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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 *)
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15 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_props".
17 include "nat_props.ma".
18 include "add_defs.ma".
20 theorem add_gen_O_2: \forall p,r. add p O r \to p = r.
21 intros. inversion H; clear H; intros;
23 | lapply eq_gen_O_S to H2 as H0. apply H0
27 theorem add_gen_S_2: \forall p,q,r. add p (S q) r \to
28 \exists s. r = (S s) \land add p q s.
29 intros. inversion H; clear H; intros;
30 [ lapply eq_gen_S_O to H as H0. apply H0
31 | lapply eq_gen_S_S to H2 as H0. clear H2.
32 rewrite > H0. clear H0.
33 apply ex_intro; [| auto ] (**)
37 theorem add_O_1: \forall q. add O q q.
38 intros. elim q; clear q; auto.
41 theorem add_S_1: \forall p,q,r. add p q r \to add (S p) q (S r).
42 intros 2. elim q; clear q;
43 [ lapply add_gen_O_2 to H as H0. clear H.
44 rewrite > H0. clear H0. clear p
45 | lapply add_gen_S_2 to H1 as H0. clear H1.
47 rewrite > H2. clear H2. clear r
51 theorem add_sym: \forall p,q,r. add p q r \to add q p r.
52 intros 2. elim q; clear q;
53 [ lapply add_gen_O_2 to H as H0. clear H.
54 rewrite > H0. clear H0. clear p
55 | lapply add_gen_S_2 to H1 as H0. clear H1.
57 rewrite > H2. clear H2. clear r
61 theorem add_shift_S_sx: \forall p,q,r. add p (S q) r \to add (S p) q r.
63 lapply add_gen_S_2 to H as H0. clear H.
65 rewrite > H1. clear H1. clear r.
69 theorem add_gen_O_1: \forall q,r. add O q r \to q = r.
73 theorem add_gen_S_1: \forall p,q,r. add (S p) q r \to
74 \exists s. r = (S s) \land add p q s.
78 theorem add_shift_S_dx: \forall p,q,r. add (S p) q r \to add p (S q) r.
80 lapply add_gen_S_1 to H as H0. clear H.
82 rewrite > H1. clear H1. clear r.
86 theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
87 \forall q2,r2. add r1 q2 r2 \to
88 \exists q. add q1 q2 q \land add p q r2.
89 intros 2; elim q1; clear q1; intros;
90 [ lapply add_gen_O_2 to H as H0. clear H.
91 rewrite > H0. clear H0. clear p
92 | lapply add_gen_S_2 to H1 as H0. clear H1.
94 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
95 lapply add_gen_S_1 to H2 as H0. clear H2.
97 rewrite > H2. clear H2. clear r2.
98 lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
100 ]; apply ex_intro; [| auto || auto ]. (**)
103 theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to
104 \forall p2,r2. add p2 r1 r2 \to
105 \exists p. add p1 p2 p \land add p q r2.
106 intros 2; elim q; clear q; intros;
107 [ lapply add_gen_O_2 to H as H0. clear H.
108 rewrite > H0. clear H0. clear p1
109 | lapply add_gen_S_2 to H1 as H0. clear H1.
111 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
112 lapply add_gen_S_2 to H2 as H0. clear H2.
114 rewrite > H2. clear H2. clear r2.
115 lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
117 ]; apply ex_intro; [| auto || auto ]. (**)
120 theorem add_conf: \forall p,q,r1. add p q r1 \to
121 \forall r2. add p q r2 \to r1 = r2.
122 intros 2. elim q; clear q; intros;
123 [ lapply add_gen_O_2 to H as H0. clear H.
124 rewrite > H0 in H1. clear H0. clear p
125 | lapply add_gen_S_2 to H1 as H0. clear H1.
127 rewrite > H3. clear H3. clear r1.
128 lapply add_gen_S_2 to H2 as H0. clear H2.
130 rewrite > H2. clear H2. clear r2.