`BookProof.ShiftedQuadratic.shiftedHPoly_apply` (a k : Vd d) (c b b' : Fin d → ℝ) (p : MvPolynomial (Fin d) ℂ) : shiftedHPoly a k c b b' p = ∑ i, (((c i : ℝ) : ℂ) • oscTPoly a k i

Proved

BookProof.ShiftedQuadratic.shiftedHPoly_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShiftedQuadraticEsa.

BookProof.ShiftedQuadratic.shiftedHPoly_apply (a k : Vd d) (c b b' : Fin d → ℝ) (p : MvPolynomial (Fin d) ℂ) : shiftedHPoly a k c b b' p = ∑ i, (((c i : ℝ) : ℂ) • oscTPoly a k i p + ((b i : ℝ) : ℂ) • mulXTPoly a i p + ((b' i : ℝ) : ℂ) • momTPoly k i p)

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

Preamble

-- Generated from ChapterShiftedQuadraticEsa.lean — theorem BookProof.ShiftedQuadratic.shiftedHPoly_apply
import Definitions.Def_ChapterHermiteProductBasis
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_ChapterNavierStokesDifferentialL2
import Definitions.Def_ChapterShiftedHermiteCore
open BookProof.HermiteProductCore
open BookProof.NavierStokesFlow.DifferentialL2
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.shiftedHPoly_apply (a k : Vd d) (c b b' : Fin d → ℝ) (p : MvPolynomial (Fin d) ℂ) :
    shiftedHPoly a k c b b' p
      = ∑ i, (((c i : ℝ) : ℂ) • oscTPoly a k i p + ((b i : ℝ) : ℂ) • mulXTPoly a i p
          + ((b' i : ℝ) : ℂ) • momTPoly k i p) := by sorry

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