Learning.IsDeterministicEnv.feedback_ae_eq
Lemma
No docstring.
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 ๐จMeasurableEq ๐จTypeclass for a measurable spaceฮฑfor which the diagonal ofฮฑ ร ฮฑis measurable. -
ฮฉ : Type u_4mฮฉ : MeasurableSpace ฮฉ
Given
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alg : Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm. -
env : Environment ๐ ๐ ๐จA stochastic environment.h_det : IsDeterministicEnv envAn environment is deterministic if its feedbacks are determined by measurable functions of the history, the observation and the action (and not possibly random kernels). -
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. -
O : โ โ ฮฉ โ ๐ -
A : โ โ ฮฉ โ ๐ -
Y : โ โ ฮฉ โ ๐จ -
n : โ
Assuming
Then
Y n =แต[P] fun ฯ => feedbackFun env n ((history O A Y n ฯ, O n ฯ), A n ฯ)f and g are *eventually equal* along a filter l if the set of x such that f x = g x belongs to l.MeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
MeasurableEq : (ฮฑ : Type u_1) โ [MeasurableSpace ฮฑ] โ PropTypeclass for a measurable space `ฮฑ` for which the diagonal of `ฮฑ ร ฮฑ` is measurable.
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
Learning.IsDeterministicEnv : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ {m๐ : MeasurableSpace ๐} โ {m๐จ : MeasurableSpace ๐จ} โ Learning.Environment ๐ ๐ ๐จ โ PropAn environment is deterministic if its feedbacks are determined by measurable functions of the history, the observation and the action (and not possibly random kernels).Go to its page
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
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.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
Filter.EventuallyEq : {ฮฑ : Type u_1} โ {ฮฒ : Type u_2} โ Filter ฮฑ โ (ฮฑ โ ฮฒ) โ (ฮฑ โ ฮฒ) โ PropTwo functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that `f x = g x` belongs to `l`.
Learning.feedbackFun : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
(env : Learning.Environment ๐ ๐ ๐จ) โ
[h_det : Learning.IsDeterministicEnv env] โ (n : โ) โ (Learning.Hist ๐ ๐ ๐จ n ร ๐) ร ๐ โ ๐จThe feedback function of a deterministic environment at step `n`.Go to its page
Prod.mk : {ฮฑ : Type u} โ {ฮฒ : Type v} โ ฮฑ โ ฮฒ โ ฮฑ ร ฮฒConstructs 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 : {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 feedback_ae_eq [MeasurableEq ๐จ] [h_det : IsDeterministicEnv env]
(h : IsAlgEnvSeq O A Y alg env P) (n : โ) :
Y n =แต[P] fun ฯ โฆ feedbackFun env n ((history O A Y n ฯ, O n ฯ), A n ฯ)Proof
by
have hO := h.measurable_obs
have hA := h.measurable_action
have hY := h.measurable_feedback
exact ae_eq_of_hasCondDistrib_deterministic (measurable_feedbackFun _ _) (by fun_prop)
(by fun_prop) (hasCondDistrib_feedback h n)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: 8 project declarations, 32 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.