Learning.IsAlgEnvSeq.hasCondDistrib_obs_comapAction
Lemma
From the authors
The environment does not use the part of the action that it ignores: the conditional distribution of the observation given the transported history is its own observation kernel.
Types
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๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_4m๐ : MeasurableSpace ๐ -
๐' : Type u_5m๐' : MeasurableSpace ๐' -
๐จ : Type u_7m๐จ : MeasurableSpace ๐จ -
ฮฉ : Type u_10mฮฉ : MeasurableSpace ฮฉ
Given
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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.IsFiniteMeasure PA measureฮผis called finite ifฮผ univ < โ. -
Y : โ โ ฮฉ โ ๐จ -
env : Environment ๐ ๐ ๐จA stochastic environment. -
f : ๐' โ ๐ -
O : โ โ ฮฉ โ ๐ -
A' : โ โ ฮฉ โ ๐' -
alg : Algorithm ๐ ๐' ๐จA stochastic, sequential algorithm. -
n : โ
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 (env.comapAction f hf) PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
Then
ProbabilityTheory.HasCondDistrib (O n) (history O (fun n ฯ => f (A' n ฯ)) Y n) (env.obs n) PY given X under the measure P is equal to the kernel ฮบ.MeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
MeasureTheory.IsFiniteMeasure : {ฮฑ : Type u_1} โ {m0 : MeasurableSpace ฮฑ} โ MeasureTheory.Measure ฮฑ โ PropA measure `ฮผ` is called finite if `ฮผ univ < โ`.
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.
Learning.Environment : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ 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
Learning.Algorithm : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `๐`, then takes an action in `๐`, and finally receives feedback in `๐จ`. The action is a random function of the past rounds and the current observation.Go to its page
Measurable : {ฮฑ : Type u_1} โ {ฮฒ : Type u_2} โ [MeasurableSpace ฮฑ] โ [MeasurableSpace ฮฒ] โ (ฮฑ โ ฮฒ) โ PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
Learning.IsAlgEnvSeq : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{ฮฉ : Type u_4} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
{mฮฉ : MeasurableSpace ฮฉ} โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐จ) โ
Learning.Algorithm ๐ ๐ ๐จ โโฆAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.Go to its page
Learning.Environment.comapAction : {๐ : Type u_1} โ
{๐ : Type u_4} โ
{๐' : Type u_5} โ
{๐จ : Type u_7} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐' : MeasurableSpace ๐'} โ
{m๐จ : MeasurableSpace ๐จ} โ
Learning.Environment ๐ ๐ ๐จ โ
(f : ๐' โ ๐) โ
autoParam (Measurable f) Learning.Environment.comapAction._auto_1 โโฆThe environment that reads `f a` when the algorithm plays `a`, both in the current round and in the past rounds.Go to its page
ProbabilityTheory.HasCondDistrib : {ฮฉ : Type u_1} โ
{๐ง : Type u_2} โ
{๐จ : Type u_3} โ
{mฮฉ : MeasurableSpace ฮฉ} โ
{m๐ง : MeasurableSpace ๐ง} โ
{m๐จ : MeasurableSpace ๐จ} โ (ฮฉ โ ๐จ) โ (ฮฉ โ ๐ง) โ ProbabilityTheory.Kernel ๐ง ๐จ โ MeasureTheory.Measure ฮฉ โ PropPredicate stating that the conditional distribution of `Y` given `X` under the measure `P` is equal to the kernel `ฮบ`.
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
Code
lemma hasCondDistrib_obs_comapAction {alg : Algorithm ๐ ๐' ๐จ}
(h : IsAlgEnvSeq O A' Y alg (env.comapAction f hf) P) (n : โ) :
HasCondDistrib (O n) (history O (fun n ฯ โฆ f (A' n ฯ)) Y n) (env.obs n) PProof
HasCondDistrib.comp_right (f := Hist.mapAction (๐ := ๐) (๐จ := ๐จ) f (n := n))
(hf := by fun_prop) (h.hasCondDistrib_obs 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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 14 project declarations, 27 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.