LeanMachineLearning

Learning.randomSampling.iIndep_action๐Ÿ”—

Lemma

Actions are mutually independent.

๐Ÿ”—theorem
Learning.randomSampling.iIndep_action.{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} {env : Environment ๐“ ๐“จ}
  (h : IsAlgEnvSeq A Y (randomSampling ฮผ) env P) :
  ProbabilityTheory.iIndepFun A P
Learning.randomSampling.iIndep_action.{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}
  {env : Environment ๐“ ๐“จ}
  (h :
    IsAlgEnvSeq A Y (randomSampling ฮผ) env
      P) :
  ProbabilityTheory.iIndepFun A P

Code

lemma iIndep_action (h : IsAlgEnvSeq A Y (randomSampling ฮผ) env P) :
    iIndepFun A P
Proof
by
  have hA := h.measurable_action
  rw [iIndepFun_nat_iff_forall_indepFun (by fun_prop)]
  intro n
  have map_eq := (h.hasCondDistrib_action n).map_eq
  simp only [randomSampling_policy, Measure.compProd_const] at map_eq
  have law_eq : P.map (A (n + 1)) = ฮผ := (hasLaw_action h (n + 1)).map_eq
  rw [โ† law_eq, โ† indepFun_iff_map_prod_eq_prod_map_map] at map_eq
  ยท change A (n + 1) โŸ‚แตข[P] (fun (f : Iic n โ†’ ๐“ ร— ๐“จ) โ†ฆ (fun i โ†ฆ (f i).1))โˆ˜ (history A Y n)
    refine map_eq.symm.comp measurable_id (by fun_prop)
  ยท exact (h.measurable_history n).aemeasurable
  ยท exact (h.measurable_action (n + 1)).aemeasurable

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: 5 project declarations, 43 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.