`BookProof.ChapterA3o.sum_signed_permMat_sq` {N : ℕ} : (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ) * (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ) = (Nat.factorial N :...

Proved

BookProof.ChapterA3o.sum_signed_permMat_sq

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3o.

BookProof.ChapterA3o.sum_signed_permMat_sq {N : ℕ} : (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ) * (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ) = (Nat.factorial N : ℂ) • ∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ

Formalization note: Lean 4 identifier BookProof.ChapterA3o.sum_signed_permMat_sq.

Preamble

-- Generated from ChapterA3o.lean — theorem BookProof.ChapterA3o.sum_signed_permMat_sq
import Definitions.Def_ChapterA3
import Definitions.Def_ChapterA3j
import Mathlib
import Definitions.Def_ChapterA3o
import Definitions.Def_ChapterA3n
open BookProof.ChapterA3n
open BookProof.ChapterA3o


open Matrix
open scoped BigOperators


open BookProof.ChapterA3 BookProof.ChapterA3j BookProof.ChapterA3n

Formal statement

theorem BookProof.ChapterA3o.sum_signed_permMat_sq {N : ℕ} :
    (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ) *
        (∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ)
      = (Nat.factorial N : ℂ) • ∑ σ : Equiv.Perm (Fin N), signC σ • permMat σ := by sorry

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