ProbabilityTheory.Kernel.leftUnitor_inv🔗

ProbabilityTheory.Kernel.leftUnitor_inv.{u_1, u_5, u_8, u_9}
  {X : Type u_1} [MeasurableSpace X] (SX : SFinKer)
  (ex : SFinKer.carrier SX ≃ᵐ X) {X₀ : Type u_5} [MeasurableSpace X₀]
  (ex₀ : X ≃ᵐ X₀) :
  CategoryTheory.Iso.inv
      (CategoryTheory.MonoidalCategoryStruct.leftUnitor SX) =
    hom (lift (map Kernel.id fun x => (PUnit.unit, x)))
ProbabilityTheory.Kernel.leftUnitor_inv.{u_1,
    u_5, u_8, u_9}
  {X : Type u_1} [MeasurableSpace X]
  (SX : SFinKer)
  (ex : SFinKer.carrier SX ≃ᵐ X)
  {X₀ : Type u_5} [MeasurableSpace X₀]
  (ex₀ : X ≃ᵐ X₀) :
  CategoryTheory.Iso.inv
      (CategoryTheory.MonoidalCategoryStruct.leftUnitor
        SX) =
    hom
      (lift
        (map Kernel.id fun x =>
          (PUnit.unit, x)))

lemma leftUnitor_inv : (λ_ SX).inv = hom (ex := ex) (ey := punit.prodCongr ex)
    (lift (Kernel.id.map (fun x ↦ (PUnit.unit, x))) (ex := ex₀) (ey := punit.prodCongr ex₀))

by
  ext; dsimp
  rw [hom_apply', lift_apply', id_map (by fun_prop), id_map (by fun_prop), deterministic_apply',
    deterministic_apply']
  · refine Set.indicator_eq_indicator ?_ rfl
    simp [Set.image, MeasurableEquiv.prodCongr]
    constructor
    all_goals simp_all
  all_goals measurability

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: 6 project declarations, 122 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.