`BookProof.QuantumGravityBrstCharge.brst_full_nilpotent` (hCAR : GhostCAR χ β) (hCA : ConstraintAlgebra f G χ β) (hf12 : ∀ a b c, f a b c = -f b a c) (hjac : ∀ a b c h : Fin n, ∑ e

Proved

BookProof.QuantumGravityBrstCharge.brst_full_nilpotent

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityBrstCharge.

BookProof.QuantumGravityBrstCharge.brst_full_nilpotent (hCAR : GhostCAR χ β) (hCA : ConstraintAlgebra f G χ β) (hf12 : ∀ a b c, f a b c = -f b a c) (hjac : ∀ a b c h : Fin n, ∑ e, (f a b e * f e c h + f b c e * f e a h + f c a e * f e b h) = 0) : brstCharge f G χ β * brstCharge f G χ β = 0

Formalization note: Lean 4 identifier BookProof.QuantumGravityBrstCharge.brst_full_nilpotent.

Preamble

-- Generated from ChapterQuantumGravityBrstCharge.lean — theorem BookProof.QuantumGravityBrstCharge.brst_full_nilpotent
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterQuantumGravity3DGauge
import Mathlib
import Definitions.Def_ChapterQuantumGravityBrstCharge
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterSmBrstGhost
open BookProof.BRSTNilpotent
open BookProof.SmBrstGhost
open BookProof.QuantumGravityBrstCharge



open MvPolynomial BookProof.BRSTNilpotent BookProof.YangMillsHermite
open BookProof.QuantumGravity3DGauge

noncomputable section

variable {R : Type*} [Ring R] [Algebra ℝ R] {n : ℕ}
variable {f : Fin n → Fin n → Fin n → ℝ} {G χ β : Fin n → R}

Formal statement

theorem BookProof.QuantumGravityBrstCharge.brst_full_nilpotent (hCAR : GhostCAR χ β) (hCA : ConstraintAlgebra f G χ β)
    (hf12 : ∀ a b c, f a b c = -f b a c)
    (hjac : ∀ a b c h : Fin n,
      ∑ e, (f a b e * f e c h + f b c e * f e a h + f c a e * f e b h) = 0) :
    brstCharge f G χ β * brstCharge f G χ β = 0 := by sorry

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