`Learning.IsAlgEnvSeq.hasLaw_feedback_zero_cond`🔗

Lemma

From the authors

Conditionally on the event (O 0, A 0) = p, the first feedback has law env.ν0 p.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.MeasurableSingletonClass 𝓞A typeclass mixin for MeasurableSpaces such that each singleton is measurable.
𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐MeasurableSingletonClass 𝓐
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

O : ℕ → Ω → 𝓞
A : ℕ → Ω → 𝓐
Y : ℕ → Ω → 𝓨
alg : Algorithm 𝓞 𝓐 𝓨A stochastic, sequential algorithm.
env : Environment 𝓞 𝓐 𝓨A stochastic environment.
P : MeasureTheory.Measure ΩA 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 measure μ is called finite if μ univ < ∞.
p : 𝓞 × 𝓐

Assuming

h : IsAlgEnvSeq O A Y alg env PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
hP : P ((fun ω => (O 0 ω, A 0 ω)) ⁻¹' {p}) ≠ 0

Then

ProbabilityTheory.HasLaw (Y 0) (env.ν0 p) P[|(fun ω => (O 0 ω, A 0 ω)) ⁻¹' {p}]

The predicate HasLaw X μ P registers the fact that the random variable X has law μ under the measure P, in other words that P.map X = μ.

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

MeasurableSingletonClass : (α : Type u_6) → [MeasurableSpace α] → Prop

A typeclass mixin for `MeasurableSpace`s such that each singleton is measurable.

Nat : Type

The 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.Algorithm : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)

A stochastic, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation.

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Learning.Environment : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → 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.

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MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called finite if `μ univ < ∞`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

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.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Prod : Type u → Type v → Type (max u v)

The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`.

Learning.IsAlgEnvSeq : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated
by an algorithm interacting with an environment.

Go to its page

Ne : {α : Sort u} → α → α → Prop

`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`.

Prod.mk : {α : Type u} → {β : Type v} → α → β → α × β

Constructs 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`.

Set.preimage : {α : Type u} → {β : Type v} → (α → β) → Set β → Set α

The preimage of `s : Set β` by `f : α → β`, written `f ⁻¹' s`,
is the set of `x : α` such that `f x ∈ s`.

Singleton.singleton : {α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β

`singleton x` is a collection with the single element `x` (notation: `{x}`). 

Conventions for notations in identifiers:

 * The recommended spelling of `{x}` in identifiers is `singleton`.

ProbabilityTheory.HasLaw : {Ω : Type u_1} →
  {𝓧 : Type u_2} →
    {mΩ : MeasurableSpace Ω} →
      {m𝓧 : MeasurableSpace 𝓧} →
        (Ω → 𝓧) → MeasureTheory.Measure 𝓧 → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasLaw._auto_1 → Prop

The predicate `HasLaw X μ P` registers the fact that the random variable `X` has law `μ` under
the measure `P`, in other words that `P.map X = μ`. 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.Environment.ν0 : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {m𝓞 : MeasurableSpace 𝓞} →
        {m𝓐 : MeasurableSpace 𝓐} →
          {m𝓨 : MeasurableSpace 𝓨} → Learning.Environment 𝓞 𝓐 𝓨 → ProbabilityTheory.Kernel (𝓞 × 𝓐) 𝓨

Distribution of the first feedback given the first observation and action: the feedback kernel
at time `0` applied to the empty history.

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ProbabilityTheory.cond : {Ω : Type u_1} → {m : MeasurableSpace Ω} → MeasureTheory.Measure Ω → Set Ω → MeasureTheory.Measure Ω

The conditional probability measure of measure `μ` on set `s` is `μ` restricted to `s`
and scaled by the inverse of `μ s` (to make it a probability measure):
`(μ s)⁻¹ • μ.restrict s`.

Code

lemma IsAlgEnvSeq.hasLaw_feedback_zero_cond [MeasurableSingletonClass 𝓞]
    [MeasurableSingletonClass 𝓐] (h : IsAlgEnvSeq O A Y alg env P) {p : 𝓞 × 𝓐}
    (hP : P ((fun ω ↦ (O 0 ω, A 0 ω)) ⁻¹' {p}) ≠ 0) :
    HasLaw (Y 0) (env.ν0 p) P[|(fun ω ↦ (O 0 ω, A 0 ω)) ⁻¹' {p}]

Proof

h.hasCondDistrib_feedback_zero.hasLaw_cond (h.measurable_feedback 0)
    (measurableSet_singleton p) (fun a ha ↦ by rw [Set.mem_singleton_iff.1 ha]) hP

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 3th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 7 project declarations, 45 external constants

✓ 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.

Learning.IsAlgEnvSeq.hasLaw_feedback_zero_cond🔗

`Learning.IsAlgEnvSeq.hasLaw_feedback_zero_cond`🔗