`BookProof.ChapterAttentionSink.headOutput_sink` (beta s0 : ℝ) (s : Fin m → ℝ) (v0 : E) (v : Fin m → E) : headOutput beta (Fin.cons s0 s) (Fin.cons v0 v) = sinkWeight beta s0 s • v

Proved

BookProof.ChapterAttentionSink.headOutput_sink

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSink.

BookProof.ChapterAttentionSink.headOutput_sink (beta s0 : ℝ) (s : Fin m → ℝ) (v0 : E) (v : Fin m → E) : headOutput beta (Fin.cons s0 s) (Fin.cons v0 v) = sinkWeight beta s0 s • v0 + (1 - sinkWeight beta s0 s) • headOutput beta s v

Formalization note: Lean 4 identifier BookProof.ChapterAttentionSink.headOutput_sink.

Preamble

-- Generated from ChapterAttentionSink.lean — theorem BookProof.ChapterAttentionSink.headOutput_sink
import Definitions.Def_ChapterObservableExpectation
import Mathlib
import Definitions.Def_ChapterAttentionSink
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionOutput
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionSink


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionOutput

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

Formal statement

theorem BookProof.ChapterAttentionSink.headOutput_sink (beta s0 : ℝ) (s : Fin m → ℝ) (v0 : E) (v : Fin m → E) :
    headOutput beta (Fin.cons s0 s) (Fin.cons v0 v)
      = sinkWeight beta s0 s • v0 + (1 - sinkWeight beta s0 s) • headOutput beta s v := by sorry

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