`BookProof.ShiftedQuadratic.pgFunT_apply_sum` {ι : Type*} (a k : Vd d) (s : Finset ι) (f : ι → MvPolynomial (Fin d) ℂ) (x : Vd d) : pgFunT a k (∑ v ∈ s, f v) x = ∑ v ∈ s, pgFunT a

Proved

BookProof.ShiftedQuadratic.pgFunT_apply_sum

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShiftedQuadraticEsa.

BookProof.ShiftedQuadratic.pgFunT_apply_sum {ι : Type*} (a k : Vd d) (s : Finset ι) (f : ι → MvPolynomial (Fin d) ℂ) (x : Vd d) : pgFunT a k (∑ v ∈ s, f v) x = ∑ v ∈ s, pgFunT a k (f v) x

Formalization note: Lean 4 identifier BookProof.ShiftedQuadratic.pgFunT_apply_sum.

Preamble

-- Generated from ChapterShiftedQuadraticEsa.lean — theorem BookProof.ShiftedQuadratic.pgFunT_apply_sum
import Definitions.Def_ChapterHermiteProductBasis
import Definitions.Def_ChapterNavierStokesDifferentialL2
import Definitions.Def_ChapterHyperbolicQuadraticEsa
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterStoneBridge
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterShiftedQuadraticEsa
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterShiftedHermiteCore
open BookProof.HermiteProductCore
open BookProof.ShiftedHermiteCore
open BookProof.ShiftedQuadratic



open MeasureTheory MvPolynomial
open BookProof.HermiteProductCore BookProof.HermiteProductBasis
open BookProof.NavierStokesFlow.DifferentialL2
open BookProof.HyperbolicQuadratic
open BookProof.ShiftedHermiteCore
open BookProof.FarisLavine
open BookProof.StoneBridge BookProof.EsaClosure BookProof.ChapterStoneResolvent

noncomputable section

variable {d : ℕ}

Formal statement

theorem BookProof.ShiftedQuadratic.pgFunT_apply_sum {ι : Type*} (a k : Vd d) (s : Finset ι)
    (f : ι → MvPolynomial (Fin d) ℂ) (x : Vd d) :
    pgFunT a k (∑ v ∈ s, f v) x = ∑ v ∈ s, pgFunT a k (f v) x := by sorry

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