`Bandits.ArrayModel.indepFun_snd_apply_truncRow`🔗

Lemma

From the authors

The reward ω.2 m a is independent of truncRow a m, which reads only other coordinates.

Types

𝓐 : Type u_1m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.DecidableEq 𝓐
𝓡 : Type u_2m𝓡 : MeasurableSpace 𝓡

Given

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.
a : 𝓐
m : ℕ

Then

ProbabilityTheory.IndepFun (fun ω => ω.2 m a) (truncRow a m) (arrayMeasure 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`.

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)`.

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.

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`).

Bandits.ArrayModel.truncRow : {𝓐 : Type u_1} →
  {𝓡 : Type u_2} → [DecidableEq 𝓐] → 𝓐 → ℕ → Bandits.ArrayModel.probSpace 𝓐 𝓡 → Bandits.ArrayModel.probSpace 𝓐 𝓡

Modification of `ω` in which the rewards of action `a` are read only up to index `m - 1`:
in row `a` of the reward array, the entry at index `i` is kept if `i < m` and replaced by the entry
at index `m + 1 + i` otherwise. The result does not depend on the coordinate `(m, a)` of the
array.

Go to its page

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

Probability measure for the array model of stochastic bandits.

Go to its page

Code

lemma indepFun_snd_apply_truncRow (ν : Kernel 𝓐 𝓡) [IsMarkovKernel ν] (a : 𝓐) (m : ℕ) :
    (fun ω ↦ ω.2 m a) ⟂ᵢ[arrayMeasure ν] (truncRow a m)

Proof

by
  let T : (ℕ → 𝓐 → 𝓡) → (ℕ → 𝓐 → 𝓡) :=
    fun ω k b ↦ if b = a then ω (if k < m then k else m + 1 + k) b else ω k b
  have hT : Measurable[⨆ p ∈ {p : ℕ × 𝓐 | p ≠ (m, a)},
      MeasurableSpace.comap (fun ω : ℕ → 𝓐 → 𝓡 ↦ ω p.1 p.2) inferInstance] T := by
    rw [measurable_iff_comap_le, MeasurableSpace.comap_pi]
    refine iSup_le fun k ↦ ?_
    rw [MeasurableSpace.comap_pi]
    refine iSup_le fun b ↦ ?_
    by_cases hb : b = a
    · simp only [T, hb, ↓reduceIte]
      refine le_iSup₂_of_le (if k < m then k else m + 1 + k, a) ?_ (le_of_eq rfl)
      simp only [Set.mem_ofPred_eq, ne_eq, Prod.mk.injEq, and_true]
      split_ifs <;> omega
    · simp only [T, hb, ↓reduceIte]
      exact le_iSup₂_of_le (k, b) (by simp [hb]) (le_of_eq rfl)
  have hTm : Measurable T :=
    hT.mono (iSup₂_le fun p _ ↦ Measurable.comap_le (by fun_prop)) le_rfl
  have h := (iIndepFun_eval_streamMeasure ν).indepFun_of_measurable_iSup_comap
    (fun _ ↦ by fun_prop) (i := (m, a)) (by simp) hT
  exact h.snd_prod (μ := Measure.infinitePi fun _ ↦ volume) (by fun_prop) hTm

Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), 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: 4 project declarations, 38 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.ArrayModel.indepFun_snd_apply_truncRow🔗

`Bandits.ArrayModel.indepFun_snd_apply_truncRow`🔗