`BookProof.GaussCoordCombo.gaussInt_prod_coordFactor` {S : Finset (Fin d)} {W : Fin d → MvPolynomial (Fin d) ℂ} {s : Fin d → ℝ} (hW : ∀ j ∈ S, CoordFactor j (W j) (s j)) {R : MvPol

Proved

BookProof.GaussCoordCombo.gaussInt_prod_coordFactor

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaussCoordCombo.

BookProof.GaussCoordCombo.gaussInt_prod_coordFactor {S : Finset (Fin d)} {W : Fin d → MvPolynomial (Fin d) ℂ} {s : Fin d → ℝ} (hW : ∀ j ∈ S, CoordFactor j (W j) (s j)) {R : MvPolynomial (Fin d) ℂ} (hR : ∀ j ∈ S, pderiv j R = 0) : gaussInt ((∏ j ∈ S, W j) * ((∏ j ∈ S, W j) * R)) = ((∏ j ∈ S, s j : ℝ) : ℂ) * gaussInt R

Formalization note: Lean 4 identifier BookProof.GaussCoordCombo.gaussInt_prod_coordFactor.

Preamble

-- Generated from ChapterGaussCoordCombo.lean — theorem BookProof.GaussCoordCombo.gaussInt_prod_coordFactor
import Mathlib
import Definitions.Def_ChapterGaussCoordCombo
import Definitions.Def_ChapterHermiteProductCore
open BookProof.HermiteProductCore
open BookProof.GaussCoordCombo



open MvPolynomial BookProof.HermiteProductCore

noncomputable section

variable {d : ℕ}

Formal statement

theorem BookProof.GaussCoordCombo.gaussInt_prod_coordFactor {S : Finset (Fin d)} {W : Fin d → MvPolynomial (Fin d) ℂ}
    {s : Fin d → ℝ} (hW : ∀ j ∈ S, CoordFactor j (W j) (s j))
    {R : MvPolynomial (Fin d) ℂ} (hR : ∀ j ∈ S, pderiv j R = 0) :
    gaussInt ((∏ j ∈ S, W j) * ((∏ j ∈ S, W j) * R))
      = ((∏ j ∈ S, s j : ℝ) : ℂ) * gaussInt R := by sorry

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