Prove2Me

`BookProof.FriedrichsSquare.mem_flipGraph_orthogonal_iff` {A : D →ₗ[ℂ] F} {q : WithLp 2 (F × F)} : q ∈ (flipGraph A)ᗮ ↔ WithLp.ofLp q ∈ adjGraph A

Proved
BookProof.FriedrichsSquare.mem_flipGraph_orthogonal_iff

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFriedrichsSquareFactorization.

BookProof.FriedrichsSquare.mem_flipGraph_orthogonal_iff {A : D →ₗ[ℂ] F} {q : WithLp 2 (F × F)} : q ∈ (flipGraph A)ᗮ ↔ WithLp.ofLp q ∈ adjGraph A

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

Preamble
-- Generated from ChapterFriedrichsSquareFactorization.lean — theorem BookProof.FriedrichsSquare.mem_flipGraph_orthogonal_iff
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterFriedrichsSquareFactorization
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterEsaClosureCore
open BookProof.ClosureUniqueness
open BookProof.EsaClosure
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.mem_flipGraph_orthogonal_iff {A : D →ₗ[ℂ] F} {q : WithLp 2 (F × F)} :
    q ∈ (flipGraph A)ᗮ ↔ WithLp.ofLp q ∈ adjGraph A := by sorry

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