The Lean 4 theorem `commForm_eq_im` in the `ChapterSqSumFarisLavine` chapter of the timepiece formalization

-- Generated from ChapterSqSumFarisLavine.lean — theorem BookProof.SqSumFarisLavine.commForm_eq_im
import Definitions.Def_ChapterQgHermiteCore
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterHermiteQuadraticEsa
import Mathlib
import Definitions.Def_ChapterSqSumFarisLavine
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterGaussCoreQuadBounds
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterQgHermiteFriedrichs
import Definitions.Def_ChapterQgHermiteOscillatorEsa
import Definitions.Def_ChapterQgOuterFockEsa
open BookProof.GaussCoreQuadBounds
open BookProof.HermiteProductCore
open BookProof.QgHermiteFriedrichs
open BookProof.QgHermiteOscillator
open BookProof.QgOuterFock
open BookProof.SqSumFarisLavine

variable {D : ℕ} {R : Type*} [Fintype R]



open Finset MvPolynomial
open BookProof.HermiteProductCore BookProof.QgHermiteCore BookProof.QgHermiteFriedrichs
open BookProof.QgHermiteOscillator BookProof.FarisLavine
open BookProof.HermiteQuadraticEsa
open BookProof.QgOuterFock
open BookProof.GaussCoreQuadBounds

noncomputable section

theorem BookProof.SqSumFarisLavine.commForm_eq_im (kappa : Fin D → ℝ) (v : R → Fin D → ℝ) (p : MvPolynomial (Fin D) ℂ) :
    commForm (sqSumOp kappa v) harmCore ⟨pgLp p, pgLp_mem_core p⟩
      = -(gaussInt (cpoly p * commPoly kappa v p)).im := by sorry