`Bandits.indepFun_update_rewardByCountUntil_eval`🔗

Lemma

From the authors

The array rewardByCountUntil A R t with the entry (b, k) erased is independent of the entry (k + 1, b) of the auxiliary array.

Types

𝓐 : Type u_1m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.DecidableEq 𝓐MeasurableSingletonClass 𝓐A typeclass mixin for MeasurableSpaces such that each singleton is measurable.Countable 𝓐A type α is countable if there exists an injective map α → ℕ.
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.
t : ℕ
b : 𝓐
k : ℕ

Assuming

hA : ∀ (n : ℕ), Measurable (A n)A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.
hR : ∀ (n : ℕ), Measurable (R n)

Then

ProbabilityTheory.IndepFun (fun ω => Function.update (Learning.rewardByCountUntil A R t ω) (b, k) 0)
  (fun ω => ω.2 (k + 1) b) (μ.prod (streamMeasure ν))

Two functions are independent if the two measurable space structures they generate are independent.

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`.

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

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

Countable : Sort u → Prop

A type `α` is countable if there exists an injective map `α → ℕ`.

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`.

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

ProbabilityTheory.IndepFun : {Ω : Type u_1} →
  {β : Type u_6} →
    {γ : Type u_7} →
      {_mΩ : MeasurableSpace Ω} →
        [MeasurableSpace β] →
          [MeasurableSpace γ] →
            (Ω → β) → (Ω → γ) → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.IndepFun._auto_1 → Prop

Two functions are independent if the two measurable space structures they generate are
independent. For a function `f` with codomain having measurable space structure `m`, the generated
measurable space structure is `MeasurableSpace.comap f m`.
We use the notation `f ⟂ᵢ[μ] g` for `IndepFun f g μ` (scoped in `ProbabilityTheory`).

Function.update : {α : Sort u} → {β : α → Sort v} → [DecidableEq α] → ((a : α) → β a) → (a' : α) → β a' → (a : α) → β a

Replacing the value of a function at a given point by a given value.

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

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`.

HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ

`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.

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.

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Code

lemma indepFun_update_rewardByCountUntil_eval [Countable 𝓐] (hA : ∀ n, Measurable (A n))
    (hR : ∀ n, Measurable (R n)) (μ : Measure Ω) [IsProbabilityMeasure μ] (t : ℕ) (b : 𝓐) (k : ℕ) :
    (fun ω ↦ Function.update (rewardByCountUntil A R t ω) (b, k) 0)
      ⟂ᵢ[μ.prod (streamMeasure ν)] (fun ω ↦ ω.2 (k + 1) b)

Proof

by
  refine ((indepFun_snd_apply_prod_streamMeasure_update μ ν (k + 1) b 0).of_measurable_right
    ?_).symm
  have h_eq : (fun ω : Ω × (ℕ → 𝓐 → ℝ) ↦ Function.update (rewardByCountUntil A R t ω) (b, k) 0)
      = (fun ω ↦ Function.update (rewardByCountUntil A R t ω) (b, k) 0)
        ∘ (fun ω ↦ (ω.1, fun i c ↦ if i = k + 1 ∧ c = b then 0 else ω.2 i c)) := by
    ext ⟨x, z⟩ p
    simp only [Function.comp_apply]
    by_cases hp : p = (b, k)
    · rw [hp, Function.update_self, Function.update_self]
    · rw [Function.update_of_ne hp, Function.update_of_ne hp]
      refine rewardByCountUntil_congr t p ?_
      split_ifs with hc
      · exact absurd (Prod.ext hc.2 (by have := hc.1; omega)) hp
      · rfl
  rw [h_eq]
  exact measurable_comp_comap _
    (measurable_update_left.comp (measurable_rewardByCountUntil hA hR t))

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, 42 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.indepFun_update_rewardByCountUntil_eval🔗

`Bandits.indepFun_update_rewardByCountUntil_eval`🔗