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. clear p
27 | lapply add_gen_S_2 to H1 as H0. clear H1.
29 rewrite > H2. clear H2. clear 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. clear p
37 | lapply add_gen_S_2 to H1 as H0. clear H1.
39 rewrite > H2. clear H2. clear 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. clear r.
52 theorem add_shift_S_dx: \forall p,q,r. add (S p) q r \to add p (S q) r.
54 lapply add_gen_S_1 to H as H0. clear H.
56 rewrite > H1. clear H1. clear r.
60 theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
61 \forall q2,r2. add r1 q2 r2 \to
62 \exists q. add q1 q2 q \land add p q r2.
63 intros 2; elim q1; clear q1; intros;
64 [ lapply add_gen_O_2 to H as H0. clear H.
65 rewrite > H0. clear H0. clear p
66 | lapply add_gen_S_2 to H1 as H0. clear H1.
68 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
69 lapply add_gen_S_1 to H2 as H0. clear H2.
71 rewrite > H2. clear H2. clear r2.
72 lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
74 ]; apply ex_intro; [| auto || auto ]. (**)
77 theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to
78 \forall p2,r2. add p2 r1 r2 \to
79 \exists p. add p1 p2 p \land add p q r2.
80 intros 2; elim q; clear q; intros;
81 [ lapply add_gen_O_2 to H as H0. clear H.
82 rewrite > H0. clear H0. clear p1
83 | lapply add_gen_S_2 to H1 as H0. clear H1.
85 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
86 lapply add_gen_S_2 to H2 as H0. clear H2.
88 rewrite > H2. clear H2. clear r2.
89 lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
91 ]; apply ex_intro; [| auto || auto ]. (**)
94 theorem add_conf: \forall p,q,r1. add p q r1 \to
95 \forall r2. add p q r2 \to r1 = r2.
96 intros 2. elim q; clear q; intros;
97 [ lapply add_gen_O_2 to H as H0. clear H.
98 rewrite > H0 in H1. clear H0. clear p
99 | lapply add_gen_S_2 to H1 as H0. clear H1.
101 rewrite > H3. clear H3. clear r1.
102 lapply add_gen_S_2 to H2 as H0. clear H2.
104 rewrite > H2. clear H2. clear r2.
110 theorem add_gen_eq_2_3: \forall p,q. add p q q \to p = O.
111 intros 2. elim q; clear q; intros;
112 [ lapply add_gen_O_2 to H as H0. clear H.
113 rewrite > H0. clear H0. clear p
114 | lapply add_gen_S_2 to H1 as H0. clear H1.
116 lapply eq_gen_S_S to H2 as H0. clear H2.
117 rewrite < H0 in H3. clear H0. clear a
121 theorem add_gen_eq_1_3: \forall p,q. add p q p \to q = O.
123 lapply add_sym to H. clear H.