`BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq` (A : E →L[ℂ] E) (x : E) (t : ℝ) : HasDerivAt (fun s : ℝ => ‖(NormedSpace.exp (s • A)) x‖ ^ 2) (2 * (inner ℂ (NormedSpac

Proved

BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeSemigroup.

BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq (A : E →L[ℂ] E) (x : E) (t : ℝ) : HasDerivAt (fun s : ℝ => ‖(NormedSpace.exp (s • A)) x‖ ^ 2) (2 * (inner ℂ (NormedSpace.exp (t • A) x) (A (NormedSpace.exp (t • A) x))).re) t

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq.

Preamble

-- Generated from ChapterNumericalRangeSemigroup.lean — theorem BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq
import Mathlib
import Definitions.Def_ChapterNumericalRangeSemigroup
open BookProof.ChapterNumericalRangeSemigroup


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_normSq (A : E →L[ℂ] E) (x : E) (t : ℝ) :
    HasDerivAt (fun s : ℝ => ‖(NormedSpace.exp (s • A)) x‖ ^ 2)
      (2 * (inner ℂ (NormedSpace.exp (t • A) x) (A (NormedSpace.exp (t • A) x))).re) t := by sorry

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