`Bandits.ArrayModel.measurable_hist_truncRowPullCount`🔗

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 𝓐Countable 𝓐A type α is countable if there exists an injective map α → ℕ.
𝓡 : Type u_2m𝓡 : MeasurableSpace 𝓡

Given

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

Then

Measurable fun x => hist alg x n

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`.

Countable : Sort u → Prop

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

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.

Bandits.ArrayModel.hist : {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 → Bandits.ArrayModel.probSpace 𝓐 𝓡 → (n : ℕ) → Learning.Hist Unit 𝓐 𝓡 n

History of actions and rewards before time `n` in the array model.

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Code

lemma measurable_hist_truncRowPullCount [Countable 𝓐] (alg : Algorithm Unit 𝓐 𝓡) (a : 𝓐)
    (n : ℕ) :
    Measurable[MeasurableSpace.comap (truncRowPullCount alg a n) inferInstance] (hist alg · n)

Proof

by
  have h_eq : (hist alg · n) = (hist alg · n) ∘ (truncRowPullCount alg a n) := by
    ext ω : 1
    refine hist_congr alg n (fun _ _ ↦ rfl) fun i b hi ↦ ?_
    by_cases hb : b = a
    · subst hb
      simp [truncRowPullCount, truncRow, hi]
    · simp [truncRowPullCount, truncRow, hb]
  rw [h_eq]
  exact (measurable_hist alg n).comp (Measurable.of_comap_le le_rfl)

Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22), the 5th 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: 13 project declarations, 68 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_hist_truncRowPullCount🔗

`Bandits.ArrayModel.measurable_hist_truncRowPullCount`🔗