`BookProof.ChapterAttentionStreaming.snoc_denom` (beta sn : ℝ) (s : Fin m → ℝ) : ∑ l, Real.exp (beta * (Fin.snoc s sn : Fin (m + 1) → ℝ) l) = (∑ l, Real.exp (beta * s l)) + Real.ex

Proved

BookProof.ChapterAttentionStreaming.snoc_denom

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionStreaming.

BookProof.ChapterAttentionStreaming.snoc_denom (beta sn : ℝ) (s : Fin m → ℝ) : ∑ l, Real.exp (beta * (Fin.snoc s sn : Fin (m + 1) → ℝ) l) = (∑ l, Real.exp (beta * s l)) + Real.exp (beta * sn)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionStreaming.snoc_denom.

Preamble

-- Generated from ChapterAttentionStreaming.lean — theorem BookProof.ChapterAttentionStreaming.snoc_denom
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionStreaming
open BookProof.ChapterAttentionStreaming


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

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

Formal statement

theorem BookProof.ChapterAttentionStreaming.snoc_denom (beta sn : ℝ) (s : Fin m → ℝ) :
    ∑ l, Real.exp (beta * (Fin.snoc s sn : Fin (m + 1) → ℝ) l)
      = (∑ l, Real.exp (beta * s l)) + Real.exp (beta * sn) := by sorry

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