`Bandits.ArrayModel.action_eq_and_pullCount_eq_congr_aux`🔗

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.
a : 𝓐
m : ℕ
n : ℕ
ω : probSpace 𝓐 𝓡Probability space for the array model of stochastic bandits.
ω' : probSpace 𝓐 𝓡

Assuming

hω1 : ∀ (i : ℕ), ω.1 i = ω'.1 i
hω2_ne : ∀ (i : ℕ) (b : 𝓐), b ≠ a → ω.2 i b = ω'.2 i b
hω2_eq : ∀ (i : ℕ), i + 1 ≤ m → ω.2 i a = ω'.2 i a
h_eq : action alg n ω = a ∧ Learning.pullCount (action alg) a n ω = m

Then

action alg n ω' = a ∧ Learning.pullCount (action alg) a n ω' = m

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.

Bandits.ArrayModel.probSpace : Type u_1 → Type u_2 → Type (max u_1 u_2)

Probability space for the array model of stochastic bandits.

Go to its page

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`.

Ne : {α : Sort u} → α → α → Prop

`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`.

HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ

`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.

LE.le : {α : Type u} → [self : LE α] → α → α → Prop

The less-equal relation: `x ≤ y` 

Conventions for notations in identifiers:

 * The recommended spelling of `≤` in identifiers is `le`.

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`.

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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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 action_eq_and_pullCount_eq_congr_aux (alg : Algorithm Unit 𝓐 𝓡)
    (a : 𝓐) (m n : ℕ) {ω ω' : probSpace 𝓐 𝓡}
    (hω1 : ∀ i, ω.1 i = ω'.1 i) (hω2_ne : ∀ i b, b ≠ a → ω.2 i b = ω'.2 i b)
    (hω2_eq : ∀ i, i + 1 ≤ m → ω.2 i a = ω'.2 i a)
    (h_eq : action alg n ω = a ∧ pullCount (action alg) a n ω = m) :
    action alg n ω' = a ∧ pullCount (action alg) a n ω' = m

Proof

by
  obtain ⟨h_action, h_pc⟩ := h_eq
  have h_hist : hist alg ω n = hist alg ω' n := by
    refine hist_congr alg n (fun i _ ↦ hω1 i) fun i b hi ↦ ?_
    rcases eq_or_ne b a with (rfl | hba)
    · rw [h_pc] at hi
      exact hω2_eq i hi
    · exact hω2_ne i b hba
  constructor
  · rw [← h_action]
    simp only [action, h_hist, hω1]
  · rw [← h_pc, pullCount_action_eq, pullCount_action_eq, h_hist]

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: 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.action_eq_and_pullCount_eq_congr_aux🔗

`Bandits.ArrayModel.action_eq_and_pullCount_eq_congr_aux`🔗