`BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE` {A : E →L[ℂ] E} {r : ℝ} (hr : 0 ≤ r) (h : NumRadiusLE A r) (n : ℕ) (hn : 0 < n) : ‖A ^ n‖ ≤ 2 * r ^ n

Proved

BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE {A : E →L[ℂ] E} {r : ℝ} (hr : 0 ≤ r) (h : NumRadiusLE A r) (n : ℕ) (hn : 0 < n) : ‖A ^ n‖ ≤ 2 * r ^ n

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE.

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE
import Mathlib
import Definitions.Def_ChapterNumericalRangeCrouzeix
open BookProof.ChapterNumericalRangeCrouzeix


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterNumericalRangeCrouzeix.norm_pow_le_of_numRadiusLE {A : E →L[ℂ] E} {r : ℝ} (hr : 0 ≤ r)
    (h : NumRadiusLE A r) (n : ℕ) (hn : 0 < n) : ‖A ^ n‖ ≤ 2 * r ^ n := by sorry

View graph