`BookProof.ChapterResidualStream.norm_iterate_residual_sub_le` {f : E → E} {C : ℝ} (hf : ∀ x, ‖f x‖ ≤ C) (n : ℕ) (x : E) : ‖(residual f)^[n] x - x‖ ≤ n * C

Proved

BookProof.ChapterResidualStream.norm_iterate_residual_sub_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterResidualStream.

BookProof.ChapterResidualStream.norm_iterate_residual_sub_le {f : E → E} {C : ℝ} (hf : ∀ x, ‖f x‖ ≤ C) (n : ℕ) (x : E) : ‖(residual f)^[n] x - x‖ ≤ n * C

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

Preamble

-- Generated from ChapterResidualStream.lean — theorem BookProof.ChapterResidualStream.norm_iterate_residual_sub_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_iterate_residual_sub_le {f : E → E} {C : ℝ} (hf : ∀ x, ‖f x‖ ≤ C) (n : ℕ)
    (x : E) :
    ‖(residual f)^[n] x - x‖ ≤ n * C := by sorry

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