`BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift` {Dom : Submodule ℂ F} (H : D →ₗ[ℂ] F) (c : ℝ) (A : Dom →ₗ[ℂ] F) (hA : IsPositiveSelfAdjointExtension (H + (c : ℂ)

Proved

BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFriedrichsCanonical.

BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift {Dom : Submodule ℂ F} (H : D →ₗ[ℂ] F) (c : ℝ) (A : Dom →ₗ[ℂ] F) (hA : IsPositiveSelfAdjointExtension (H + (c : ℂ) • D.subtype) A) : IsSemiboundedSelfAdjointExtension c H (A - (c : ℂ) • Dom.subtype)

Formalization note: Lean 4 identifier BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift.

Preamble

-- Generated from ChapterFriedrichsCanonical.lean — theorem BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterComplexShiftCore
import Mathlib
import Definitions.Def_ChapterFriedrichsCanonical
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterFriedrichsExtension
import Definitions.Def_ChapterYangMillsFriedrichs
open BookProof.FriedrichsExtension
open BookProof.FriedrichsExtension.FormDom
open BookProof.YangMillsFriedrichs
open BookProof.FriedrichsCanonical



open BookProof.FarisLavine BookProof.YangMillsFriedrichs BookProof.HashimotoShiftInvert
open BookProof.FriedrichsExtension BookProof.FriedrichsExtension.FormDom

noncomputable section

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

variable {D : Submodule ℂ F}

Formal statement

theorem BookProof.FriedrichsCanonical.isSemibounded_of_isPositive_shift {Dom : Submodule ℂ F} (H : D →ₗ[ℂ] F) (c : ℝ)
    (A : Dom →ₗ[ℂ] F)
    (hA : IsPositiveSelfAdjointExtension (H + (c : ℂ) • D.subtype) A) :
    IsSemiboundedSelfAdjointExtension c H (A - (c : ℂ) • Dom.subtype) := by sorry

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