Prove2Me

`BookProof.ScalaronHermiteEsa.moments_of_monomial_moments` {u : Vd d → ℂ} (hu : GaussExpDecay u) (hmon : ∀ a : Fin d →₀ ℕ, ∫ x : Vd d, pgFun (monomial a (1 : ℂ)) x * u x = 0) (p :

Proved
BookProof.ScalaronHermiteEsa.moments_of_monomial_moments

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterScalaronHermiteEsa.

BookProof.ScalaronHermiteEsa.moments_of_monomial_moments {u : Vd d → ℂ} (hu : GaussExpDecay u) (hmon : ∀ a : Fin d →₀ ℕ, ∫ x : Vd d, pgFun (monomial a (1 : ℂ)) x * u x = 0) (p : MvPolynomial (Fin d) ℂ) : ∫ x : Vd d, pgFun p x * u x = 0

Formalization note: Lean 4 identifier BookProof.ScalaronHermiteEsa.moments_of_monomial_moments.

Preamble
-- Generated from ChapterScalaronHermiteEsa.lean — theorem BookProof.ScalaronHermiteEsa.moments_of_monomial_moments
import Definitions.Def_ChapterQgHermiteCore
import Definitions.Def_ChapterQgHermiteFriedrichs
import Definitions.Def_ChapterQgHermiteOscillatorEsa
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterStarobinskyPotential
import Definitions.Def_ChapterStoneBridge
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterScalaronHermiteEsa
import Definitions.Def_ChapterHermiteProductCore
open BookProof.HermiteProductCore
open BookProof.ScalaronHermiteEsa



open MeasureTheory Complex MvPolynomial FourierTransform
open BookProof.HermiteProductCore BookProof.QgHermiteCore BookProof.QgHermiteFriedrichs
open BookProof.QgHermiteOscillator BookProof.FarisLavine BookProof.Starobinsky
open BookProof.StoneBridge BookProof.EsaClosure BookProof.ChapterStoneResolvent

noncomputable section

variable {d : ℕ}
Formal statement
theorem BookProof.ScalaronHermiteEsa.moments_of_monomial_moments {u : Vd d → ℂ} (hu : GaussExpDecay u)
    (hmon : ∀ a : Fin d →₀ ℕ, ∫ x : Vd d, pgFun (monomial a (1 : ℂ)) x * u x = 0)
    (p : MvPolynomial (Fin d) ℂ) : ∫ x : Vd d, pgFun p x * u x = 0 := by sorry

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