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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_props".
17 include "add_defs.ma".
19 axiom add_gen_O_2: \forall p,r. add p O r \to p = r.
21 axiom add_gen_S_2: \forall p,q,r. add p (S q) r \to
22 \exists s. r = (S s) \land add p q s.
24 theorem add_O_1: \forall q. add O q q.
25 intros. elim q; clear q; auto.
28 theorem add_S_1: \forall p,q,r. add p q r \to add (S p) q (S r).
29 intros 2. elim q; clear q;
30 [ lapply add_gen_O_2 to H using H0. clear H.
31 rewrite > H0. clear H0. clear p
32 | lapply add_gen_S_2 to H1 using H0. clear H1.
34 rewrite > H2. clear H2. clear r
38 theorem add_sym: \forall p,q,r. add p q r \to add q p r.
39 intros 2. elim q; clear q;
40 [ lapply add_gen_O_2 to H using H0. clear H.
41 rewrite > H0. clear H0. clear p
42 | lapply add_gen_S_2 to H1 using H0. clear H1.
44 rewrite > H2. clear H2. clear r
48 theorem add_shift_S_sx: \forall p,q,r. add p (S q) r \to add (S p) q r.
50 lapply add_gen_S_2 to H using H0. clear H.
52 rewrite > H1. clear H1. clear r.
56 theorem add_gen_O_1: \forall q,r. add O q r \to q = r.
60 theorem add_gen_S_1: \forall p,q,r. add (S p) q r \to
61 \exists s. r = (S s) \land add p q s.
65 theorem add_shift_S_dx: \forall p,q,r. add (S p) q r \to add p (S q) r.
67 lapply add_gen_S_1 to H using H0. clear H.
69 rewrite > H1. clear H1. clear r.
73 theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
74 \forall q2,r2. add r1 q2 r2 \to
75 \exists q. add q1 q2 q \land add p q r2.
76 intros 2; elim q1; clear q1; intros;
77 [ lapply add_gen_O_2 to H using H0. clear H.
78 rewrite > H0. clear H0. clear p
79 | lapply add_gen_S_2 to H1 using H0. clear H1.
81 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
82 lapply add_gen_S_1 to H2 using H0. clear H2.
84 rewrite > H2. clear H2. clear r2.
85 lapply H to H4, H3 using H0. clear H. clear H4. clear H3.
87 ]; apply ex_intro; [| auto || auto ].
90 theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to
91 \forall p2,r2. add p2 r1 r2 \to
92 \exists p. add p1 p2 p \land add p q r2.
93 intros 2; elim q; clear q; intros;
94 [ lapply add_gen_O_2 to H using H0. clear H.
95 rewrite > H0. clear H0. clear p1
96 | lapply add_gen_S_2 to H1 using H0. clear H1.
98 rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
99 lapply add_gen_S_2 to H2 using H0. clear H2.
101 rewrite > H2. clear H2. clear r2.
102 lapply H to H4, H3 using H0. clear H. clear H4. clear H3.
104 ]; apply ex_intro; [| auto || auto ].
107 theorem add_conf: \forall p,q,r1. add p q r1 \to
108 \forall r2. add p q r2 \to r1 = r2.
109 intros 2. elim q; clear q; intros;
110 [ lapply add_gen_O_2 to H using H0. clear H.
111 rewrite > H0 in H1. clear H0. clear p
112 | lapply add_gen_S_2 to H1 using H0. clear H1.
114 rewrite > H3. clear H3. clear r1.
115 lapply add_gen_S_2 to H2 using H0. clear H2.
117 rewrite > H2. clear H2. clear r2.