`Learning.klDiv_map_trajectory_eq_iSup`🔗

Lemma

From the authors

The divergence between the laws of two trajectories is the supremum over n of the divergences between the laws of the histories up to time n.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.
𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Ω : Type u_4mΩ : MeasurableSpace Ω
Ω' : Type u_5mΩ' : MeasurableSpace Ω'

Given

P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsProbabilityMeasure PA measure μ is called a probability measure if μ univ = 1.
P' : MeasureTheory.Measure Ω'MeasureTheory.IsProbabilityMeasure P'
O : ℕ → Ω → 𝓞
A : ℕ → Ω → 𝓐
Y : ℕ → Ω → 𝓨
O' : ℕ → Ω' → 𝓞
A' : ℕ → Ω' → 𝓐
Y' : ℕ → Ω' → 𝓨

Assuming

hO : ∀ (n : ℕ), Measurable (O n)A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.
hA : ∀ (n : ℕ), Measurable (A n)
hY : ∀ (n : ℕ), Measurable (Y n)
hO' : ∀ (n : ℕ), Measurable (O' n)
hA' : ∀ (n : ℕ), Measurable (A' n)
hY' : ∀ (n : ℕ), Measurable (Y' n)

Then

InformationTheory.klDiv (MeasureTheory.Measure.map (trajectory O A Y) P)
    (MeasureTheory.Measure.map (trajectory O' A' Y') P') =
  ⨆ n,
    InformationTheory.klDiv (MeasureTheory.Measure.map (history O A Y n) P)
      (MeasureTheory.Measure.map (history O' A' Y' n) P')

MeasurableSpace : Type u_6 → Type u_6

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

MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called a probability measure if `μ univ = 1`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Nat : Type

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.

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

Eq : {α : Sort u_1} → α → α → Prop

The 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`.

InformationTheory.klDiv : {α : Type u_2} → {mα : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → ENNReal

Kullback-Leibler divergence between two measures.

MeasureTheory.Measure.map : {α : Type u_4} →
  {β : Type u_5} →
    [inst : MeasurableSpace α] →
      [inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure β

The pushforward of a measure. If `f` is not an almost everywhere measurable function,
we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way
we always have `map f 0 = 0`, and the push-forward of a probability measure is always a
probability measure.

Learning.trajectory : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → Ω → ℕ → Learning.Round 𝓞 𝓐 𝓨

A random variable that gives the sequence of rounds.

Go to its page

Learning.history : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → (n : ℕ) → Ω → Learning.Hist 𝓞 𝓐 𝓨 n

History of the algorithm-environment sequence before time `n`: the rounds at
times `0, ..., n - 1`.

Go to its page

Code

lemma klDiv_map_trajectory_eq_iSup (hO : ∀ n, Measurable (O n)) (hA : ∀ n, Measurable (A n))
    (hY : ∀ n, Measurable (Y n)) (hO' : ∀ n, Measurable (O' n)) (hA' : ∀ n, Measurable (A' n))
    (hY' : ∀ n, Measurable (Y' n)) :
    klDiv (P.map (trajectory O A Y)) (P'.map (trajectory O' A' Y')) =
      ⨆ n, klDiv (P.map (history O A Y n)) (P'.map (history O' A' Y' n))

Proof

by
  have hg : ∀ n, Measurable fun f : ℕ → Round 𝓞 𝓐 𝓨 ↦ fun i : Fin n ↦ f i.1 := fun n ↦
    .of_eval fun i ↦ measurable_pi_apply i.1
  rw [klDiv_eq_iSup_map hg ?_ MeasurableSpace.iSup_comap_restrictFin]
  · refine iSup_congr fun n ↦ ?_
    rw [Measure.map_map (hg n) (measurable_trajectory hO hA hY),
      Measure.map_map (hg n) (measurable_trajectory hO' hA' hY')]
    rfl
  · intro n m hnm
    have : (fun f : ℕ → Round 𝓞 𝓐 𝓨 ↦ fun i : Fin n ↦ f i.1) =
        (fun h : Fin m → Round 𝓞 𝓐 𝓨 ↦ fun i : Fin n ↦ h (Fin.castLE hnm i)) ∘
          fun f : ℕ → Round 𝓞 𝓐 𝓨 ↦ fun i : Fin m ↦ f i.1 := rfl
    beta_reduce
    rw [this, ← MeasurableSpace.comap_comp]
    exact MeasurableSpace.comap_mono (Measurable.of_eval fun i ↦
      measurable_pi_apply (Fin.castLE hnm i)).comap_le

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 2th recorded change.

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: 4 project declarations, 23 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.

Learning.klDiv_map_trajectory_eq_iSup🔗

`Learning.klDiv_map_trajectory_eq_iSup`🔗