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1 (**************************************************************************)
2 (*       __                                                               *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU Lesser General Public License Version 2.1         *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 set "baseuri" "cic:/matita/nat/times".
16
17 include "nat/plus.ma".
18
19 let rec times n m \def 
20  match n with 
21  [ O \Rightarrow O
22  | (S p) \Rightarrow m+(times p m) ].
23
24 (*CSC: the URI must disappear: there is a bug now *)
25 interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
26
27 theorem times_n_O: \forall n:nat. O = n*O.
28 intros.elim n.
29 simplify.reflexivity.
30 simplify.assumption.
31 qed.
32
33 theorem times_n_Sm : 
34 \forall n,m:nat. n+(n*m) = n*(S m).
35 intros.elim n.
36 simplify.reflexivity.
37 simplify.apply eq_f.rewrite < H.
38 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
39 transitivity ((m+n1)+n1*m).
40 apply eq_f2.
41 apply sym_plus.
42 reflexivity.
43 apply assoc_plus.
44 qed.
45
46 theorem times_n_SO : \forall n:nat. n = n * S O.
47 intros.
48 rewrite < times_n_Sm.
49 rewrite < times_n_O.
50 rewrite < plus_n_O.
51 reflexivity.
52 qed.
53
54 theorem symmetric_times : symmetric nat times. 
55 unfold symmetric.
56 intros.elim x.
57 simplify.apply times_n_O.
58 simplify.rewrite > H.apply times_n_Sm.
59 qed.
60
61 variant sym_times : \forall n,m:nat. n*m = m*n \def
62 symmetric_times.
63
64 theorem distributive_times_plus : distributive nat times plus.
65 unfold distributive.
66 intros.elim x.
67 simplify.reflexivity.
68 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
69 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
70 rewrite > assoc_plus.reflexivity.
71 qed.
72
73 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
74 \def distributive_times_plus.
75
76 theorem associative_times: associative nat times.
77 unfold associative.intros.
78 elim x.simplify.apply refl_eq.
79 simplify.rewrite < sym_times.
80 rewrite > distr_times_plus.
81 rewrite < sym_times.
82 rewrite < (sym_times (times n y) z).
83 rewrite < H.apply refl_eq.
84 qed.
85
86 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
87 associative_times.