`Bandits.hasLaw_rewardByCountUntil_zero`🔗

Lemma

From the authors

The law of the array rewardByCountUntil A R 0, which is a sub-array of the auxiliary array, is the product measure ⨂ (a, m), ν a.

Types

𝓐 : Type u_1m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.DecidableEq 𝓐
Ω : Type u_2mΩ : MeasurableSpace Ω

Given

A : ℕ → Ω → 𝓐
R : ℕ → Ω → ℝ
ν : ProbabilityTheory.Kernel 𝓐 ℝA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
μ : 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.IsProbabilityMeasure μA measure μ is called a probability measure if μ univ = 1.

Then

ProbabilityTheory.HasLaw (Learning.rewardByCountUntil A R 0) (MeasureTheory.Measure.infinitePi fun p => ν p.1)
  (μ.prod (streamMeasure ν))

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.

DecidableEq : Sort u → Sort (max 1 u)

Propositional equality is `Decidable` for all elements of a type.

In other words, an instance of `DecidableEq α` is a means of deciding the proposition `a = b` is
for all `a b : α`.

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.

Real : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is a Markov kernel if every measure in its image is a probability measure.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

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

A measure `μ` is called a probability measure if `μ univ = 1`.

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`.

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.rewardByCountUntil : {𝓐 : Type u_2} →
  {R : Type u_3} → {Ω : Type u_4} → [DecidableEq 𝓐] → (ℕ → Ω → 𝓐) → (ℕ → Ω → R) → ℕ → Ω × (ℕ → 𝓐 → R) → 𝓐 × ℕ → R

Array of rewards by count, truncated at time `t`: the entry `(a, m)` is the reward received at
the `(m + 1)`-th pull of action `a` if that pull happened before time `t`, and the entry
`(m + 1, a)` of the auxiliary array `ω.2` otherwise.

This is an auxiliary definition used to prove results about the distribution of `rewardByCount`.

It is defined recursively: at time `t`, the
entry `(A t, pullCount A (A t) t)` is replaced by the reward `R' t`.
See `rewardByCountUntil_apply_of_lt_pullCount` and `rewardByCountUntil_apply_of_pullCount_le`.

Go to its page

MeasureTheory.Measure.infinitePi : {ι : Type u_1} →
  {X : ι → Type u_2} →
    {mX : (i : ι) → MeasurableSpace (X i)} →
      ((i : ι) → MeasureTheory.Measure (X i)) → MeasureTheory.Measure ((i : ι) → X i)

The product measure of an arbitrary family of probability measures. It is defined as the unique
extension of the function which gives to cylinders the measure given by the associated product
measure.

It is defined via an `if ... then ... else` so that it can be manipulated without carrying
a proof that the measures are probability measures.

MeasureTheory.Measure.prod : {α : Type u_4} →
  {β : Type u_5} →
    [inst : MeasurableSpace α] →
      [inst_1 : MeasurableSpace β] → MeasureTheory.Measure α → MeasureTheory.Measure β → MeasureTheory.Measure (α × β)

The binary product of measures. They are defined for arbitrary measures, but we basically
prove all properties under the assumption that at least one of them is s-finite.

Bandits.streamMeasure : {𝓐 : Type u_1} →
  {𝓡 : Type u_2} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓡 : MeasurableSpace 𝓡} → ProbabilityTheory.Kernel 𝓐 𝓡 → MeasureTheory.Measure (ℕ → 𝓐 → 𝓡)

Measure of an infinite stream of rewards from each action.

Go to its page

Code

lemma hasLaw_rewardByCountUntil_zero (μ : Measure Ω) [IsProbabilityMeasure μ] :
    HasLaw (rewardByCountUntil A R 0) (Measure.infinitePi fun p : 𝓐 × ℕ ↦ ν p.1)
      (μ.prod (streamMeasure ν))

Proof

have h_indep : iIndepFun (fun (p : 𝓐 × ℕ) (ω : Ω × (ℕ → 𝓐 → ℝ)) ↦ ω.2 (p.2 + 1) p.1)
      (μ.prod (streamMeasure ν)) :=
    (iIndepFun_snd_apply_prod_streamMeasure μ ν).precomp (g := fun p : 𝓐 × ℕ ↦ (p.2 + 1, p.1))
      fun p q hpq ↦ Prod.ext (Prod.mk.inj hpq).2 (by have := (Prod.mk.inj hpq).1; omega)
  h_indep.hasLaw_infinitePi (fun p ↦ hasLaw_snd_apply_prod_streamMeasure μ ν _ _)
    (by fun_prop : Measurable fun (ω : Ω × (ℕ → 𝓐 → ℝ)) (p : 𝓐 × ℕ) ↦
      ω.2 (p.2 + 1) p.1).aemeasurable

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 · open it in the Lean web editor.

Dependency graph

Audit surface: 3 project declarations, 37 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.

Bandits.hasLaw_rewardByCountUntil_zero🔗

`Bandits.hasLaw_rewardByCountUntil_zero`🔗