`BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup` (f : ℤ → ℝ) (x : (mulSA f).domain) : HasDerivAt (fun t : ℝ => (phaseGroup f).U t (x : L2Z)) ((-Complex.I) • (mulSA f).op x)

Proved

BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup

by leonardopedro · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterStoneSeparable.

BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup (f : ℤ → ℝ) (x : (mulSA f).domain) : HasDerivAt (fun t : ℝ => (phaseGroup f).U t (x : L2Z)) ((-Complex.I) • (mulSA f).op x) 0

Formalization note: Lean 4 identifier BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup.

Preamble

-- Generated from ChapterStoneSeparable.lean — theorem BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup
import Definitions.Def_ChapterSirkTrotterKato
import Definitions.Def_ChapterStoneConverse
import Mathlib
import Definitions.Def_ChapterStoneSeparable
import Definitions.Def_ChapterContinuityUnitaryInfinite
import Definitions.Def_ChapterF7
import Definitions.Def_ChapterUnboundedPosition
open BookProof.ChapterContinuityUnitaryInfinite
open BookProof.ChapterF7
open BookProof.ChapterUnboundedPosition
open BookProof.ChapterStoneSeparable


open scoped InnerProductSpace
open Filter Topology


open BookProof.ChapterStoneResolvent BookProof.ChapterStoneMeasurable

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

Formal statement

theorem BookProof.ChapterStoneSeparable.hasDerivAt_phaseGroup (f : ℤ → ℝ) (x : (mulSA f).domain) :
    HasDerivAt (fun t : ℝ => (phaseGroup f).U t (x : L2Z))
      ((-Complex.I) • (mulSA f).op x) 0 := by sorry

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