`BookProof.ChapterObservableExpectation.attention_eq_expectation` (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (j₀ : Fin m) : (∀ j, 0 ≤ bor

Proved

BookProof.ChapterObservableExpectation.attention_eq_expectation

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterObservableExpectation.

BookProof.ChapterObservableExpectation.attention_eq_expectation (q : EuclideanSpace ℝ (Fin n)) (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (j₀ : Fin m) : (∀ j, 0 ≤ bornWeight q k j) ∧ (∑ j, bornWeight q k j = 1) ∧ attentionOutput q k v = observableExpectation (bornWeight q k) v

Formalization note: Lean 4 identifier BookProof.ChapterObservableExpectation.attention_eq_expectation.

Preamble

-- Generated from ChapterObservableExpectation.lean — theorem BookProof.ChapterObservableExpectation.attention_eq_expectation
import Mathlib
import Definitions.Def_ChapterObservableExpectation
import Definitions.Def_ChapterSoftmaxBorn
open BookProof.ChapterSoftmaxBorn
open BookProof.ChapterObservableExpectation


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxBorn

variable {m n : ℕ}
variable {E : Type*} [AddCommGroup E] [Module ℝ E]

Formal statement

theorem BookProof.ChapterObservableExpectation.attention_eq_expectation (q : EuclideanSpace ℝ (Fin n))
    (k : Fin m → EuclideanSpace ℝ (Fin n)) (v : Fin m → E) (j₀ : Fin m) :
    (∀ j, 0 ≤ bornWeight q k j) ∧ (∑ j, bornWeight q k j = 1) ∧
      attentionOutput q k v = observableExpectation (bornWeight q k) v := by sorry

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