New version of the library, a bit more structured.

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

diff --git a/helm/matita/library/nat/times.ma b/helm/matita/library/nat/times.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/times.ma".
+
+include "logic/equality.ma".
+include "nat/nat.ma".
+include "nat/plus.ma".
+
+let rec times n m \def 
+ match n with 
+ [ O \Rightarrow O
+ | (S p) \Rightarrow (plus m (times p m)) ].
+
+theorem times_n_O: \forall n:nat. eq nat O (times n O).
+intros.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem times_n_Sm : 
+\forall n,m:nat. eq nat (plus n (times n  m)) (times n (S m)).
+intros.elim n.
+simplify.reflexivity.
+simplify.apply eq_f.rewrite < H.
+transitivity (plus (plus e1 m) (times e1 m)).symmetry.apply assoc_plus.
+transitivity (plus (plus m e1) (times e1 m)).
+apply eq_f2.
+apply sym_plus.
+reflexivity.
+apply assoc_plus.
+qed.
+
+(* same problem with symmetric: see plus 
+theorem symmetric_times : symmetric nat times. *)
+
+theorem sym_times : 
+\forall n,m:nat. eq nat (times n m) (times m n).
+intros.elim n.
+simplify.apply times_n_O.
+simplify.rewrite > H.apply times_n_Sm.
+qed.
+
+theorem times_plus_distr: \forall n,m,p:nat.
+eq nat (times n (plus m p)) (plus (times n m) (times n p)).
+intros.elim n.
+simplify.reflexivity.
+simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
+apply eq_f.rewrite < assoc_plus. rewrite < sym_plus ? p.
+rewrite > assoc_plus.reflexivity.
+qed.