`BookProof.ChapterResidualStream.norm_residual_sub_self_le` [NormedSpace ℝ E] (beta : ℝ) (s : Fin m → ℝ) {v : Fin m → E} {C : ℝ} (hv : ∀ j, ‖v j‖ ≤ C) (i : Fin m) (x : E) :...

Proved

BookProof.ChapterResidualStream.norm_residual_sub_self_le

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterResidualStream.

BookProof.ChapterResidualStream.norm_residual_sub_self_le [NormedSpace ℝ E] (beta : ℝ) (s : Fin m → ℝ) {v : Fin m → E} {C : ℝ} (hv : ∀ j, ‖v j‖ ≤ C) (i : Fin m) (x : E) : ‖residual (fun _ => headOutput beta s v) x - x‖ ≤ C

Formalization note: Lean 4 identifier BookProof.ChapterResidualStream.norm_residual_sub_self_le.

Preamble

-- Generated from ChapterResidualStream.lean — theorem BookProof.ChapterResidualStream.norm_residual_sub_self_le
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionOutput
import Mathlib
import Definitions.Def_ChapterResidualStream
open BookProof.ChapterResidualStream


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterAttentionOutput

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

Formal statement

theorem BookProof.ChapterResidualStream.norm_residual_sub_self_le [NormedSpace ℝ E] (beta : ℝ) (s : Fin m → ℝ)
    {v : Fin m → E} {C : ℝ} (hv : ∀ j, ‖v j‖ ≤ C) (i : Fin m) (x : E) :
    ‖residual (fun _ => headOutput beta s v) x - x‖ ≤ C := by sorry

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