LeanMachineLearning

Learning.RoundRobin.action_ae_eq๐Ÿ”—

Lemma

The action chosen at time n is the action n % K.

๐Ÿ”—theorem
Learning.RoundRobin.action_ae_eq.{u_1, u_2} {๐“จ : Type u_1}
  {m๐“จ : MeasurableSpace ๐“จ} {K : โ„•} {hK : 0 < K}
  {ฮฝ : ProbabilityTheory.Kernel (Fin K) ๐“จ}
  [ProbabilityTheory.IsMarkovKernel ฮฝ] {ฮฉ : Type u_2}
  {mฮฉ : MeasurableSpace ฮฉ} {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P] {A : โ„• โ†’ ฮฉ โ†’ Fin K}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} (n : โ„•)
  (h :
    IsAlgEnvSeqUntil A Y (roundRobinAlgorithm hK) (stationaryEnv ฮฝ) P
      n) :
  A n =แต[P] fun x => โŸจn % K, โ‹ฏโŸฉ
Learning.RoundRobin.action_ae_eq.{u_1,
    u_2}
  {๐“จ : Type u_1} {m๐“จ : MeasurableSpace ๐“จ}
  {K : โ„•} {hK : 0 < K}
  {ฮฝ : ProbabilityTheory.Kernel (Fin K) ๐“จ}
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  {ฮฉ : Type u_2} {mฮฉ : MeasurableSpace ฮฉ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ Fin K} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  (n : โ„•)
  (h :
    IsAlgEnvSeqUntil A Y
      (roundRobinAlgorithm hK)
      (stationaryEnv ฮฝ) P n) :
  A n =แต[P] fun x => โŸจn % K, โ‹ฏโŸฉ

Code

lemma action_ae_eq (n : โ„•)
    (h : IsAlgEnvSeqUntil A Y (roundRobinAlgorithm hK) (stationaryEnv ฮฝ) P n) :
    A n =แต[P] fun _ โ†ฆ โŸจn % K, Nat.mod_lt _ hKโŸฉ
Proof
by
  cases n with
  | zero => exact action_zero h
  | succ n =>
    filter_upwards [action_ae_eq_roundRobinNextAction n h] with h hn_eq
    rw [hn_eq, nextAction]

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: 9 project declarations, 57 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.