Learning.IsAlgEnvSeq.hasCondDistrib_action_comapObs
Lemma
From the authors
The algorithm does not use the part of the observation that it ignores: the conditional distribution of its action given the transported history and observation is its own policy.
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' -
π : Type u_4mπ : MeasurableSpace π -
π¨ : Type u_7mπ¨ : MeasurableSpace π¨ -
Ξ© : Type u_10mΞ© : MeasurableSpace Ξ©
Given
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alg : Algorithm π π π¨A stochastic, sequential algorithm. -
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.IsFiniteMeasure PA measureΞΌis called finite ifΞΌ univ < β. -
A : β β Ξ© β π -
Y : β β Ξ© β π¨ -
env : Environment π' π π¨A stochastic environment. -
f : π' β π -
O : β β Ξ© β π' -
n : β
Assuming
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hf : Measurable fimplicitA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable. -
h : IsAlgEnvSeq O A Y (alg.comapObs f hf) env PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
Then
ProbabilityTheory.HasCondDistrib (A n) (fun Ο => (history (fun n Ο => f (O n Ο)) A Y n Ο, f (O n Ο))) (alg.policy n) PY given X under the measure P is equal to the kernel ΞΊ.MeasurableSpace : Type u_6 β Type u_6A measurable space is a space equipped with a Ο-algebra.
Learning.Algorithm : (π : Type u_5) β
(π : Type u_6) β
(π¨ : Type u_7) β [MeasurableSpace π] β [MeasurableSpace π] β [MeasurableSpace π¨] β Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `π`, then takes an action in `π`, and finally receives feedback in `π¨`. The action is a random function of the past rounds and the current observation.Go to its page
MeasureTheory.IsFiniteMeasure : {Ξ± : Type u_1} β {m0 : MeasurableSpace Ξ±} β MeasureTheory.Measure Ξ± β PropA measure `ΞΌ` is called finite if `ΞΌ univ < β`.
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.Environment : (π : Type u_5) β
(π : Type u_6) β
(π¨ : Type u_7) β [MeasurableSpace π] β [MeasurableSpace π] β [MeasurableSpace π¨] β Type (max (max u_5 u_6) u_7)A stochastic environment. At each round, an observation is drawn prior to the algorithm taking an action. Then the environment provides feedback based on the observation and the action.Go to its page
Measurable : {Ξ± : Type u_1} β {Ξ² : Type u_2} β [MeasurableSpace Ξ±] β [MeasurableSpace Ξ²] β (Ξ± β Ξ²) β PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
Learning.IsAlgEnvSeq : {π : Type u_1} β
{π : Type u_2} β
{π¨ : Type u_3} β
{Ξ© : Type u_4} β
{mπ : MeasurableSpace π} 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
(β β Ξ© β π¨) β
Learning.Algorithm π π π¨ ββ¦An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.Go to its page
Learning.Algorithm.comapObs : {π : Type u_1} β
{π' : Type u_2} β
{π : Type u_4} β
{π¨ : Type u_7} β
{mπ : MeasurableSpace π} β
{mπ' : MeasurableSpace π'} β
{mπ : MeasurableSpace π} β
{mπ¨ : MeasurableSpace π¨} β
Learning.Algorithm π π π¨ β
(f : π' β π) β
autoParam (Measurable f) Learning.Algorithm.comapObs._auto_1 β Learniβ¦The algorithm that sees `f o` when the observation is `o`, both in the current round and in the past rounds.Go to its page
ProbabilityTheory.HasCondDistrib : {Ξ© : Type u_1} β
{π§ : Type u_2} β
{π¨ : Type u_3} β
{mΞ© : MeasurableSpace Ξ©} β
{mπ§ : MeasurableSpace π§} β
{mπ¨ : MeasurableSpace π¨} β (Ξ© β π¨) β (Ξ© β π§) β ProbabilityTheory.Kernel π§ π¨ β MeasureTheory.Measure Ξ© β PropPredicate stating that the conditional distribution of `Y` given `X` under the measure `P` is equal to the kernel `ΞΊ`.
Prod.mk : {Ξ± : Type u} β {Ξ² : Type v} β Ξ± β Ξ² β Ξ± Γ Ξ²Constructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`. Conventions for notations in identifiers: * The recommended spelling of `(a, b)` in identifiers is `mk`.
Learning.history : {π : Type u_1} β
{π : Type u_2} β
{π¨ : Type u_3} β {Ξ© : Type u_4} β (β β Ξ© β π) β (β β Ξ© β π) β (β β Ξ© β π¨) β (n : β) β Ξ© β Learning.Hist π π π¨ nHistory of the algorithm-environment sequence before time `n`: the rounds at times `0, ..., n - 1`.Go to its page
Code
lemma hasCondDistrib_action_comapObs (h : IsAlgEnvSeq O A Y (alg.comapObs f hf) env P) (n : β) :
HasCondDistrib (A n)
(fun Ο β¦ (history (fun n Ο β¦ f (O n Ο)) A Y n Ο, f (O n Ο))) (alg.policy n) PProof
HasCondDistrib.comp_right (f := fun p : Hist π' π π¨ n Γ π' β¦ (Hist.mapObs f p.1, f p.2))
(hf := by fun_prop) (h.hasCondDistrib_action n)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 Β· open it in the Lean web editor.
Dependency graph
Audit surface: 14 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.