Variable n' : nat.
Local Notation n := n'.+1.
Canonical matrix_algType :=
Eval hnf in AlgType R 'M[R]_n (fun k => scalemxAr k).
End MatrixAlgType.

Lemma det_perm n (s : 'S_n) : \det (perm_mx s) = (-1) ^+ s :> R.