Section CommMx.
(***********************************************************************)
(************* Commutation property specialized to 'M[R]_n *************)
(***********************************************************************)
(* GRing.comm is bound to (non trivial) rings, and matrices form a     *)
(* (non trivial) ring only when they are square and of manifestly      *)
(* positive size. However during proofs in endomorphism reduction, we  *)
(* take restrictions, which are matrices of size #|V| (with V a matrix *)
(* space) and it becomes cumbersome to state commutation between       *)
(* restrictions, unless we relax the setting, and this relaxation      *)
(* corresponds to comm_mx A B := A *m B = B *m A.                      *)
(* As witnessed by comm_mxE, when A and B have type 'M_n.+1,           *)
(*   comm_mx A B is convertible to GRing.comm A B.                     *)
(* The boolean version comm_mxb is designed to be used with seq.allrel *)
(***********************************************************************)
Context {R : ringType} {n : nat}.
Implicit Types (f g p : 'M[R]_n) (fs : seq 'M[R]_n) (d : 'rV[R]_n) (I : Type).
Definition comm_mx  f g : Prop := f *m g =  g *m f.
Definition comm_mxb f g : bool := f *m g == g *m f.

Lemma scalemxAr a A B : a *: (A *m B) = A *m (a *: B).