Learning.feedbackCondAction_obliviousEnv🔗

Learning.feedbackCondAction_obliviousEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [hν : ∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (ν n)] (n : ℕ) :
  feedbackCondAction (obliviousEnv ν) n = ν n
Learning.feedbackCondAction_obliviousEnv.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [hν :
    ∀ (n : ℕ),
      ProbabilityTheory.IsMarkovKernel
        (ν n)]
  (n : ℕ) :
  feedbackCondAction (obliviousEnv ν) n =
    ν n

lemma feedbackCondAction_obliviousEnv (ν : ℕ → Kernel 𝓐 𝓨) [hν : ∀ n, IsMarkovKernel (ν n)]
    (n : ℕ) :
    feedbackCondAction (obliviousEnv ν) n = ν n

by
  rcases isEmpty_or_nonempty 𝓐 with h𝓐 | h𝓐
  · ext a : 1
    exact h𝓐.elim a
  rcases isEmpty_or_nonempty 𝓨 with hR | hR
  · refine absurd (hν 0) ?_
    simp only [Subsingleton.eq_zero ν, Pi.zero_apply]
    exact Kernel.not_isMarkovKernel_zero
  have : Nonempty (Iic n → 𝓐 × 𝓨) := ⟨fun _ ↦ (h𝓐.some, hR.some)⟩
  have h_eq_zero := ν0_eq_feedbackCondAction (obliviousEnv ν)
  have h_eq := feedback_eq_feedbackCondAction (obliviousEnv ν) (n - 1)
  cases n with
  | zero => exact h_eq_zero.symm
  | succ n =>
    simp only [Nat.add_one_sub_one, obliviousEnv_feedback, add_tsub_cancel_right] at h_eq
    rw [← Kernel.prodMkLeft_inj (γ := Iic n → 𝓐 × 𝓨)]
    exact h_eq.symm

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 2th 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: 5 project declarations, 31 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.