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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 include "nat/nat.ma".
16
17 let rec plus n m \def 
18  match n with 
19  [ O \Rightarrow m
20  | (S p) \Rightarrow S (plus p m) ].
21
22 (*CSC: the URI must disappear: there is a bug now *)
23 interpretation "natural plus" 'plus x y = (cic:/matita/nat/plus/plus.con x y).
24
25 theorem plus_n_O: \forall n:nat. n = n+O.
26 intros.elim n.
27 simplify.reflexivity.
28 simplify.apply eq_f.assumption.
29 qed.
30
31 theorem plus_n_Sm : \forall n,m:nat. S (n+m) = n+(S m).
32 intros.elim n.
33 simplify.reflexivity.
34 simplify.apply eq_f.assumption.
35 qed.
36
37 theorem plus_n_SO : \forall n:nat. S n = n+(S O).
38 intro.rewrite > plus_n_O.
39 apply plus_n_Sm.
40 qed.
41
42 theorem sym_plus: \forall n,m:nat. n+m = m+n.
43 intros.elim n.
44 simplify.apply plus_n_O.
45 simplify.rewrite > H.apply plus_n_Sm.
46 qed.
47
48 theorem associative_plus : associative nat plus.
49 unfold associative.intros.elim x.
50 simplify.reflexivity.
51 simplify.apply eq_f.assumption.
52 qed.
53
54 theorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
55 \def associative_plus.
56
57 theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.n+m).
58 intro.simplify.intros 2.elim n.
59 exact H.
60 apply H.apply inj_S.apply H1.
61 qed.
62
63 theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
64 \def injective_plus_r.
65
66 theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.n+m).
67 intro.simplify.intros.
68 apply (injective_plus_r m).
69 rewrite < sym_plus.
70 rewrite < (sym_plus y).
71 assumption.
72 qed.
73
74 theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
75 \def injective_plus_l.