1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/relevant_equations.ma".
17 include "nat/times.ma".
19 theorem times_plus_l: \forall n,m,p:nat. (n+m)*p = n*p + m*p.
21 apply trans_eq ? ? (p*(n+m)).
23 apply trans_eq ? ? (p*n+p*m).
24 apply distr_times_plus.
30 theorem times_plus_plus: \forall n,m,p,q:nat. (n + m)*(p + q) =
31 n*p + n*q + m*p + m*q.
33 apply trans_eq nat ? ((n*(p+q) + m*(p+q))).
35 rewrite > distr_times_plus.
36 rewrite > distr_times_plus.
37 rewrite < assoc_plus.reflexivity.