LeanMachineLearning

Bandits.ArrayModel.hasCondDistrib_action๐Ÿ”—

Lemma

No docstring.

๐Ÿ”—theorem
Bandits.ArrayModel.hasCondDistrib_action.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] [DecidableEq ๐“] [Countable ๐“]
  [StandardBorelSpace R] [Nonempty R] (alg : Learning.Algorithm ๐“ R)
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] (n : โ„•) :
  ProbabilityTheory.HasCondDistrib (action alg (n + 1))
    (fun ฯ‰ i => (action alg (โ†‘i) ฯ‰, reward alg (โ†‘i) ฯ‰))
    (Learning.Algorithm.policy alg n) (arrayMeasure ฮฝ)
Bandits.ArrayModel.hasCondDistrib_action.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  [Countable ๐“] [StandardBorelSpace R]
  [Nonempty R]
  (alg : Learning.Algorithm ๐“ R)
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  (n : โ„•) :
  ProbabilityTheory.HasCondDistrib
    (action alg (n + 1))
    (fun ฯ‰ i =>
      (action alg (โ†‘i) ฯ‰,
        reward alg (โ†‘i) ฯ‰))
    (Learning.Algorithm.policy alg n)
    (arrayMeasure ฮฝ)

Code

lemma hasCondDistrib_action (alg : Algorithm ๐“ R) (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] (n : โ„•) :
    HasCondDistrib (action alg (n + 1))
      (fun ฯ‰ (i : Iic n) โ†ฆ (action alg i ฯ‰, reward alg i ฯ‰))
      (alg.policy n) (arrayMeasure ฮฝ)
Proof
by
  convert hasCondDistrib_action' alg ฮฝ n with ฯ‰ i
  ยท simp only [action]
    rw [hist_eq _ _ n]
  ยท simp only [reward]
    rw [hist_eq _ _ n]

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 5th 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: 13 project declarations, 91 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.