LeanMachineLearning

ProbabilityTheory.Kernel.compProd_lift🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.Kernel.compProd_lift.{x, y, z, w} {X : Type x}
  [MeasurableSpace X] {Y : Type y} [MeasurableSpace Y] {X' : Type w}
  [MeasurableSpace X'] {Y' : Type w} [MeasurableSpace Y'] (ex : X' ≃ᵐ X)
  (ey : Y' ≃ᵐ Y) {Z : Type z} [MeasurableSpace Z] {Z' : Type w}
  [MeasurableSpace Z'] (ez : Z' ≃ᵐ Z) (κ : Kernel X Y)
  (η : Kernel (X × Y) Z) :
  compProd (lift κ) (lift η) = lift (compProd κ η)
ProbabilityTheory.Kernel.compProd_lift.{x,
    y, z, w}
  {X : Type x} [MeasurableSpace X]
  {Y : Type y} [MeasurableSpace Y]
  {X' : Type w} [MeasurableSpace X']
  {Y' : Type w} [MeasurableSpace Y']
  (ex : X' ≃ᵐ X) (ey : Y' ≃ᵐ Y)
  {Z : Type z} [MeasurableSpace Z]
  {Z' : Type w} [MeasurableSpace Z']
  (ez : Z' ≃ᵐ Z) (κ : Kernel X Y)
  (η : Kernel (X × Y) Z) :
  compProd (lift κ) (lift η) =
    lift (compProd κ η)

Code

lemma compProd_lift (κ : Kernel X Y) (η : Kernel (X × Y) Z) :
    κ.lift (ex := ex) (ey := ey) ⊗ₖ η.lift (ex := ex.prodCongr ey) (ey := ez) =
      lift (ex := ex) (ey := ey.prodCongr ez) (κ ⊗ₖ η)
Proof
by
  by_cases hκ : IsSFiniteKernel <| lift (ex := ex) (ey := ey) κ
  swap
  · simp only [hκ, not_false_eq_true, compProd_of_not_isSFiniteKernel_left,
    (isSFinite_lift ex ey κ).not.mpr hκ]
    simp [lift]
  · by_cases hη : IsSFiniteKernel <| lift (ex := ex.prodCongr ey) (ey := ez) η
    swap
    · simp [hη, (isSFinite_lift (ex.prodCongr ey) ez η).not.mpr hη]
      simp [lift]
    · simp only [compProd]
      rw [← comp_lift (ex := ex.prodCongr ex) (ey := ey.prodCongr ez), ← copy_lift,
        ← comp_lift (ex := ex.prodCongr ey), ← parallelComp_lift, ← id_lift,
        ← comp_lift (ex := ex.prodCongr (ey.prodCongr ey)), ← parallelComp_lift, ← id_lift,
        ← copy_lift,
        ← comp_lift (ex := (ex.prodCongr ey).prodCongr ey),
        ← comp_lift (ex := ez.prodCongr ey), ← parallelComp_lift, ← id_lift, ← swap_lift]
      congr
      simp only [lift]
      rw [deterministic_map (MeasurableEquiv.measurable _) (MeasurableEquiv.measurable _)]
      ext _ : 1
      simp [comap_apply, deterministic_apply, MeasurableEquiv.prodCongr, prodAssoc]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23).

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: 1 project declarations, 14 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.