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

diff --git a/matita/library/nat/times.ma b/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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "nat/plus.ma".
+
+let rec times n m \def 
+ match n with 
+ [ O \Rightarrow O
+ | (S p) \Rightarrow m+(times p m) ].
+
+interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
+
+theorem times_n_O: \forall n:nat. O = n*O.
+intros.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem times_n_Sm : 
+\forall n,m:nat. n+(n*m) = n*(S m).
+intros.elim n.
+simplify.reflexivity.
+simplify.apply eq_f.rewrite < H.
+transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
+transitivity ((m+n1)+n1*m).
+apply eq_f2.
+apply sym_plus.
+reflexivity.
+apply assoc_plus.
+qed.
+
+theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
+apply nat_elim2;intros
+  [left.reflexivity
+  |right.reflexivity
+  |apply False_ind.
+   simplify in H1.
+   apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
+  ]
+qed.
+
+theorem times_n_SO : \forall n:nat. n = n * S O.
+intros.
+rewrite < times_n_Sm.
+rewrite < times_n_O.
+rewrite < plus_n_O.
+reflexivity.
+qed.
+
+theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
+intros.
+simplify.
+rewrite < plus_n_O.
+reflexivity.
+qed.
+
+alias num (instance 0) = "natural number".
+lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
+intro.simplify.rewrite < plus_n_Sm.reflexivity.
+qed.
+
+theorem or_eq_eq_S: \forall n.\exists m. 
+n = (S(S O))*m \lor n = S ((S(S O))*m).
+intro.elim n
+  [apply (ex_intro ? ? O).
+   left.reflexivity
+  |elim H.elim H1
+    [apply (ex_intro ? ? a).
+     right.apply eq_f.assumption
+    |apply (ex_intro ? ? (S a)).
+     left.rewrite > H2.
+     apply sym_eq.
+     apply times_SSO
+    ]
+  ]
+qed.
+
+theorem symmetric_times : symmetric nat times. 
+unfold symmetric.
+intros.elim x.
+simplify.apply times_n_O.
+simplify.rewrite > H.apply times_n_Sm.
+qed.
+
+variant sym_times : \forall n,m:nat. n*m = m*n \def
+symmetric_times.
+
+theorem distributive_times_plus : distributive nat times plus.
+unfold distributive.
+intros.elim x.
+simplify.reflexivity.
+simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
+apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
+rewrite > assoc_plus.reflexivity.
+qed.
+
+variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
+\def distributive_times_plus.
+
+theorem associative_times: associative nat times.
+unfold associative.intros.
+elim x.simplify.apply refl_eq.
+simplify.rewrite < sym_times.
+rewrite > distr_times_plus.
+rewrite < sym_times.
+rewrite < (sym_times (times n y) z).
+rewrite < H.apply refl_eq.
+qed.
+
+variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
+associative_times.