ProbabilityTheory.Kernel.iicOfFin
From the authors
Kernels indexed by Iic n (as needed for Kernel.traj), obtained from kernels indexed by
Fin n: the kernel κ' (n + 1) on Π i : Fin (n + 1), X i is seen as a kernel on
Π i : Iic n, X i.
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X : ℕ → Type u_2(n : ℕ) → MeasurableSpace (X n)A measurable space is a space equipped with a σ-algebra. -
κ' : (n : ℕ) → Kernel ((i : Fin n) → X ↑i) (X n)A kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β. -
n : ℕ
Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))(κ' (n + 1)).comap ⇑(MeasurableEquiv.finSuccPiIic X n).symm ⋯MeasurableSpace : 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.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
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.
Finset.Iic : {α : Type u_1} → [inst : Preorder α] → [LocallyFiniteOrderBot α] → α → Finset αThe finset $(-∞, b]$ of elements `x` such that `x ≤ b`. Basically `Set.Iic b` as a finset.
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`.
ProbabilityTheory.Kernel.comap : {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} →
{mβ : MeasurableSpace β} →
{γ : Type u_4} →
{mγ : MeasurableSpace γ} →
ProbabilityTheory.Kernel α β → (g : γ → α) → Measurable g → ProbabilityTheory.Kernel γ βPullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`. We include measurability in the assumptions instead of using junk values to make sure that typeclass inference can infer that the `comap` of a Markov kernel is again a Markov kernel.
MeasurableEquiv.finSuccPiIic : (X : ℕ → Type u_2) →
[inst : (n : ℕ) → MeasurableSpace (X n)] → (n : ℕ) → ((i : Fin (n + 1)) → X ↑i) ≃ᵐ ((i : ↥(Finset.Iic n)) → X ↑i)Measurable equivalence between `Π i : Fin (n + 1), X i` and `Π i : Iic n, X i`.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.
Code
noncomputable
def iicOfFin (κ' : (n : ℕ) → Kernel (Π i : Fin n, X i) (X n)) (n : ℕ) :
Kernel (Π i : Iic n, X i) (X (n + 1)) :=
(κ' (n + 1)).comap (MeasurableEquiv.finSuccPiIic X n).symm (by fun_prop)Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).
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, 32 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.