`BookProof.ChapterNumericalRangeCrouzeix.crouzeix_ball` [CompleteSpace E] {A : E →L[ℂ] E} {c : ℂ} {r M R : ℝ} (hr : 0 ≤ r) (hM : 0 ≤ M) (hR : r < R) (h : NumBallLE A c r) {a : ℕ →

Proved

BookProof.ChapterNumericalRangeCrouzeix.crouzeix_ball

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.crouzeix_ball [CompleteSpace E] {A : E →L[ℂ] E} {c : ℂ} {r M R : ℝ} (hr : 0 ≤ r) (hM : 0 ≤ M) (hR : r < R) (h : NumBallLE A c r) {a : ℕ → ℂ} (ha : ∀ n, ‖a n‖ ≤ M / R ^ n) : ‖analyticFC a (A - c • (1 : E →L[ℂ] E))‖ ≤ (1 + 2 * r / (R - r)) * M

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

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.crouzeix_ball
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.crouzeix_ball [CompleteSpace E] {A : E →L[ℂ] E} {c : ℂ} {r M R : ℝ} (hr : 0 ≤ r)
    (hM : 0 ≤ M) (hR : r < R) (h : NumBallLE A c r) {a : ℕ → ℂ}
    (ha : ∀ n, ‖a n‖ ≤ M / R ^ n) :
    ‖analyticFC a (A - c • (1 : E →L[ℂ] E))‖ ≤ (1 + 2 * r / (R - r)) * M := by sorry

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