`BookProof.QgCouplingDGammaSum.coupling_quadForm_le` {s : Finset ι} {cols : ι → ℕ → (ℕ →₀ ℂ)} (hpos : ∀ i ∈ s, IsPosCol (cols i)) {i : ι} (hi : i ∈ s) (x : lpFiniteModes...

Proved

BookProof.QgCouplingDGammaSum.coupling_quadForm_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQgCouplingDGammaSum.

BookProof.QgCouplingDGammaSum.coupling_quadForm_le {s : Finset ι} {cols : ι → ℕ → (ℕ →₀ ℂ)} (hpos : ∀ i ∈ s, IsPosCol (cols i)) {i : ι} (hi : i ∈ s) (x : lpFiniteModes Conf) : quadForm (dGammaOp (cols i)) x ≤ quadForm (dGammaOp (fun k => ∑ j ∈ s, cols j k)) x

Formalization note: Lean 4 identifier BookProof.QgCouplingDGammaSum.coupling_quadForm_le.

Preamble

-- Generated from ChapterQgCouplingDGammaSum.lean — theorem BookProof.QgCouplingDGammaSum.coupling_quadForm_le
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Definitions.Def_ChapterNavierStokesIkebeKato
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterFriedrichsExtension
import Definitions.Def_ChapterYangMillsFriedrichs
import Mathlib
import Definitions.Def_ChapterQgCouplingDGammaSum
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterFockSecondQuantization
import Definitions.Def_ChapterNavierStokesEsa
open BookProof.FockSecondQuantization
open BookProof.QgCouplingDGammaSum



open BookProof.FockSecondQuantization
open BookProof.NavierStokesFlow BookProof.NavierStokesFlow.IkebeKato
open BookProof.FarisLavine BookProof.FriedrichsExtension BookProof.YangMillsFriedrichs

noncomputable section

variable {ι : Type*}

Formal statement

theorem BookProof.QgCouplingDGammaSum.coupling_quadForm_le {s : Finset ι} {cols : ι → ℕ → (ℕ →₀ ℂ)}
    (hpos : ∀ i ∈ s, IsPosCol (cols i)) {i : ι} (hi : i ∈ s) (x : lpFiniteModes Conf) :
    quadForm (dGammaOp (cols i)) x
      ≤ quadForm (dGammaOp (fun k => ∑ j ∈ s, cols j k)) x := by sorry

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