`BookProof.ChapterAttentionCoarseGrain.headOutput_merge` (beta : ℝ) (s : Fin m → ℝ) (f : Fin m → Fin r) (v : Fin r → E) : observableExpectation (mergeWeights f (scoreSoftmax beta s

Proved

BookProof.ChapterAttentionCoarseGrain.headOutput_merge

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionCoarseGrain.

BookProof.ChapterAttentionCoarseGrain.headOutput_merge (beta : ℝ) (s : Fin m → ℝ) (f : Fin m → Fin r) (v : Fin r → E) : observableExpectation (mergeWeights f (scoreSoftmax beta s)) v = headOutput beta s (fun x => v (f x))

Formalization note: Lean 4 identifier BookProof.ChapterAttentionCoarseGrain.headOutput_merge.

Preamble

-- Generated from ChapterAttentionCoarseGrain.lean — theorem BookProof.ChapterAttentionCoarseGrain.headOutput_merge
import Mathlib
import Definitions.Def_ChapterAttentionCoarseGrain
import Definitions.Def_ChapterObservableExpectation
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionOutput
open BookProof.ChapterObservableExpectation
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionCoarseGrain


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionOutput

variable {m r : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionCoarseGrain.headOutput_merge (beta : ℝ) (s : Fin m → ℝ) (f : Fin m → Fin r) (v : Fin r → E) :
    observableExpectation (mergeWeights f (scoreSoftmax beta s)) v
      = headOutput beta s (fun x => v (f x)) := by sorry

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