Learning.IsAlgEnvSeq.identDistrib_trajectory
Theorem
The law of the sequence of actions and observations generated by an algorithm-environment pair is unique: it does not depend on the probability space used.
Types
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π : Type u_1mπ : MeasurableSpace πA measurable space is a space equipped with a Ο-algebra. -
π¨ : Type u_2mπ¨ : MeasurableSpace π¨ -
Ξ© : Type u_4mΞ© : MeasurableSpace Ξ© -
Ξ©' : Type u_5mΞ©' : MeasurableSpace Ξ©'
Given
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alg : Algorithm π π¨A stochastic, sequential algorithm. -
env : Environment π π¨A stochastic environment. -
P : MeasureTheory.Measure Ξ©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.MeasureTheory.IsProbabilityMeasure PA measureΞΌis called a probability measure ifΞΌ univ = 1. -
P' : MeasureTheory.Measure Ξ©'MeasureTheory.IsProbabilityMeasure P' -
Aβ : β β Ξ© β π -
Rβ : β β Ξ© β π¨ -
Aβ : β β Ξ©' β π -
Rβ : β β Ξ©' β π¨
Assuming
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h1 : IsAlgEnvSeq Aβ Rβ alg env PAn algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment. -
h2 : IsAlgEnvSeq Aβ Rβ alg env P'
Then
ProbabilityTheory.IdentDistrib (trajectory Aβ Rβ) (trajectory Aβ Rβ) P P'MeasurableSpace : Type u_6 β Type u_6A measurable space is a space equipped with a Ο-algebra.
Learning.Algorithm : (π : Type u_4) β (π¨ : Type u_5) β [MeasurableSpace π] β [MeasurableSpace π¨] β Type (max u_4 u_5)A stochastic, sequential algorithm.Go to its page
Learning.Environment : (π : Type u_4) β (π¨ : Type u_5) β [MeasurableSpace π] β [MeasurableSpace π¨] β Type (max u_4 u_5)A stochastic environment.Go to its page
MeasureTheory.IsProbabilityMeasure : {Ξ± : Type u_1} β {m0 : MeasurableSpace Ξ±} β MeasureTheory.Measure Ξ± β PropA measure `ΞΌ` is called a probability measure if `ΞΌ univ = 1`.
MeasureTheory.Measure : (Ξ± : Type u_5) β [MeasurableSpace Ξ±] β 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 `ΞΌ s`, is an extended nonnegative real. The real-valued version is written `ΞΌ.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.
Learning.IsAlgEnvSeq : {π : Type u_1} β
{π¨ : Type u_2} β
{Ξ© : Type u_3} β
{mπ : MeasurableSpace π} β
{mπ¨ : MeasurableSpace π¨} β
{mΞ© : MeasurableSpace Ξ©} β
(β β Ξ© β π) β
(β β Ξ© β π¨) β
Learning.Algorithm π π¨ β
Learning.Environment π π¨ β (P : MeasureTheory.Measure Ξ©) β [MeasureTheory.IsFiniteMeasure P] β PropAn algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment.Go to its page
ProbabilityTheory.IdentDistrib : {Ξ± : Type u_1} β
{Ξ² : Type u_2} β
{Ξ³ : Type u_3} β
[inst : MeasurableSpace Ξ±] β
[inst_1 : MeasurableSpace Ξ²] β
[MeasurableSpace Ξ³] β
(Ξ± β Ξ³) β
(Ξ² β Ξ³) β
autoParam (MeasureTheory.Measure Ξ±) ProbabilityTheory.IdentDistrib._auto_1 β
autoParam (MeasureTheory.Measure Ξ²) ProbabilityTheory.IdentDistrib._auto_3 β Proβ¦Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition.
Learning.trajectory : {π : Type u_1} β {π¨ : Type u_2} β {Ξ© : Type u_3} β (β β Ξ© β π) β (β β Ξ© β π¨) β Ξ© β β β π Γ π¨A random variable that gives the sequence of action-feedback pairs.Go to its page
Code
theorem IsAlgEnvSeq.identDistrib_trajectory (h1 : IsAlgEnvSeq Aβ Rβ alg env P)
(h2 : IsAlgEnvSeq Aβ Rβ alg env P') :
IdentDistrib (trajectory Aβ Rβ) (trajectory Aβ Rβ) P P' where
aemeasurable_fstProof
h1.measurable_trajectory.aemeasurable aemeasurable_snd := h2.measurable_trajectory.aemeasurable map_eq := isAlgEnvSeq_unique h1 h2
Actions: Source Β· Open Issue
Meaning last changed in v4.34.0-rc2-39-gb743f31 (2026-09-08), the 5th recorded change.
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
Audit surface: 5 project declarations, 27 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.