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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 *)
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15 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_fwd".
18 include "add_defs.ma".
20 (* primitive generation lemmas proved by elimination and inversion *)
22 theorem add_gen_O_1: \forall q,r. add O q r \to q = r.
23 intros. elim H; clear H q r; intros;
29 theorem add_gen_S_1: \forall p,q,r. add (S p) q r \to
30 \exists s. r = (S s) \land add p q s.
31 intros. elim H; clear H q r; intros;
35 rewrite > H1. clear H1 n2
36 ]; apply ex_intro; [| auto || auto ]. (**)
39 theorem add_gen_O_2: \forall p,r. add p O r \to p = r.
40 intros. inversion H; clear H; intros;
43 lapply eq_gen_O_S to H2 as H0. apply H0
47 theorem add_gen_S_2: \forall p,q,r. add p (S q) r \to
48 \exists s. r = (S s) \land add p q s.
49 intros. inversion H; clear H; intros;
50 [ lapply eq_gen_S_O to H as H0. apply H0
52 lapply linear eq_gen_S_S to H2 as H0.
53 rewrite > H0. clear H0 q.
54 apply ex_intro; [| auto ] (**)
58 theorem add_gen_O_3: \forall p,q. add p q O \to p = O \land q = O.
59 intros. inversion H; clear H; intros;
60 [ rewrite < H1. clear H1 p.
63 lapply eq_gen_O_S to H3 as H0. apply H0
67 theorem add_gen_S_3: \forall p,q,r. add p q (S r) \to
68 \exists s. p = S s \land add s q r \lor
69 q = S s \land add p s r.
70 intros. inversion H; clear H; intros;
71 [ rewrite < H1. clear H1 p
73 lapply linear eq_gen_S_S to H3 as H0.
74 rewrite > H0. clear H0 r.
75 ]; apply ex_intro; [| auto || auto ] (**)
78 (* alternative proofs invoking add_gen_2 *)
80 variant add_gen_O_3_alt: \forall p,q. add p q O \to p = O \land q = O.
81 intros 2. elim q; clear q; intros;
82 [ lapply linear add_gen_O_2 to H as H0.
83 rewrite > H0. clear H0 p.
86 lapply linear add_gen_S_2 to H1 as H0.
88 lapply linear eq_gen_O_S to H1 as H0. apply H0
92 variant add_gen_S_3_alt: \forall p,q,r. add p q (S r) \to
93 \exists s. p = S s \land add s q r \lor
94 q = S s \land add p s r.
95 intros 2. elim q; clear q; intros;
96 [ lapply linear add_gen_O_2 to H as H0.
97 rewrite > H0. clear H0 p
99 lapply linear add_gen_S_2 to H1 as H0.
101 lapply linear eq_gen_S_S to H1 as H0.
102 rewrite > H0. clear H0 r.
103 ]; apply ex_intro; [| auto || auto ]. (**)
106 (* other simplification lemmas *)
108 theorem add_gen_eq_2_3: \forall p,q. add p q q \to p = O.
109 intros 2. elim q; clear q; intros;
110 [ lapply linear add_gen_O_2 to H as H0.
111 rewrite > H0. clear H0 p
112 | lapply linear add_gen_S_2 to H1 as H0.
114 lapply linear eq_gen_S_S to H2 as H0.
115 rewrite < H0 in H3. clear H0 a
119 theorem add_gen_eq_1_3: \forall p,q. add p q p \to q = O.
120 intros 1. elim p; clear p; intros;
121 [ lapply linear add_gen_O_1 to H as H0.
122 rewrite > H0. clear H0 q
123 | lapply linear add_gen_S_1 to H1 as H0.
125 lapply linear eq_gen_S_S to H2 as H0.
126 rewrite < H0 in H3. clear H0 a