LeanMachineLearning

ProbabilityTheory.Kernel.associator_inv🔗

Lemma

No docstring.

🔗theorem
ProbabilityTheory.Kernel.associator_inv.{u_1, u_2, u_4, u_5, u_6, u_7,
    u_8}
  {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) {X₀ : Type u_5} {Y₀ : Type u_6}
  {Z₀ : Type u_7} [MeasurableSpace X₀] [MeasurableSpace Y₀]
  [MeasurableSpace Z₀] (ex₀ : X ≃ᵐ X₀) (ey₀ : Y ≃ᵐ Y₀) (ez₀ : Z ≃ᵐ Z₀) :
  CategoryTheory.Iso.inv
      (CategoryTheory.MonoidalCategoryStruct.associator SX SY SZ) =
    hom
      (lift
        (deterministic ⇑(MeasurableEquiv.symm MeasurableEquiv.prodAssoc)
          ⋯))
ProbabilityTheory.Kernel.associator_inv.{u_1,
    u_2, u_4, u_5, u_6, u_7, u_8}
  {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)
  {X₀ : Type u_5} {Y₀ : Type u_6}
  {Z₀ : Type u_7} [MeasurableSpace X₀]
  [MeasurableSpace Y₀]
  [MeasurableSpace Z₀] (ex₀ : X ≃ᵐ X₀)
  (ey₀ : Y ≃ᵐ Y₀) (ez₀ : Z ≃ᵐ Z₀) :
  CategoryTheory.Iso.inv
      (CategoryTheory.MonoidalCategoryStruct.associator
        SX SY SZ) =
    hom
      (lift
        (deterministic
          ⇑(MeasurableEquiv.symm
              MeasurableEquiv.prodAssoc)
          ⋯))

Code

lemma associator_inv : (α_ SX SY SZ).inv =
    hom (ex := ex.prodCongr (ey.prodCongr ez)) (ey := (ex.prodCongr ey).prodCongr ez)
      (lift (Kernel.deterministic prodAssoc.symm (by fun_prop))
        (ex := ex₀.prodCongr (ey₀.prodCongr ez₀)) (ey := (ex₀.prodCongr ey₀).prodCongr ez₀))
Proof
by
  ext; dsimp
  simp only [hom]
  rw [comap_apply', map_apply', lift_apply', deterministic_apply', deterministic_apply']
  · refine Set.indicator_eq_indicator ?_ rfl
    simp [MeasurableEquiv.prodCongr, prodAssoc]
  all_goals measurability

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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: 5 project declarations, 118 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.