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/library_autobatch/nat/relevant_equations".
17 include "auto/nat/times.ma".
18 include "auto/nat/minus.ma".
19 include "auto/nat/gcd.ma".
20 (* if gcd is compiled before this, the applys will take too much *)
22 theorem times_plus_l: \forall n,m,p:nat. (n+m)*p = n*p + m*p.
24 apply (trans_eq ? ? (p*(n+m)))
26 | apply (trans_eq ? ? (p*n+p*m));autobatch
27 (*[ apply distr_times_plus
34 theorem times_minus_l: \forall n,m,p:nat. (n-m)*p = n*p - m*p.
36 apply (trans_eq ? ? (p*(n-m)))
38 | apply (trans_eq ? ? (p*n-p*m));autobatch
39 (*[ apply distr_times_minus
46 theorem times_plus_plus: \forall n,m,p,q:nat. (n + m)*(p + q) =
47 n*p + n*q + m*p + m*q.
50 (*apply (trans_eq nat ? ((n*(p+q) + m*(p+q))))
52 | rewrite > distr_times_plus.
53 rewrite > distr_times_plus.