MeasurableEquiv.finSuccProd
From the authors
Measurable equivalence between Fin (n + 1) → X and (Fin n → X) × X.
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X : Type u_2MeasurableSpace XA measurable space is a space equipped with a σ-algebra.
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n : ℕ
(Fin (n + 1) → X) ≃ᵐ (Fin n → X) × X(piFinSuccAbove (fun x => X) (Fin.last n)).trans prodCommMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
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.
Fin : ℕ → TypeNatural numbers less than some upper bound. In particular, a `Fin n` is a natural number `i` with the constraint that `i < n`. It is the canonical type with `n` elements.
HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ`a + b` computes the sum of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `+` in identifiers is `add`.
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.piFinSuccAbove : {n : ℕ} →
(α : Fin (n + 1) → Type u_8) →
[inst : (i : Fin (n + 1)) → MeasurableSpace (α i)] →
(i : Fin (n + 1)) → ((j : Fin (n + 1)) → α j) ≃ᵐ α i × ((j : Fin n) → α (i.succAbove j))Measurable equivalence between `Π j : Fin (n + 1), α j` and `α i × Π j : Fin n, α (Fin.succAbove i j)`. Measurable version of `Fin.insertNthEquiv`.
Fin.last : (n : ℕ) → Fin (n + 1)The greatest value of `Fin (n+1)`, namely `n`. Examples: * `Fin.last 4 = (4 : Fin 5)` * `(Fin.last 0).val = (0 : Nat)`
MeasurableEquiv.trans : {α : Type u_1} →
{β : Type u_2} →
{γ : Type u_3} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → [inst_2 : MeasurableSpace γ] → α ≃ᵐ β → β ≃ᵐ γ → α ≃ᵐ γThe composition of equivalences between measurable spaces.
MeasurableEquiv.prodComm : {α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → α × β ≃ᵐ β × αProducts of measurable spaces are symmetric.
Code
def finSuccProd (X : Type*) [MeasurableSpace X] (n : ℕ) :
(Fin (n + 1) → X) ≃ᵐ (Fin n → X) × X :=
(piFinSuccAbove (fun _ ↦ X) (Fin.last n)).trans prodCommNew 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, 16 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.