Learning.IsAlgEnvSeq.map_trajectory🔗

Learning.IsAlgEnvSeq.map_trajectory.{u_1, u_2, u_4} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {Ω : Type u_4} {mΩ : MeasurableSpace Ω} {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {A₁ : ℕ → Ω → 𝓐}
  {R₁ : ℕ → Ω → 𝓨} (h : IsAlgEnvSeq A₁ R₁ alg env P) :
  MeasureTheory.Measure.map (trajectory A₁ R₁) P = trajMeasure alg env
Learning.IsAlgEnvSeq.map_trajectory.{u_1,
    u_2, u_4}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ω : Type u_4}
  {mΩ : MeasurableSpace Ω}
  {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A₁ : ℕ → Ω → 𝓐} {R₁ : ℕ → Ω → 𝓨}
  (h : IsAlgEnvSeq A₁ R₁ alg env P) :
  MeasureTheory.Measure.map
      (trajectory A₁ R₁) P =
    trajMeasure alg env

lemma IsAlgEnvSeq.map_trajectory (h : IsAlgEnvSeq A₁ R₁ alg env P) :
    P.map (trajectory A₁ R₁) = trajMeasure alg env

by
  rw [trajMeasure]
  have h := (Kernel.hasLaw_trajMeasure (Y := fun n ω ↦ (A₁ n ω, R₁ n ω)) (P := P)
    (μ₀ := alg.p0 ⊗ₘ env.ν0) (κ := stepKernel alg env) (fun n ↦ ?_) ?_ (fun n ↦ ?_)).map_eq
  · exact h
  · have hA := h.measurable_action n
    have hR := h.measurable_feedback n
    fun_prop
  · exact h.hasLaw_step_zero
  · exact h.hasCondDistrib_step n

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 · open it in the Lean web editor.

Dependency graph

Audit surface: 7 project declarations, 46 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.