MeasurableEquiv.finSuccPiIic🔗

((i : Fin (n + 1)) → X ↑i) ≃ᵐ ((i : ↥(Finset.Iic n)) → X ↑i)

{ toFun := fun h i => h ⟨↑i, ⋯⟩, invFun := fun h i => h ⟨↑i, ⋯⟩, left_inv := ⋯, right_inv := ⋯, measurable_toFun := ⋯,
  measurable_invFun := ⋯ }

A measurable space is a space equipped with a σ-algebra.

The 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.

Equivalences between measurable spaces. Main application is the simplification of measurability
statements along measurable equivalences.

Natural 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.

`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`.

The finset $(-∞, b]$ of elements `x` such that `x ≤ b`. Basically `Set.Iic b` as a finset.

Creates a `Fin n` from `i : Nat` and a proof that `i < n`.

def finSuccPiIic (X : ℕ → Type*) [∀ n, MeasurableSpace (X n)] (n : ℕ) :
    (Π i : Fin (n + 1), X i) ≃ᵐ (Π i : Iic n, X i) where
  toFun h i := h ⟨i.1, Nat.lt_succ_of_le (mem_Iic.mp i.2)⟩
  invFun h i := h ⟨i.1, mem_Iic.mpr (Nat.le_of_lt_succ i.2)⟩
  left_inv _ := rfl
  right_inv _ := rfl
  measurable_toFun := .of_eval fun _ ↦ measurable_pi_apply _
  measurable_invFun := .of_eval fun _ ↦ measurable_pi_apply _

New in v4.34.0-rc2-39-gb743f31 (2026-09-08), and its meaning has not changed since. Its file was edited 2026-09-10 (357e9dd, “Update Mathlib”) without changing what it means.

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, 26 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.