`BookProof.NavierStokesGaugeY.genY_genY_expand` (j k : Fin 3) (p : NSAlg) : genY j (genY k p) = pderiv (NSVar.y j) (pderiv (NSVar.y k) p) - ∑ i : Fin 3, X (NSVar.uD i j) * pderiv (

Proved

BookProof.NavierStokesGaugeY.genY_genY_expand

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNavierStokesGaugeY.

BookProof.NavierStokesGaugeY.genY_genY_expand (j k : Fin 3) (p : NSAlg) : genY j (genY k p) = pderiv (NSVar.y j) (pderiv (NSVar.y k) p) - ∑ i : Fin 3, X (NSVar.uD i j) * pderiv (NSVar.y k) (pderiv (NSVar.u i) p) - ∑ i : Fin 3, X (NSVar.uD i k) * pderiv (NSVar.y j) (pderiv (NSVar.u i) p) + ∑ i : Fin 3, ∑ m : Fin 3, X (NSVar.uD i j) * X (NSVar.uD m k) * pderiv (NSVar.u i) (pderiv (NSVar.u m) p)

Formalization note: Lean 4 identifier BookProof.NavierStokesGaugeY.genY_genY_expand.

Preamble

-- Generated from ChapterNavierStokesGaugeY.lean — theorem BookProof.NavierStokesGaugeY.genY_genY_expand
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Mathlib
import Definitions.Def_ChapterNavierStokesGaugeY
open BookProof.NavierStokesGaugeY



open MvPolynomial BookProof.NavierStokesFlow

Formal statement

theorem BookProof.NavierStokesGaugeY.genY_genY_expand (j k : Fin 3) (p : NSAlg) :
    genY j (genY k p) =
      pderiv (NSVar.y j) (pderiv (NSVar.y k) p)
      - ∑ i : Fin 3, X (NSVar.uD i j) * pderiv (NSVar.y k) (pderiv (NSVar.u i) p)
      - ∑ i : Fin 3, X (NSVar.uD i k) * pderiv (NSVar.y j) (pderiv (NSVar.u i) p)
      + ∑ i : Fin 3, ∑ m : Fin 3,
          X (NSVar.uD i j) * X (NSVar.uD m k) * pderiv (NSVar.u i) (pderiv (NSVar.u m) p) := by sorry

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