`BookProof.BrstLeakage.hasDerivAt_duhamel` (X Y : E →L[ℂ] E) (t s : ℝ) (x : E) : HasDerivAt (fun u : ℝ => (exp ((t - u) • X)) ((exp (u • Y)) x)) ((exp ((t - s) • X)) ((Y - X) ((exp

Proved

BookProof.BrstLeakage.hasDerivAt_duhamel

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstTruncationLeakage.

BookProof.BrstLeakage.hasDerivAt_duhamel (X Y : E →L[ℂ] E) (t s : ℝ) (x : E) : HasDerivAt (fun u : ℝ => (exp ((t - u) • X)) ((exp (u • Y)) x)) ((exp ((t - s) • X)) ((Y - X) ((exp (s • Y)) x))) s

Formalization note: Lean 4 identifier BookProof.BrstLeakage.hasDerivAt_duhamel.

Preamble

-- Generated from ChapterBrstTruncationLeakage.lean — theorem BookProof.BrstLeakage.hasDerivAt_duhamel
import Mathlib
import Definitions.Def_ChapterBrstTruncationLeakage
open BookProof.BrstLeakage


open NormedSpace


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]

Formal statement

theorem BookProof.BrstLeakage.hasDerivAt_duhamel (X Y : E →L[ℂ] E) (t s : ℝ) (x : E) :
    HasDerivAt (fun u : ℝ => (exp ((t - u) • X)) ((exp (u • Y)) x))
      ((exp ((t - s) • X)) ((Y - X) ((exp (s • Y)) x))) s := by sorry

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