LeanMachineLearning

Bandits.UCB.arm_ae_all_eqπŸ”—

Lemma

No docstring.

πŸ”—theorem
Bandits.UCB.arm_ae_all_eq.{u_1} {K : β„•} {hK : 0 < K} {c : ℝ}
  {Ξ½ : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel Ξ½] {Ξ© : Type u_1}
  {mΞ© : MeasurableSpace Ξ©} {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P] {A : β„• β†’ Ξ© β†’ Fin K}
  {R : β„• β†’ Ξ© β†’ ℝ}
  (h :
    Learning.IsAlgEnvSeq A R (ucbAlgorithm hK c)
      (Learning.stationaryEnv Ξ½) P) :
  βˆ€α΅ (h : Ξ©) βˆ‚P,
    A 0 h = ⟨0, hK⟩ ∧
      βˆ€ (n : β„•), A (n + 1) h = nextArm hK c n (Learning.history A R n h)
Bandits.UCB.arm_ae_all_eq.{u_1} {K : β„•}
  {hK : 0 < K} {c : ℝ}
  {Ξ½ : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel Ξ½]
  {Ξ© : Type u_1} {mΞ© : MeasurableSpace Ξ©}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : β„• β†’ Ξ© β†’ Fin K} {R : β„• β†’ Ξ© β†’ ℝ}
  (h :
    Learning.IsAlgEnvSeq A R
      (ucbAlgorithm hK c)
      (Learning.stationaryEnv Ξ½) P) :
  βˆ€α΅ (h : Ξ©) βˆ‚P,
    A 0 h = ⟨0, hK⟩ ∧
      βˆ€ (n : β„•),
        A (n + 1) h =
          nextArm hK c n
            (Learning.history A R n h)

Code

lemma arm_ae_all_eq (h : IsAlgEnvSeq A R (ucbAlgorithm hK c) (stationaryEnv Ξ½) P) :
    βˆ€α΅ h βˆ‚P, A 0 h = ⟨0, hK⟩ ∧ βˆ€ n, A (n + 1) h = nextArm hK c n (history A R n h)
Proof
by
  rw [eventually_and, ae_all_iff]
  exact ⟨arm_zero h, arm_ae_eq_ucbNextArm h⟩

Actions: Source Β· 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 Β· open it in the Lean web editor.

Dependency graph

Audit surface: 17 project declarations, 110 external constants

βœ“ 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.