`BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone` (B : H →L[ℂ] H) (t : ℝ) (x : H) (hdom : ∀ s : ℝ, flow B s x ∈ T.domain) (s : ℝ) : HasDerivAt (fun u : ℝ => T.stoneU (t - u

Proved

BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstUnboundedLeakage.

BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone (B : H →L[ℂ] H) (t : ℝ) (x : H) (hdom : ∀ s : ℝ, flow B s x ∈ T.domain) (s : ℝ) : HasDerivAt (fun u : ℝ => T.stoneU (t - u) (flow B u x)) (T.stoneU (t - s) ((-Complex.I) • (B (flow B s x) - T.op ⟨flow B s x, hdom s⟩))) s

Formalization note: Lean 4 identifier BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone.

Preamble

-- Generated from ChapterBrstUnboundedLeakage.lean — theorem BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterBrstUnboundedLeakage
import Definitions.Def_ChapterBrstTruncationLeakage
import Definitions.Def_ChapterStoneResolvent
import Definitions.Def_ChapterStoneUnitary
open BookProof.BrstLeakage
open BookProof.BrstUnboundedLeakage


open NormedSpace Filter Topology
open scoped InnerProductSpace


open BookProof.BrstLeakage BookProof.ChapterStoneResolvent

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

variable (T : UnboundedSelfAdjoint H)

Formal statement

theorem BookProof.BrstUnboundedLeakage.hasDerivAt_duhamel_stone (B : H →L[ℂ] H) (t : ℝ) (x : H)
    (hdom : ∀ s : ℝ, flow B s x ∈ T.domain) (s : ℝ) :
    HasDerivAt (fun u : ℝ => T.stoneU (t - u) (flow B u x))
      (T.stoneU (t - s) ((-Complex.I) • (B (flow B s x) - T.op ⟨flow B s x, hdom s⟩))) s := by sorry

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