`BookProof.FriedrichsSquare.factorRel_quadForm_nonneg` {A : D →ₗ[ℂ] F} {p : F × F} (hp : p ∈ factorRel A) : 0 ≤ (inner ℂ p.1 p.2 : ℂ).re ∧ (inner ℂ p.1 p.2 : ℂ).im = 0

Proved

BookProof.FriedrichsSquare.factorRel_quadForm_nonneg

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFriedrichsSquareFactorization.

BookProof.FriedrichsSquare.factorRel_quadForm_nonneg {A : D →ₗ[ℂ] F} {p : F × F} (hp : p ∈ factorRel A) : 0 ≤ (inner ℂ p.1 p.2 : ℂ).re ∧ (inner ℂ p.1 p.2 : ℂ).im = 0

Formalization note: Lean 4 identifier BookProof.FriedrichsSquare.factorRel_quadForm_nonneg.

Preamble

-- Generated from ChapterFriedrichsSquareFactorization.lean — theorem BookProof.FriedrichsSquare.factorRel_quadForm_nonneg
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterClosureUniqueness
import Mathlib
import Definitions.Def_ChapterFriedrichsSquareFactorization
open BookProof.FriedrichsSquare



open BookProof.FarisLavine BookProof.EsaClosure BookProof.ClosureUniqueness

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable {D : Submodule ℂ F}

variable [CompleteSpace F]

Formal statement

theorem BookProof.FriedrichsSquare.factorRel_quadForm_nonneg {A : D →ₗ[ℂ] F} {p : F × F} (hp : p ∈ factorRel A) :
    0 ≤ (inner ℂ p.1 p.2 : ℂ).re ∧ (inner ℂ p.1 p.2 : ℂ).im = 0 := by sorry

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