`BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation` {col : ℕ → (ℕ →₀ ℂ)} {mu : ℝ} (hmu : 0 < mu) (hgap : IsPosCol (shiftCol col mu)) {f : ℕ →₀ ℂ} (hf : l2norm...

Proved

BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockFieldPerturbation.

BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation {col : ℕ → (ℕ →₀ ℂ)} {mu : ℝ} (hmu : 0 < mu) (hgap : IsPosCol (shiftCol col mu)) {f : ℕ →₀ ℂ} (hf : l2norm f ≤ mu) {u : FockAlg} (h0 : u 0 = 0) : (mu - 2 * l2norm f) * ‖toLp u‖ ^ 2 ≤ (inner ℂ (toLp u) (toLp (dGamma col u)) : ℂ).re + (inner ℂ (toLp u) (toLp (fieldVec f u)) : ℂ).re

Formalization note: Lean 4 identifier BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation.

Preamble

-- Generated from ChapterFockFieldPerturbation.lean — theorem BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation
import Definitions.Def_ChapterFockOneParticleGap
import Definitions.Def_ChapterFockNumberPreservingGap
import Definitions.Def_ChapterFockInteractionStability
import Mathlib
import Definitions.Def_ChapterFockFieldPerturbation
import Definitions.Def_ChapterFockSecondQuantization
open BookProof.FockSecondQuantization
open BookProof.FockFieldPerturbation


noncomputable section


open BookProof.FockSecondQuantization BookProof.FockOneParticleGap
open BookProof.FockNumberPreservingGap BookProof.FockInteractionStability

Formal statement

theorem BookProof.FockFieldPerturbation.fock_gap_of_field_perturbation {col : ℕ → (ℕ →₀ ℂ)} {mu : ℝ} (hmu : 0 < mu)
    (hgap : IsPosCol (shiftCol col mu)) {f : ℕ →₀ ℂ} (hf : l2norm f ≤ mu)
    {u : FockAlg} (h0 : u 0 = 0) :
    (mu - 2 * l2norm f) * ‖toLp u‖ ^ 2
      ≤ (inner ℂ (toLp u) (toLp (dGamma col u)) : ℂ).re
        + (inner ℂ (toLp u) (toLp (fieldVec f u)) : ℂ).re := by sorry

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