Learning.hasLaw_history_zero
Lemma
From the authors
The history before time 0 is a constant.
Types
-
๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_2m๐ : MeasurableSpace ๐ -
๐จ : Type u_3m๐จ : MeasurableSpace ๐จ -
ฮฉ : Type u_4mฮฉ : MeasurableSpace ฮฉ
Given
-
P : MeasureTheory.Measure ฮฉA 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 measureฮผis called a probability measure ifฮผ univ = 1. -
O : โ โ ฮฉ โ ๐ -
A : โ โ ฮฉ โ ๐ -
Y : โ โ ฮฉ โ ๐จ
Then
ProbabilityTheory.HasLaw (history O A Y 0) (MeasureTheory.Measure.dirac default) PHasLaw X ฮผ P registers the fact that the random variable X has law ฮผ under the measure P, in other words that P.map X = ฮผ.MeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
MeasureTheory.IsProbabilityMeasure : {ฮฑ : Type u_1} โ {m0 : MeasurableSpace ฮฑ} โ MeasureTheory.Measure ฮฑ โ PropA measure `ฮผ` is called a probability measure if `ฮผ univ = 1`.
MeasureTheory.Measure : (ฮฑ : Type u_5) โ [MeasurableSpace ฮฑ] โ Type u_5A 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 `ฮผ s`, is an extended nonnegative real. The real-valued version is written `ฮผ.real s`.
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.HasLaw : {ฮฉ : Type u_1} โ
{๐ง : Type u_2} โ
{mฮฉ : MeasurableSpace ฮฉ} โ
{m๐ง : MeasurableSpace ๐ง} โ
(ฮฉ โ ๐ง) โ MeasureTheory.Measure ๐ง โ autoParam (MeasureTheory.Measure ฮฉ) ProbabilityTheory.HasLaw._auto_1 โ PropThe predicate `HasLaw X ฮผ P` registers the fact that the random variable `X` has law `ฮผ` under the measure `P`, in other words that `P.map X = ฮผ`. We also require `X` to be `AEMeasurable`, to allow for nice interactions with operations on the codomain of `X`. See for instance `HasLaw.comp`, `IndepFun.hasLaw_mul` and `IndepFun.hasLaw_add`.
Learning.history : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ {ฮฉ : Type u_4} โ (โ โ ฮฉ โ ๐) โ (โ โ ฮฉ โ ๐) โ (โ โ ฮฉ โ ๐จ) โ (n : โ) โ ฮฉ โ Learning.Hist ๐ ๐ ๐จ nHistory of the algorithm-environment sequence before time `n`: the rounds at times `0, ..., n - 1`.Go to its page
MeasureTheory.Measure.dirac : {ฮฑ : Type u_1} โ [inst : MeasurableSpace ฮฑ] โ ฮฑ โ MeasureTheory.Measure ฮฑThe dirac measure.
Inhabited.default : {ฮฑ : Sort u} โ [self : Inhabited ฮฑ] โ ฮฑ`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
lemma hasLaw_history_zero (O : โ โ ฮฉ โ ๐) (A : โ โ ฮฉ โ ๐) (Y : โ โ ฮฉ โ ๐จ) :
HasLaw (history O A Y 0) (Measure.dirac default) P where
aemeasurableProof
by rw [history_zero]; exact aemeasurable_const map_eq := by rw [history_zero, Measure.map_const, measure_univ, one_smul]
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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 3 project declarations, 18 external constants
โ 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.