`Bandits.ArrayModel.measurable_pullCount_action_hist`🔗

Lemma

No docstring.

Types

𝓐 : Type u_1m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.Nonempty 𝓐StandardBorelSpace 𝓐A standard Borel space is a measurable space arising as the Borel sets of some Polish topology.DecidableEq 𝓐
𝓡 : Type u_2m𝓡 : MeasurableSpace 𝓡

Given

alg : Learning.Algorithm Unit 𝓐 𝓡A stochastic, sequential algorithm.
n : ℕ

Then

Measurable fun ω => Learning.pullCount (action alg) (action alg n ω) n ω

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

MeasurableSpace : Type u_6 → Type u_6

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

Nonempty : Sort u → Prop

`Nonempty α` is a typeclass that says that `α` is not an empty type,
that is, there exists an element in the type. It differs from `Inhabited α`
in that `Nonempty α` is a `Prop`, which means that it does not actually carry
an element of `α`, only a proof that *there exists* such an element.
Given `Nonempty α`, you can construct an element of `α` *nonconstructively*
using `Classical.choice`.

StandardBorelSpace : (α : Type u_1) → [MeasurableSpace α] → Prop

A standard Borel space is a measurable space arising as the Borel sets of some Polish topology.
This is useful in situations where a space has no natural topology or
the natural topology in a space is non-Polish.

To endow a standard Borel space `α` with a compatible Polish topology, use
`letI := upgradeStandardBorel α`. One can then use `eq_borel_upgradeStandardBorel α` to
rewrite the `MeasurableSpace α` instance to `borel α t`, where `t` is the new topology.

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`.

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.

Go to its page

Unit : Type

The canonical type with one element. This element is written `()`.

`Unit` has a number of uses:
* It can be used to model control flow that returns from a function call without providing other
  information.
* Monadic actions that return `Unit` have side effects without computing values.
* In polymorphic types, it can be used to indicate that no data is to be stored in a particular
  field.

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.

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.

Learning.pullCount : {𝓐 : Type u_2} → {Ω : Type u_4} → [DecidableEq 𝓐] → (ℕ → Ω → 𝓐) → 𝓐 → ℕ → Ω → ℕ

Number of times action `a` was chosen up to time `t` (excluding `t`).

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Bandits.ArrayModel.action : {𝓐 : Type u_1} →
  {𝓡 : Type u_2} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓡 : MeasurableSpace 𝓡} →
        [Nonempty 𝓐] →
          [StandardBorelSpace 𝓐] →
            [DecidableEq 𝓐] → Learning.Algorithm Unit 𝓐 𝓡 → ℕ → Bandits.ArrayModel.probSpace 𝓐 𝓡 → 𝓐

Action taken at time `n` in the array model.

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Code

lemma measurable_pullCount_action_hist (alg : Algorithm Unit 𝓐 𝓡) (n : ℕ) :
    Measurable[MeasurableSpace.comap (fun ω ↦ (hist alg ω n, action alg n ω)) inferInstance]
      (fun ω ↦ pullCount (action alg) (action alg n ω) n ω)

Proof

by
  simp_rw [pullCount_action_eq]
  change Measurable[MeasurableSpace.comap (fun ω ↦ (hist alg ω n, action alg n ω)) inferInstance]
    ((fun p : Hist Unit 𝓐 𝓡 n × 𝓐 ↦ pullCount' n p.1 p.2) ∘
      (fun ω ↦ (hist alg ω n, action alg n ω)))
  exact measurable_comp_comap _ (measurable_uncurry_pullCount' n)

Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), the 4th 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: 11 project declarations, 64 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.measurable_pullCount_action_hist🔗

`Bandits.ArrayModel.measurable_pullCount_action_hist`🔗