`BookProof.ChapterNumericalRangeCrouzeix.re_inner_sub_smul_nonneg` {A : E →L[ℂ] E} (h : NumRadiusLE A 1) {z : ℂ} (hz : ‖z‖ ≤ 1) (x : E) : 0 ≤ (⟪x, x - z • A x⟫_ℂ).re

Proved

BookProof.ChapterNumericalRangeCrouzeix.re_inner_sub_smul_nonneg

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.re_inner_sub_smul_nonneg {A : E →L[ℂ] E} (h : NumRadiusLE A 1) {z : ℂ} (hz : ‖z‖ ≤ 1) (x : E) : 0 ≤ (⟪x, x - z • A x⟫_ℂ).re

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

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.re_inner_sub_smul_nonneg
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.re_inner_sub_smul_nonneg {A : E →L[ℂ] E} (h : NumRadiusLE A 1) {z : ℂ}
    (hz : ‖z‖ ≤ 1) (x : E) : 0 ≤ (⟪x, x - z • A x⟫_ℂ).re := by sorry

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