1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_props".
19 theorem add_O_1: \forall q. add O q q.
20 intros. elim q; clear q; auto.
23 theorem add_S_1: \forall p,q,r. add p q r \to add (S p) q (S r).
24 intros 2. elim q; clear q;
25 [ lapply add_gen_O_2 to H as H0. clear H.
26 rewrite > H0. clear H0 p
27 | lapply add_gen_S_2 to H1 as H0. clear H1.
29 rewrite > H2. clear H2 r
33 theorem add_sym: \forall p,q,r. add p q r \to add q p r.
34 intros 2. elim q; clear q;
35 [ lapply add_gen_O_2 to H as H0. clear H.
36 rewrite > H0. clear H0 p
37 | lapply add_gen_S_2 to H1 as H0. clear H1.
39 rewrite > H2. clear H2 r
43 theorem add_shift_S_sx: \forall p,q,r. add p (S q) r \to add (S p) q r.
45 lapply add_gen_S_2 to H as H0. clear H.
47 rewrite > H1. clear H1 r.
51 theorem add_shift_S_dx: \forall p,q,r. add (S p) q r \to add p (S q) r.
53 lapply add_gen_S_1 to H as H0. clear H.
55 rewrite > H1. clear H1 r.
59 theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
60 \forall q2,r2. add r1 q2 r2 \to
61 \exists q. add q1 q2 q \land add p q r2.
62 intros 2; elim q1; clear q1; intros;
63 [ lapply add_gen_O_2 to H as H0. clear H.
64 rewrite > H0. clear H0 p
65 | lapply add_gen_S_2 to H1 as H0. clear H1.
67 rewrite > H3. rewrite > H3 in H2. clear H3 r1.
68 lapply add_gen_S_1 to H2 as H0. clear H2.
70 rewrite > H2. clear H2 r2.
71 lapply H to H4, H3 as H0. clear H H4 H3.
73 ]; apply ex_intro; [| auto || auto ]. (**)
76 theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to
77 \forall p2,r2. add p2 r1 r2 \to
78 \exists p. add p1 p2 p \land add p q r2.
79 intros 2; elim q; clear q; intros;
80 [ lapply add_gen_O_2 to H as H0. clear H.
81 rewrite > H0. clear H0 p1
82 | lapply add_gen_S_2 to H1 as H0. clear H1.
84 rewrite > H3. rewrite > H3 in H2. clear H3 r1.
85 lapply add_gen_S_2 to H2 as H0. clear H2.
87 rewrite > H2. clear H2 r2.
88 lapply H to H4, H3 as H0. clear H H4 H3.
90 ]; apply ex_intro; [| auto || auto ]. (**)
93 theorem add_conf: \forall p,q,r1. add p q r1 \to
94 \forall r2. add p q r2 \to r1 = r2.
95 intros 2. elim q; clear q; intros;
96 [ lapply add_gen_O_2 to H as H0. clear H.
97 rewrite > H0 in H1. clear H0 p
98 | lapply add_gen_S_2 to H1 as H0. clear H1.
100 rewrite > H3. clear H3 r1.
101 lapply add_gen_S_2 to H2 as H0. clear H2.
103 rewrite > H2. clear H2 r2.