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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/NPlus/props".
17 include "NPlus/fwd.ma".
19 theorem nplus_zero_1: \forall q. zero + q == q.
20 intros. elim q; clear q; auto.
23 theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
24 (succ p) + q == (succ r).
25 intros 2. elim q; clear q;
26 [ lapply linear nplus_gen_zero_2 to H as H0.
27 rewrite > H0. clear H0 p
28 | lapply linear nplus_gen_succ_2 to H1 as H0.
30 rewrite > H2. clear H2 r
34 theorem nplus_sym: \forall p,q,r. (p + q == r) \to q + p == r.
35 intros 2. elim q; clear q;
36 [ lapply linear nplus_gen_zero_2 to H as H0.
37 rewrite > H0. clear H0 p
38 | lapply linear nplus_gen_succ_2 to H1 as H0.
40 rewrite > H2. clear H2 r
44 theorem nplus_shift_succ_sx: \forall p,q,r.
45 (p + (succ q) == r) \to (succ p) + q == r.
47 lapply linear nplus_gen_succ_2 to H as H0.
49 rewrite > H1. clear H1 r.
53 theorem nplus_shift_succ_dx: \forall p,q,r.
54 ((succ p) + q == r) \to p + (succ q) == r.
56 lapply linear nplus_gen_succ_1 to H as H0.
58 rewrite > H1. clear H1 r.
62 theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
63 \forall q2,r2. (r1 + q2 == r2) \to
64 \exists q. (q1 + q2 == q) \land p + q == r2.
65 intros 2; elim q1; clear q1; intros;
66 [ lapply linear nplus_gen_zero_2 to H as H0.
67 rewrite > H0. clear H0 p
68 | lapply linear nplus_gen_succ_2 to H1 as H0.
70 rewrite > H3. rewrite > H3 in H2. clear H3 r1.
71 lapply linear nplus_gen_succ_1 to H2 as H0.
73 rewrite > H2. clear H2 r2.
74 lapply linear H to H4, H3 as H0.
76 ]; apply ex_intro; [| auto || auto ]. (**)
79 theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
80 \forall p2,r2. (p2 + r1 == r2) \to
81 \exists p. (p1 + p2 == p) \land p + q == r2.
82 intros 2; elim q; clear q; intros;
83 [ lapply linear nplus_gen_zero_2 to H as H0.
84 rewrite > H0. clear H0 p1
85 | lapply linear nplus_gen_succ_2 to H1 as H0.
87 rewrite > H3. rewrite > H3 in H2. clear H3 r1.
88 lapply linear nplus_gen_succ_2 to H2 as H0.
90 rewrite > H2. clear H2 r2.
91 lapply linear H to H4, H3 as H0.
93 ]; apply ex_intro; [| auto || auto ]. (**)
96 theorem nplus_conf: \forall p,q,r1. (p + q == r1) \to
97 \forall r2. (p + q == r2) \to r1 = r2.
98 intros 2. elim q; clear q; intros;
99 [ lapply linear nplus_gen_zero_2 to H as H0.
100 rewrite > H0 in H1. clear H0 p
101 | lapply linear nplus_gen_succ_2 to H1 as H0.
103 rewrite > H3. clear H3 r1.
104 lapply linear nplus_gen_succ_2 to H2 as H0.
106 rewrite > H2. clear H2 r2.