1 (**************************************************************************)
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/inv".
17 include "NPlus/defs.ma".
19 (* primitive generation lemmas proved by elimination and inversion *)
21 theorem nplus_gen_zero_1: \forall q,r. (zero + q == r) \to q = r.
22 intros. elim H; clear H q r; intros;
24 | clear H1. auto new timeout=30
28 theorem nplus_gen_succ_1: \forall p,q,r. ((succ p) + q == r) \to
29 \exists s. r = (succ s) \land p + q == s.
30 intros. elim H; clear H q r; intros;
35 ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
38 theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
39 intros. inversion H; clear H; intros;
46 theorem nplus_gen_succ_2: \forall p,q,r. (p + (succ q) == r) \to
47 \exists s. r = (succ s) \land p + q == s.
48 intros. inversion H; clear H; intros;
51 destruct H2; clear H2.
53 apply ex_intro; [| auto new timeout=30 ] (**)
57 theorem nplus_gen_zero_3: \forall p,q. (p + q == zero) \to
58 p = zero \land q = zero.
59 intros. inversion H; clear H; intros;
60 [ subst. auto new timeout=30
66 theorem nplus_gen_succ_3: \forall p,q,r. (p + q == (succ r)) \to
67 \exists s. p = succ s \land (s + q == r) \lor
68 q = succ s \land p + s == r.
69 intros. inversion H; clear H; intros;
72 destruct H3. clear H3.
74 ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ] (**)
77 (* alternative proofs invoking nplus_gen_2 *)
79 variant nplus_gen_zero_3_alt: \forall p,q. (p + q == zero) \to
80 p = zero \land q = zero.
81 intros 2. elim q; clear q; intros;
82 [ lapply linear nplus_gen_zero_2 to H as H0.
83 subst. auto new timeout=30
85 lapply linear nplus_gen_succ_2 to H1 as H0.
87 lapply linear eq_gen_zero_succ to H1 as H0. apply H0
91 variant nplus_gen_succ_3_alt: \forall p,q,r. (p + q == (succ r)) \to
92 \exists s. p = succ s \land (s + q == r) \lor
93 q = succ s \land p + s == r.
94 intros 2. elim q; clear q; intros;
95 [ lapply linear nplus_gen_zero_2 to H as H0.
98 lapply linear nplus_gen_succ_2 to H1 as H0.
100 lapply linear eq_gen_succ_succ to H1 as H0.
102 ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
105 (* other simplification lemmas *)
107 theorem nplus_gen_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
108 intros 2. elim q; clear q; intros;
109 [ lapply linear nplus_gen_zero_2 to H as H0.
111 | lapply linear nplus_gen_succ_2 to H1 as H0.
113 destruct H2. clear H2.
115 ]; auto new timeout=30.
118 theorem nplus_gen_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
119 intros 1. elim p; clear p; intros;
120 [ lapply linear nplus_gen_zero_1 to H as H0.
122 | lapply linear nplus_gen_succ_1 to H1 as H0.
124 destruct H2. clear H2.
126 ]; auto new timeout=30.