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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/NPlus/props".
17 include "NPlus/inv.ma".
19 (* Monoidal properties *)
21 theorem nplus_conf: \forall p,q,r1. (p + q == r1) \to
22 \forall r2. (p + q == r2) \to r1 = r2.
23 intros 4. elim H; clear H q r1;
24 [ lapply linear nplus_gen_zero_2 to H1
25 | lapply linear nplus_gen_succ_2 to H3. decompose
29 theorem nplus_zero_1: \forall q. zero + q == q.
30 intros. elim q; clear q; auto.
33 theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
34 (succ p) + q == (succ r).
35 intros. elim H; clear H q r; auto.
38 theorem nplus_comm: \forall p,q,r. (p + q == r) \to q + p == r.
39 intros. elim H; clear H q r; auto.
44 theorem nplus_comm_1: \forall p1,q,r1. (p1 + q == r1) \to
45 \forall p2,r2. (p2 + q == r2) \to
46 \forall s. (p1 + r2 == s) \to (p2 + r1 == s).
47 intros 4. elim H; clear H q r1;
48 [ lapply linear nplus_gen_zero_2 to H1. subst
49 | lapply linear nplus_gen_succ_2 to H3. decompose. subst.
50 lapply linear nplus_gen_succ_2 to H4. decompose. subst
55 theorem nplus_shift_succ_sx: \forall p,q,r.
56 (p + (succ q) == r) \to (succ p) + q == r.
58 lapply linear nplus_gen_succ_2 to H as H0.
59 decompose. subst. auto new timeout=100.
62 theorem nplus_shift_succ_dx: \forall p,q,r.
63 ((succ p) + q == r) \to p + (succ q) == r.
65 lapply linear nplus_gen_succ_1 to H as H0.
66 decompose. subst. auto new timeout=100.
69 theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
70 \forall q2,r2. (r1 + q2 == r2) \to
71 \exists q. (q1 + q2 == q) \land p + q == r2.
72 intros 2; elim q1; clear q1; intros;
73 [ lapply linear nplus_gen_zero_2 to H as H0.
75 | lapply linear nplus_gen_succ_2 to H1 as H0.
77 lapply linear nplus_gen_succ_1 to H2 as H0.
79 lapply linear H to H4, H3 as H0.
81 ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
84 theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
85 \forall p2,r2. (p2 + r1 == r2) \to
86 \exists p. (p1 + p2 == p) \land p + q == r2.
87 intros 2; elim q; clear q; intros;
88 [ lapply linear nplus_gen_zero_2 to H as H0.
90 | lapply linear nplus_gen_succ_2 to H1 as H0.
92 lapply linear nplus_gen_succ_2 to H2 as H0.
94 lapply linear H to H4, H3 as H0.
96 ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)