test branch

[helm.git] / helm / matita / library / nat / relevant_equations.ma

diff --git a/helm/matita/library/nat/relevant_equations.ma b/helm/matita/library/nat/relevant_equations.ma
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+(**************************************************************************)
+(*       __                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/relevant_equations.ma".
+
+include "nat/times.ma".
+include "nat/minus.ma".
+
+theorem times_plus_l: \forall n,m,p:nat. (n+m)*p = n*p + m*p.
+intros.
+apply (trans_eq ? ? (p*(n+m))).
+apply sym_times.
+apply (trans_eq ? ? (p*n+p*m)).
+apply distr_times_plus.
+apply eq_f2.
+apply sym_times.
+apply sym_times.
+qed.
+
+theorem times_minus_l: \forall n,m,p:nat. (n-m)*p = n*p - m*p.
+intros.
+apply (trans_eq ? ? (p*(n-m))).
+apply sym_times.
+apply (trans_eq ? ? (p*n-p*m)).
+apply distr_times_minus.
+apply eq_f2.
+apply sym_times.
+apply sym_times.
+qed.
+
+theorem times_plus_plus: \forall n,m,p,q:nat. (n + m)*(p + q) =
+n*p + n*q + m*p + m*q.
+intros.
+apply (trans_eq nat ? ((n*(p+q) + m*(p+q)))).
+apply times_plus_l.
+rewrite > distr_times_plus.
+rewrite > distr_times_plus.
+rewrite < assoc_plus.reflexivity.
+qed.