`BookProof.ChapterA3r.trace_permMat` {N : ℕ} (σ : Equiv.Perm (Fin N)) : Matrix.trace (permMat σ) = ((Finset.univ.filter (fun a : Idx N => a ∘ σ = a)).card : ℂ)

Proved

BookProof.ChapterA3r.trace_permMat

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3r.

BookProof.ChapterA3r.trace_permMat {N : ℕ} (σ : Equiv.Perm (Fin N)) : Matrix.trace (permMat σ) = ((Finset.univ.filter (fun a : Idx N => a ∘ σ = a)).card : ℂ)

Formalization note: Lean 4 identifier BookProof.ChapterA3r.trace_permMat.

Preamble

-- Generated from ChapterA3r.lean — theorem BookProof.ChapterA3r.trace_permMat
import Definitions.Def_ChapterA3o
import Definitions.Def_ChapterA3q
import Mathlib
import Definitions.Def_ChapterA3r
import Definitions.Def_ChapterA3n
open BookProof.ChapterA3n
open BookProof.ChapterA3r


open Matrix
open scoped BigOperators


open BookProof.ChapterA3n BookProof.ChapterA3o BookProof.ChapterA3q

Formal statement

theorem BookProof.ChapterA3r.trace_permMat {N : ℕ} (σ : Equiv.Perm (Fin N)) :
    Matrix.trace (permMat σ) =
      ((Finset.univ.filter (fun a : Idx N => a ∘ σ = a)).card : ℂ) := by sorry

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