MeasurableEquiv.prodUnique
From the authors
The measurable equivalence α × β ≃ᵐ α when β has a unique element.
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α : Type u_1MeasurableSpace αA measurable space is a space equipped with a σ-algebra. -
β : Type u_2MeasurableSpace βUnique βUnique αexpresses thatαis a type with a unique termdefault.
α × β ≃ᵐ α{ toFun := Prod.fst, invFun := fun a => (a, default), left_inv := ⋯, right_inv := ⋯, measurable_toFun := ⋯,
measurable_invFun := ⋯ }MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Unique : Sort u → Sort (max 1 u)`Unique α` expresses that `α` is a type with a unique term `default`. This is implemented as a type, rather than a `Prop`-valued predicate, for good definitional properties of the default term.
MeasurableEquiv : (α : Type u_6) → (β : Type u_7) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_6 u_7)Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.
Prod : Type u → Type v → Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types. Elements of this type are pairs in which the first element is an `α` and the second element is a `β`. Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`. Conventions for notations in identifiers: * The recommended spelling of `×` in identifiers is `Prod`.
MeasurableEquiv.mk : {α : Type u_6} →
{β : Type u_7} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] →
(toEquiv : α ≃ β) →
autoParam (Measurable ⇑toEquiv) MeasurableEquiv.measurable_toFun._autoParam →
autoParam (Measurable ⇑toEquiv.symm) MeasurableEquiv.measurable_invFun._autoParam → α ≃ᵐ βProd.fst : {α : Type u} → {β : Type v} → α × β → αThe first element of a pair.
Prod.mk : {α : Type u} → {β : Type v} → α → β → α × βConstructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`. Conventions for notations in identifiers: * The recommended spelling of `(a, b)` in identifiers is `mk`.
Inhabited.default : {α : Sort u} → [self : Inhabited α] → α`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
def prodUnique (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] [Unique β] :
α × β ≃ᵐ α where
toFun := Prod.fst
invFun a := (a, default)
left_inv _ := Prod.ext rfl (Unique.eq_default _).symm
right_inv _ := rfl
measurable_toFun := measurable_fst
measurable_invFun := measurable_id.prodMk measurable_constNew in v4.34.0-rc2-39-gb743f31 (2026-09-08), and its meaning has not changed since.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.
Audit surface: 0 project declarations, 11 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.