MeasurableEquiv.finSuccProd_symm_apply
Lemma
No docstring.
Types
-
X : Type u_2MeasurableSpace XA measurable space is a space equipped with a σ-algebra.
Given
-
n : ℕ -
p : (Fin n → X) × X
Then
(finSuccProd X n).symm p = Fin.snoc p.1 p.2MeasurableSpace : 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.
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`.
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.
Eq : {α : Sort u_1} → α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.MeasurableEquiv.finSuccProd : (X : Type u_2) → [inst : MeasurableSpace X] → (n : ℕ) → (Fin (n + 1) → X) ≃ᵐ (Fin n → X) × XMeasurable equivalence between `Fin (n + 1) → X` and `(Fin n → X) × X`.Go to its page
MeasurableEquiv.symm : {α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → α ≃ᵐ β → β ≃ᵐ αThe inverse of an equivalence between measurable spaces.
Fin.snoc : {n : ℕ} → {α : Fin (n + 1) → Sort u_1} → ((i : Fin n) → α i.castSucc) → α (Fin.last n) → (i : Fin (n + 1)) → α iAdding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order.
Code
lemma finSuccProd_symm_apply {X : Type*} [MeasurableSpace X] (n : ℕ)
(p : (Fin n → X) × X) :
(finSuccProd X n).symm p = Fin.snoc p.1 p.2Proof
by simp [finSuccProd] rfl
New 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
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.