`BookProof.ShiftedQuadratic.shiftedHPoly_apply_eq_differential` (a k : Vd d) (c b b' : Fin d → ℝ) (p : MvPolynomial (Fin d) ℂ) (x : Vd d) : pgFunT a k (shiftedHPoly a k c b b' p) x

Proved

BookProof.ShiftedQuadratic.shiftedHPoly_apply_eq_differential

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShiftedQuadraticEsa.

BookProof.ShiftedQuadratic.shiftedHPoly_apply_eq_differential (a k : Vd d) (c b b' : Fin d → ℝ) (p : MvPolynomial (Fin d) ℂ) (x : Vd d) : pgFunT a k (shiftedHPoly a k c b b' p) x = ∑ i, (((c i : ℝ) : ℂ) * (-deriv (fun t : ℝ => deriv (fun s : ℝ => pgFunT a k p (sec i x s)) t) (x i) + (((x i : ℝ) : ℂ) ^ 2 / 4) * pgFunT a k p x) + ((b i : ℝ) : ℂ) * (((x i : ℝ) : ℂ) * pgFunT a k p x) + ((b' i : ℝ) : ℂ) * (-Complex.I * deriv (fun t : ℝ => pgFunT a k p (sec i x t)) (x i)))

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

Preamble

-- Generated from ChapterShiftedQuadraticEsa.lean — theorem BookProof.ShiftedQuadratic.shiftedHPoly_apply_eq_differential
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_eq_differential (a k : Vd d) (c b b' : Fin d → ℝ)
    (p : MvPolynomial (Fin d) ℂ) (x : Vd d) :
    pgFunT a k (shiftedHPoly a k c b b' p) x
      = ∑ i, (((c i : ℝ) : ℂ)
            * (-deriv (fun t : ℝ => deriv (fun s : ℝ => pgFunT a k p (sec i x s)) t) (x i)
                + (((x i : ℝ) : ℂ) ^ 2 / 4) * pgFunT a k p x)
          + ((b i : ℝ) : ℂ) * (((x i : ℝ) : ℂ) * pgFunT a k p x)
          + ((b' i : ℝ) : ℂ)
              * (-Complex.I * deriv (fun t : ℝ => pgFunT a k p (sec i x t)) (x i))) := by sorry

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