LeanMachineLearning

Learning.IsObliviousEnv.hasCondDistrib_feedback๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
Learning.IsObliviousEnv.hasCondDistrib_feedback.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {ฮฉ : Type u_3} {mฮฉ : MeasurableSpace ฮฉ}
  {alg : Algorithm ๐“ ๐“จ} {env : Environment ๐“ ๐“จ}
  {P : MeasureTheory.Measure ฮฉ} [MeasureTheory.IsFiniteMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} [IsObliviousEnv env]
  (h : IsAlgEnvSeq A Y alg env P) (n : โ„•) :
  ProbabilityTheory.HasCondDistrib (Y n) (A n)
    (feedbackCondAction env n) P
Learning.IsObliviousEnv.hasCondDistrib_feedback.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {ฮฉ : Type u_3}
  {mฮฉ : MeasurableSpace ฮฉ}
  {alg : Algorithm ๐“ ๐“จ}
  {env : Environment ๐“ ๐“จ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsFiniteMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  [IsObliviousEnv env]
  (h : IsAlgEnvSeq A Y alg env P)
  (n : โ„•) :
  ProbabilityTheory.HasCondDistrib (Y n)
    (A n) (feedbackCondAction env n) P

Code

lemma hasCondDistrib_feedback [IsObliviousEnv env] (h : IsAlgEnvSeq A Y alg env P) (n : โ„•) :
    HasCondDistrib (Y n) (A n) (feedbackCondAction env n) P
Proof
by
  have hA := h.measurable_action
  have hY := h.measurable_feedback
  cases n with
  | zero => rw [โ† ฮฝ0_eq_feedbackCondAction]; exact h.hasCondDistrib_feedback_zero
  | succ n =>
    refine โŸจby fun_prop, ?_โŸฉ
    have h_eq := (h.hasCondDistrib_feedback n).map_eq
    have : P.map (A (n + 1)) =
        (P.map (fun x โ†ฆ (history A Y n x, A (n + 1) x))).snd := by
      rw [Measure.snd_map_prodMk (by fun_prop)]
    simp only [feedback_eq_feedbackCondAction] at h_eq
    rw [this, โ† Measure.snd_prodAssoc_compProd_prodMkLeft, โ† h_eq,
      Measure.snd_map_prodMk (by fun_prop), Measure.map_map (by fun_prop) (by fun_prop)]
    congr

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: 6 project declarations, 44 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.