`BookProof.FockCubicQuarticStability.trial_cubic_quartic_bounded_below_general` (k : ℕ) (lam : ℝ) (u : FockAlg) : -(2 * lam ^ 4 + lam ^ 2 + 1 / 8) * ‖toLp u‖ ^ 2 ≤ numberQuad u + l

Proved

BookProof.FockCubicQuarticStability.trial_cubic_quartic_bounded_below_general

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockCubicQuarticStability.

BookProof.FockCubicQuarticStability.trial_cubic_quartic_bounded_below_general (k : ℕ) (lam : ℝ) (u : FockAlg) : -(2 * lam ^ 4 + lam ^ 2 + 1 / 8) * ‖toLp u‖ ^ 2 ≤ numberQuad u + 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.trial_cubic_quartic_bounded_below_general.

Preamble

-- Generated from ChapterFockCubicQuarticStability.lean — theorem BookProof.FockCubicQuarticStability.trial_cubic_quartic_bounded_below_general
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.trial_cubic_quartic_bounded_below_general (k : ℕ) (lam : ℝ) (u : FockAlg) :
    -(2 * lam ^ 4 + lam ^ 2 + 1 / 8) * ‖toLp u‖ ^ 2
      ≤ numberQuad u
        + lam * (inner ℂ (toLp u) (toLp (cubeA k u)) : ℂ).re
        + (inner ℂ (toLp u) (toLp (quartA k u)) : ℂ).re := by sorry

View graph