`BookProof.QuantumGravityBrstCharge.sum4_sub_distrib` [AddCommGroup α] (F G : Fin d → Fin d → Fin d → Fin d → α) : (∑ j, ∑ k, ∑ l, ∑ m, F j k l m) - (∑ j, ∑ k, ∑ l, ∑ m,...

Proved

BookProof.QuantumGravityBrstCharge.sum4_sub_distrib

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityBrstCharge.

BookProof.QuantumGravityBrstCharge.sum4_sub_distrib [AddCommGroup α] (F G : Fin d → Fin d → Fin d → Fin d → α) : (∑ j, ∑ k, ∑ l, ∑ m, F j k l m) - (∑ j, ∑ k, ∑ l, ∑ m, G j k l m) = ∑ j, ∑ k, ∑ l, ∑ m, (F j k l m - G j k l m)

Formalization note: Lean 4 identifier BookProof.QuantumGravityBrstCharge.sum4_sub_distrib.

Preamble

-- Generated from ChapterQuantumGravityBrstCharge.lean — theorem BookProof.QuantumGravityBrstCharge.sum4_sub_distrib
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterQuantumGravity3DGauge
import Mathlib
import Definitions.Def_ChapterQuantumGravityBrstCharge
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}
variable {d : ℕ}
variable {α : Type*}

Formal statement

theorem BookProof.QuantumGravityBrstCharge.sum4_sub_distrib [AddCommGroup α] (F G : Fin d → Fin d → Fin d → Fin d → α) :
    (∑ j, ∑ k, ∑ l, ∑ m, F j k l m) - (∑ j, ∑ k, ∑ l, ∑ m, G j k l m)
      = ∑ j, ∑ k, ∑ l, ∑ m, (F j k l m - G j k l m) := by sorry

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