`BookProof.ChapterAttentionSink.sink_denom` (beta s0 : ℝ) (s : Fin m → ℝ) : ∑ l, Real.exp (beta * (Fin.cons s0 s : Fin (m + 1) → ℝ) l) = Real.exp (beta * s0) + ∑ l, Real.exp (beta

Proved

BookProof.ChapterAttentionSink.sink_denom

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSink.

BookProof.ChapterAttentionSink.sink_denom (beta s0 : ℝ) (s : Fin m → ℝ) : ∑ l, Real.exp (beta * (Fin.cons s0 s : Fin (m + 1) → ℝ) l) = Real.exp (beta * s0) + ∑ l, Real.exp (beta * s l)

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

Preamble

-- Generated from ChapterAttentionSink.lean — theorem BookProof.ChapterAttentionSink.sink_denom
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.sink_denom (beta s0 : ℝ) (s : Fin m → ℝ) :
    ∑ l, Real.exp (beta * (Fin.cons s0 s : Fin (m + 1) → ℝ) l)
      = Real.exp (beta * s0) + ∑ l, Real.exp (beta * s l) := by sorry

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