`Learning.measurableSet_action_eq_and_pullCount_eq`🔗

Lemma

No docstring.

Types

𝓐 : Type u_2m𝓐 : 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.
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

A : ℕ → Ω → 𝓐
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.

Then

MeasurableSet {x | A t x = b ∧ pullCount A b t x = k}

MeasurableSet s means that s is measurable (in the ambient measure space on α)

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.

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.

MeasurableSet : {α : Type u_1} → [MeasurableSpace α] → Set α → Prop

`MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`)

Set.ofPred : {α : Type u} → (α → Prop) → Set α

Turn a predicate `p : α → Prop` into a set, also written as `{x | p x}`

Eq : {α : Sort u_1} → α → α → Prop

The equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.

And : Prop → Prop → Prop

`And a b`, or `a ∧ b`, is the conjunction of propositions. It can be
constructed and destructed like a pair: if `ha : a` and `hb : b` then
`⟨ha, hb⟩ : a ∧ b`, and if `h : a ∧ b` then `h.left : a` and `h.right : b`.


Conventions for notations in identifiers:

 * The recommended spelling of `∧` in identifiers is `and`.

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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Code

lemma measurableSet_action_eq_and_pullCount_eq [MeasurableSingletonClass 𝓐]
    (hA : ∀ n, Measurable (A n)) (t : ℕ) (b : 𝓐) (k : ℕ) :
    MeasurableSet {x | A t x = b ∧ pullCount A b t x = k}

Proof

((measurableSet_singleton _).preimage (hA t)).inter
    ((measurableSet_singleton _).preimage (measurable_pullCount hA b t))

Meaning last changed in v4.35.0-rc2-1-g61e506b (2026-09-22).

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: 1 project declarations, 12 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.

Learning.measurableSet_action_eq_and_pullCount_eq🔗

`Learning.measurableSet_action_eq_and_pullCount_eq`🔗