`BookProof.ChapterResidualStream.residual_injective` {f : E → E} {L : ℝ} (hL : L < 1) (hf : ∀ x y, ‖f x - f y‖ ≤ L * ‖x - y‖) : Function.Injective (residual f)

Proved

BookProof.ChapterResidualStream.residual_injective

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterResidualStream.

BookProof.ChapterResidualStream.residual_injective {f : E → E} {L : ℝ} (hL : L < 1) (hf : ∀ x y, ‖f x - f y‖ ≤ L * ‖x - y‖) : Function.Injective (residual f)

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

Preamble

-- Generated from ChapterResidualStream.lean — theorem BookProof.ChapterResidualStream.residual_injective
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.residual_injective {f : E → E} {L : ℝ} (hL : L < 1)
    (hf : ∀ x y, ‖f x - f y‖ ≤ L * ‖x - y‖) :
    Function.Injective (residual f) := by sorry

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