`Bandits.ArrayModel.indepFun_fst_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 𝓐Countable 𝓐A type α is countable if there exists an injective map α → ℕ.
𝓡 : Type u_2m𝓡 : MeasurableSpace 𝓡StandardBorelSpace 𝓡Nonempty 𝓡

Given

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

Then

ProbabilityTheory.IndepFun (fun ω => ω.1 n) (fun x => hist alg x n) (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.

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.

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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.

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.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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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.

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Code

lemma indepFun_fst_hist [Countable 𝓐] (alg : Algorithm Unit 𝓐 𝓡)
    (ν : Kernel 𝓐 𝓡) [IsMarkovKernel ν] (n : ℕ) :
    IndepFun (fun ω ↦ ω.1 n) (hist alg · n) (arrayMeasure ν)

Proof

(indepFun_fst_aux ν n).of_measurable_right (measurable_hist_comap alg 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.indepFun_fst_hist🔗

`Bandits.ArrayModel.indepFun_fst_hist`🔗