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[helm.git] / matita / contribs / library_auto / auto / nat / relevant_equations.ma

diff --git a/matita/contribs/library_auto/auto/nat/relevant_equations.ma b/matita/contribs/library_auto/auto/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/library_autobatch/nat/relevant_equations".
+
+include "auto/nat/times.ma".
+include "auto/nat/minus.ma".
+include "auto/nat/gcd.ma". 
+(* if gcd is compiled before this, the applys will take too much *)
+
+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));autobatch
+  (*[ apply distr_times_plus
+  | apply eq_f2;
+      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));autobatch
+  (*[ apply distr_times_minus
+  | apply eq_f2;
+      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.
+autobatch.
+(*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.