`Bandits.UCB.forall_arm_eq_mod_of_lt`🔗

Lemma

No docstring.

theorem

Bandits.UCB.forall_arm_eq_mod_of_lt.{u_1} {K : ℕ} {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {Ω : Type u_1}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ}
  (h :
    Learning.IsAlgEnvSeq A R (ucbAlgorithm hK c)
      (Learning.stationaryEnv ν) P) :
  ∀ᵐ (h : Ω) ∂P, ∀ n < K, A n h = ⟨n % K, ⋯⟩
Bandits.UCB.forall_arm_eq_mod_of_lt.{u_1}
  {K : ℕ} {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  (h :
    Learning.IsAlgEnvSeq A R
      (ucbAlgorithm hK c)
      (Learning.stationaryEnv ν) P) :
  ∀ᵐ (h : Ω) ∂P,
    ∀ n < K, A n h = ⟨n % K, ⋯⟩

Code

lemma forall_arm_eq_mod_of_lt (h : IsAlgEnvSeq A R (ucbAlgorithm hK c) (stationaryEnv ν) P) :
    ∀ᵐ h ∂P, ∀ n < K, A n h = ⟨n % K, Nat.mod_lt _ hK⟩

Proof

by
  simp_rw [ae_all_iff]
  intro n hn
  induction n with
  | zero => exact arm_zero h
  | succ n _ =>
    filter_upwards [arm_ae_eq_ucbNextArm h n] with h h_eq
    rw [h_eq, nextArm, ite_eq_left]
    · rfl
    · grind

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

Dependency graph

Audit surface: 17 project declarations, 110 external constants

✓ 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.

Bandits.UCB.forall_arm_eq_mod_of_lt🔗

`Bandits.UCB.forall_arm_eq_mod_of_lt`🔗