`BookProof.FockFieldPerturbation.fieldVec_relative_form_bound` {col : ℕ → (ℕ →₀ ℂ)} {mu t : ℝ} (hmu : 0 < mu) (ht : 0 < t) (hgap : IsPosCol (shiftCol col mu)) (f : ℕ →₀ ℂ)...

Proved

BookProof.FockFieldPerturbation.fieldVec_relative_form_bound

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFockFieldPerturbation.

BookProof.FockFieldPerturbation.fieldVec_relative_form_bound {col : ℕ → (ℕ →₀ ℂ)} {mu t : ℝ} (hmu : 0 < mu) (ht : 0 < t) (hgap : IsPosCol (shiftCol col mu)) (f : ℕ →₀ ℂ) (u : FockAlg) : |(inner ℂ (toLp u) (toLp (fieldVec f u)) : ℂ).re| ≤ t / mu * (inner ℂ (toLp u) (toLp (dGamma col u)) : ℂ).re + l2norm f ^ 2 / t * ‖toLp u‖ ^ 2

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

Preamble

-- Generated from ChapterFockFieldPerturbation.lean — theorem BookProof.FockFieldPerturbation.fieldVec_relative_form_bound
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.fieldVec_relative_form_bound {col : ℕ → (ℕ →₀ ℂ)} {mu t : ℝ} (hmu : 0 < mu)
    (ht : 0 < t) (hgap : IsPosCol (shiftCol col mu)) (f : ℕ →₀ ℂ) (u : FockAlg) :
    |(inner ℂ (toLp u) (toLp (fieldVec f u)) : ℂ).re|
      ≤ t / mu * (inner ℂ (toLp u) (toLp (dGamma col u)) : ℂ).re
        + l2norm f ^ 2 / t * ‖toLp u‖ ^ 2 := by sorry

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