`Bandits.ClippedUCB.ucb_eq_ucb'`🔗

Lemma

No docstring.

Types

Ω : Type u_1

Given

K : ℕ
l : ℝ
u : ℝ
σ2 : ℝ
δ : ℝ
A : ℕ → Ω → Fin K
R : ℕ → Ω → ℝ
O : ℕ → Ω → Unit
a : Fin K
n : ℕ
ω : Ω

Then

ucb A R l u σ2 δ a n ω = ucb' n (Learning.history O A R n ω) l u σ2 δ a

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.

Real : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

Fin : ℕ → Type

Natural numbers less than some upper bound.

In particular, a `Fin n` is a natural number `i` with the constraint that `i < n`. It is the
canonical type with `n` elements.

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.

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`.

Bandits.ClippedUCB.ucb : {K : ℕ} → {Ω : Type u_1} → (ℕ → Ω → Fin K) → (ℕ → Ω → ℝ) → ℝ → ℝ → ℝ → ℝ → Fin K → ℕ → Ω → ℝ

Clipped upper confidence bound used in the regret analysis of Thompson sampling.

Go to its page

Bandits.ClippedUCB.ucb' : {K : ℕ} → (n : ℕ) → Learning.Hist Unit (Fin K) ℝ n → ℝ → ℝ → ℝ → ℝ → Fin K → ℝ

Clipped upper confidence bound (history-based version).

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Learning.history : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → (n : ℕ) → Ω → Learning.Hist 𝓞 𝓐 𝓨 n

History of the algorithm-environment sequence before time `n`: the rounds at
times `0, ..., n - 1`.

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Code

lemma ucb_eq_ucb' {O : ℕ → Ω → Unit} {a : Fin K} {n : ℕ} {ω : Ω} :
    ucb A R l u σ2 δ a n ω = ucb' n (history O A R n ω) l u σ2 δ a

Proof

by
  have hp : pullCount A a n ω = pullCount' n (history O A R n ω) a := pullCount_eq_pullCount'
  have he : empMean A R a n ω = empMean' n (history O A R n ω) a := empMean_eq_empMean'
  rw [ucb, ucb', hp, he]

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 2th 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, 52 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.ClippedUCB.ucb_eq_ucb'🔗

`Bandits.ClippedUCB.ucb_eq_ucb'`🔗