLaw_feeback.{u_1, u_2, u_3} {๐“ : Type u_1}
  {๐“จ : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {mฮฉ : MeasurableSpace ฮฉ}
  {ฮผ : MeasureTheory.Measure ๐“} [MeasureTheory.IsProbabilityMeasure ฮผ]
  {P : MeasureTheory.Measure ฮฉ} [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} {f : ๐“ โ†’ ๐“จ} {hf : Measurable f}
  [StandardBorelSpace ๐“จ] [Nonempty ๐“จ]
  (h : IsAlgEnvSeq A Y (randomSampling ฮผ) (evalEnv f hf) P) (n : โ„•) :
  ProbabilityTheory.HasLaw (Y n) (MeasureTheory.Measure.map f ฮผ) P
Learning.randomSampling.hasLaw_feeback.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ}
  {ฮผ : MeasureTheory.Measure ๐“}
  [MeasureTheory.IsProbabilityMeasure ฮผ]
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  {f : ๐“ โ†’ ๐“จ} {hf : Measurable f}
  [StandardBorelSpace ๐“จ] [Nonempty ๐“จ]
  (h :
    IsAlgEnvSeq A Y (randomSampling ฮผ)
      (evalEnv f hf) P)
  (n : โ„•) :
  ProbabilityTheory.HasLaw (Y n)
    (MeasureTheory.Measure.map f ฮผ) P

Code

lemma hasLaw_feeback (h : IsAlgEnvSeq A Y (randomSampling ฮผ) (evalEnv f hf) P) (n : โ„•) :
    HasLaw (Y n) (ฮผ.map f) P
Proof
by
  refine HasLaw.congr ?_ (feedback_evalEnv_ae_eq_eval_action h n)
  have hA := h.measurable_action n
  refine โŸจby fun_prop, ?_โŸฉ
  rw [โ† Measure.map_map hf hA, (hasLaw_action h n).map_eq]

Actions: Source ยท 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 ยท open it in the Lean web editor.

Dependency graph

Audit surface: 8 project declarations, 47 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.