LeanMachineLearning

Bandits.ETC.arm_of_ge๐Ÿ”—

Lemma

For n โ‰ฅ K * m, the arm pulled at time n is the same as the arm pulled at time K * m.

๐Ÿ”—theorem
Bandits.ETC.arm_of_ge.{u_1} {K : โ„•} {hK : 0 < K} {m : โ„•}
  {ฮฝ : ProbabilityTheory.Kernel (Fin K) โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ] {ฮฉ : Type u_1}
  {mฮฉ : MeasurableSpace ฮฉ} {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P] {A : โ„• โ†’ ฮฉ โ†’ Fin K}
  {R : โ„• โ†’ ฮฉ โ†’ โ„}
  (h :
    Learning.IsAlgEnvSeq A R (etcAlgorithm hK m)
      (Learning.stationaryEnv ฮฝ) P)
  {n : โ„•} (hm : m โ‰  0) (hn : K * m โ‰ค n) : A n =แต[P] A (K * m)
Bandits.ETC.arm_of_ge.{u_1} {K : โ„•}
  {hK : 0 < K} {m : โ„•}
  {ฮฝ : ProbabilityTheory.Kernel (Fin K) โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  {ฮฉ : Type u_1} {mฮฉ : MeasurableSpace ฮฉ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ Fin K} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  (h :
    Learning.IsAlgEnvSeq A R
      (etcAlgorithm hK m)
      (Learning.stationaryEnv ฮฝ) P)
  {n : โ„•} (hm : m โ‰  0) (hn : K * m โ‰ค n) :
  A n =แต[P] A (K * m)

Code

lemma arm_of_ge (h : IsAlgEnvSeq A R (etcAlgorithm hK m) (stationaryEnv ฮฝ) P)
    {n : โ„•} (hm : m โ‰  0) (hn : K * m โ‰ค n) :
    A n =แต[P] A (K * m)
Proof
by
  have h_ae n : K * m โ‰ค n โ†’ A (n + 1) =แต[P] fun ฯ‰ โ†ฆ A n ฯ‰ := arm_add_one_of_ge h hm
  simp_rw [Filter.EventuallyEq, โ† ae_all_iff] at h_ae
  filter_upwards [h_ae] with ฯ‰ h_ae
  induction n, hn using Nat.le_induction with
  | base => rfl
  | succ n hmn h_ind => rw [h_ae n hmn, h_ind]

Actions: Source ยท Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), 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: 16 project declarations, 112 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.