`BookProof.ChapterAttentionStreaming.norm_headOutput_snoc_sub_le` (beta sn : ℝ) (s : Fin m → ℝ) (vn : E) (v : Fin m → E) : ‖headOutput beta (Fin.snoc s sn) (Fin.snoc v vn) - headOu

Proved

BookProof.ChapterAttentionStreaming.norm_headOutput_snoc_sub_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionStreaming.

BookProof.ChapterAttentionStreaming.norm_headOutput_snoc_sub_le (beta sn : ℝ) (s : Fin m → ℝ) (vn : E) (v : Fin m → E) : ‖headOutput beta (Fin.snoc s sn) (Fin.snoc v vn) - headOutput beta s v‖ = newWeight beta sn s * ‖vn - headOutput beta s v‖

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionOutput

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

Formal statement

theorem BookProof.ChapterAttentionStreaming.norm_headOutput_snoc_sub_le (beta sn : ℝ) (s : Fin m → ℝ) (vn : E) (v : Fin m → E) :
    ‖headOutput beta (Fin.snoc s sn) (Fin.snoc v vn) - headOutput beta s v‖
      = newWeight beta sn s * ‖vn - headOutput beta s v‖ := by sorry

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