MeasurableEquiv.IicSuccProd
From the authors
Measurable equivalence between a product up to n + 1 and the pair of the product up to n and
the space at n + 1.
-
X : β β Type u_2(n : β) β MeasurableSpace (X n)A measurable space is a space equipped with a Ο-algebra. -
n : β
((i : β₯(Finset.Iic (n + 1))) β X βi) βα΅ ((i : β₯(Finset.Iic n)) β X βi) Γ X (n + 1)(IicProdIoc β―).symm.trans ((refl ((i : β₯(Finset.Iic n)) β X βi)).prodCongr (piSingleton n).symm)MeasurableSpace : Type u_6 β Type u_6A measurable space is a space equipped with a Ο-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 : (Ξ± : Type u_6) β (Ξ² : Type u_7) β [MeasurableSpace Ξ±] β [MeasurableSpace Ξ²] β Type (max u_6 u_7)Equivalences between measurable spaces. Main application is the simplification of measurability statements along measurable equivalences.
Finset.Iic : {Ξ± : Type u_1} β [inst : Preorder Ξ±] β [LocallyFiniteOrderBot Ξ±] β Ξ± β Finset Ξ±The finset $(-β, b]$ of elements `x` such that `x β€ b`. Basically `Set.Iic b` as a finset.
HAdd.hAdd : {Ξ± : Type u} β {Ξ² : Type v} β {Ξ³ : outParam (Type w)} β [self : HAdd Ξ± Ξ² Ξ³] β Ξ± β Ξ² β Ξ³`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 β Type v β Type (max u v)The product type, usually written `Ξ± Γ Ξ²`. Product types are also called pair or tuple types. Elements of this type are pairs in which the first element is an `Ξ±` and the second element is a `Ξ²`. Products nest to the right, so `(x, y, z) : Ξ± Γ Ξ² Γ Ξ³` is equivalent to `(x, (y, z)) : Ξ± Γ (Ξ² Γ Ξ³)`. Conventions for notations in identifiers: * The recommended spelling of `Γ` in identifiers is `Prod`.
MeasurableEquiv.IicProdIoc : {ΞΉ : Type u_1} β
[inst : LinearOrder ΞΉ] β
[inst_1 : LocallyFiniteOrder ΞΉ] β
[DecidableLE ΞΉ] β
{X : ΞΉ β Type u_2} β
[inst_3 : LocallyFiniteOrderBot ΞΉ] β
[inst_4 : (i : ΞΉ) β MeasurableSpace (X i)] β
{a b : ΞΉ} β
a β€ b β
((i : β₯(Finset.Iic a)) β X βi) Γ ((i : β₯(Finset.Ioc a b)) β X βi) βα΅ ((i : β₯(Finset.Iic b)) ββ¦Gluing `Iic a` and `Ioc a b` into `Iic b`. This version requires `a β€ b` to get a measurable equivalence.
MeasurableEquiv.symm : {Ξ± : Type u_1} β {Ξ² : Type u_2} β [inst : MeasurableSpace Ξ±] β [inst_1 : MeasurableSpace Ξ²] β Ξ± βα΅ Ξ² β Ξ² βα΅ Ξ±The inverse of an equivalence between measurable spaces.
MeasurableEquiv.trans : {Ξ± : Type u_1} β
{Ξ² : Type u_2} β
{Ξ³ : Type u_3} β
[inst : MeasurableSpace Ξ±] β
[inst_1 : MeasurableSpace Ξ²] β [inst_2 : MeasurableSpace Ξ³] β Ξ± βα΅ Ξ² β Ξ² βα΅ Ξ³ β Ξ± βα΅ Ξ³The composition of equivalences between measurable spaces.
MeasurableEquiv.refl : (Ξ± : Type u_6) β [inst : MeasurableSpace Ξ±] β Ξ± βα΅ Ξ±Any measurable space is equivalent to itself.
MeasurableEquiv.prodCongr : {Ξ± : Type u_1} β
{Ξ² : Type u_2} β
{Ξ³ : Type u_3} β
{Ξ΄ : Type u_4} β
[inst : MeasurableSpace Ξ±] β
[inst_1 : MeasurableSpace Ξ²] β
[inst_2 : MeasurableSpace Ξ³] β [inst_3 : MeasurableSpace Ξ΄] β Ξ± βα΅ Ξ² β Ξ³ βα΅ Ξ΄ β Ξ± Γ Ξ³ βα΅ Ξ² Γ Ξ΄Products of equivalent measurable spaces are equivalent.
MeasurableEquiv.piSingleton : {X : β β Type u_1} β
[inst : (n : β) β MeasurableSpace (X n)] β (a : β) β X (a + 1) βα΅ ((i : β₯(Finset.Ioc a (a + 1))) β X βi)Identifying `{a + 1}` with `Ioc a (a + 1)`, as a measurable equiv on dependent functions.Code
def IicSuccProd (X : β β Type*) [β n, MeasurableSpace (X n)] (n : β) :
MeasurableEquiv (Ξ i : Iic (n + 1), X i) ((Ξ i : Iic n, X i) Γ X (n + 1)) :=
(IicProdIoc (Nat.le_succ n)).symm.trans
(prodCongr (refl _) (piSingleton n).symm)
Meaning unchanged since v4.33.0-rc1-29-gce231eb, the oldest
revision on record (2026-07-30). Its file was edited 2026-08-26 (ce8eda9, βhelper lemmasβ) without changing what it means.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file Β· 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 β so the graph is this declaration alone. That is the answer, not a missing picture.
Audit surface: 0 project declarations, 37 external constants
β 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.