`BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below` (k n : ℕ) (hn : 1 ≤ n) (lam c : ℝ) : -(lam ^ 4 / 4 + 2 * lam ^ 2) * ‖toLp (trial k n c)‖ ^ 2 ≤ numberQuad (trial k

Proved

BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockCubicUnbounded.

BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below (k n : ℕ) (hn : 1 ≤ n) (lam c : ℝ) : -(lam ^ 4 / 4 + 2 * lam ^ 2) * ‖toLp (trial k n c)‖ ^ 2 ≤ numberQuad (trial k n c) + lam * (inner ℂ (toLp (trial k n c)) (toLp (cubeA k (trial k n c))) : ℂ).re + (inner ℂ (toLp (trial k n c)) (toLp (quartA k (trial k n c))) : ℂ).re

Formalization note: Lean 4 identifier BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below.

Preamble

-- Generated from ChapterFockCubicUnbounded.lean — theorem BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below
import Definitions.Def_ChapterFockOneParticleGap
import Definitions.Def_ChapterFockNumberPreservingGap
import Definitions.Def_ChapterFockFieldPerturbation
import Mathlib
import Definitions.Def_ChapterFockCubicUnbounded
import Definitions.Def_ChapterFockSecondQuantization
import Definitions.Def_ChapterTempleSeparationNecessary
open BookProof.FockSecondQuantization
open BookProof.TempleSeparationNecessary
open BookProof.FockCubicUnbounded


noncomputable section


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

Formal statement

theorem BookProof.FockCubicUnbounded.trial_cubic_quartic_bounded_below (k n : ℕ) (hn : 1 ≤ n) (lam c : ℝ) :
    -(lam ^ 4 / 4 + 2 * lam ^ 2) * ‖toLp (trial k n c)‖ ^ 2
      ≤ numberQuad (trial k n c)
        + lam * (inner ℂ (toLp (trial k n c)) (toLp (cubeA k (trial k n c))) : ℂ).re
        + (inner ℂ (toLp (trial k n c)) (toLp (quartA k (trial k n c))) : ℂ).re := by sorry

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