LeanMachineLearning

ProbabilityTheory.Kernel.comp_hom🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.Kernel.comp_hom.{u_1, u_2, u_4, u_5} {X : Type u_1}
  {Y : Type u_2} {Z : Type u_4} [MeasurableSpace X] [MeasurableSpace Y]
  [MeasurableSpace Z] (SX SY SZ : SFinKer)
  (ex : SFinKer.carrier SX ≃ᵐ X) (ey : SFinKer.carrier SY ≃ᵐ Y)
  (ez : SFinKer.carrier SZ ≃ᵐ Z) (η : Kernel X Y) (κ : Kernel Z X)
  [IsSFiniteKernel η] [IsSFiniteKernel κ] :
  CategoryTheory.CategoryStruct.comp (hom κ) (hom η) = hom (comp η κ)
ProbabilityTheory.Kernel.comp_hom.{u_1,
    u_2, u_4, u_5}
  {X : Type u_1} {Y : Type u_2}
  {Z : Type u_4} [MeasurableSpace X]
  [MeasurableSpace Y] [MeasurableSpace Z]
  (SX SY SZ : SFinKer)
  (ex : SFinKer.carrier SX ≃ᵐ X)
  (ey : SFinKer.carrier SY ≃ᵐ Y)
  (ez : SFinKer.carrier SZ ≃ᵐ Z)
  (η : Kernel X Y) (κ : Kernel Z X)
  [IsSFiniteKernel η]
  [IsSFiniteKernel κ] :
  CategoryTheory.CategoryStruct.comp
      (hom κ) (hom η) =
    hom (comp η κ)

Code

lemma comp_hom (η : Kernel X Y) (κ : Kernel Z X) [IsSFiniteKernel η] [IsSFiniteKernel κ] :
    κ.hom (ex := ez) (ey := ex) ≫ η.hom (ex := ex) (ey := ey) =
      (η ∘ₖ κ).hom (ex := ez) (ey := ey)
Proof
by
  ext a s hs
  dsimp
  rw [hom_apply', comp_apply', comp_apply', hom_apply, lintegral_map]
  · congr with y
    simp [hom_apply' _ _ hs]
  all_goals try fun_prop
  all_goals try measurability
  · exact Kernel.measurable_coe η.hom.hom hs

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, 24 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.