`BookProof.ChapterAttentionSink.one_sub_sinkWeight_eq` (beta s0 : ℝ) (s : Fin m → ℝ) : 1 - sinkWeight beta s0 s = (∑ l, Real.exp (beta * s l)) / (Real.exp (beta * s0) + ∑ l, Real.e

Proved

BookProof.ChapterAttentionSink.one_sub_sinkWeight_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSink.

BookProof.ChapterAttentionSink.one_sub_sinkWeight_eq (beta s0 : ℝ) (s : Fin m → ℝ) : 1 - sinkWeight beta s0 s = (∑ l, Real.exp (beta * s l)) / (Real.exp (beta * s0) + ∑ l, Real.exp (beta * s l))

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterObservableExpectation BookProof.ChapterSoftmaxSharpness

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

Formal statement

theorem BookProof.ChapterAttentionSink.one_sub_sinkWeight_eq (beta s0 : ℝ) (s : Fin m → ℝ) :
    1 - sinkWeight beta s0 s
      = (∑ l, Real.exp (beta * s l)) / (Real.exp (beta * s0) + ∑ l, Real.exp (beta * s l)) := by sorry

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