`BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below` (S : Finset ℕ) {mu lam : ℝ} (hmu : 0 ≤ mu) (u : FockAlg) : -(S.card * (2 * lam ^ 2 + (2 * lam ^ 2 + 1 /

Proved

BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below

by leonardopedro · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockCubicQuarticStability.

BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below (S : Finset ℕ) {mu lam : ℝ} (hmu : 0 ≤ mu) (u : FockAlg) : -(S.card * (2 * lam ^ 2 + (2 * lam ^ 2 + 1 / 2 - mu) ^ 2 / 2)) * ‖toLp u‖ ^ 2 ≤ mu * numberQuad u + ∑ k ∈ S, (lam * (inner ℂ (toLp u) (toLp (cubeA k u)) : ℂ).re + (inner ℂ (toLp u) (toLp (quartA k u)) : ℂ).re)

Formalization note: Lean 4 identifier BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below.

Preamble

-- Generated from ChapterFockCubicQuarticStability.lean — theorem BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below
import Definitions.Def_ChapterFockOneParticleGap
import Definitions.Def_ChapterFockNumberPreservingGap
import Definitions.Def_ChapterFockFieldPerturbation
import Definitions.Def_ChapterFockCubicUnbounded
import Mathlib
import Definitions.Def_ChapterFockCubicQuarticStability
import Definitions.Def_ChapterFockSecondQuantization
open BookProof.FockSecondQuantization
open BookProof.FockCubicQuarticStability


noncomputable section


open BookProof.FockSecondQuantization BookProof.FockOneParticleGap
open BookProof.FockNumberPreservingGap BookProof.FockFieldPerturbation
open BookProof.FockCubicUnbounded

Formal statement

theorem BookProof.FockCubicQuarticStability.multiMode_cubic_quartic_bounded_below (S : Finset ℕ) {mu lam : ℝ} (hmu : 0 ≤ mu)
    (u : FockAlg) :
    -(S.card * (2 * lam ^ 2 + (2 * lam ^ 2 + 1 / 2 - mu) ^ 2 / 2)) * ‖toLp u‖ ^ 2
      ≤ mu * numberQuad u
        + ∑ k ∈ S, (lam * (inner ℂ (toLp u) (toLp (cubeA k u)) : ℂ).re
            + (inner ℂ (toLp u) (toLp (quartA k u)) : ℂ).re) := by sorry

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