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Full specification of find. Added notation for If_Then_Else. Probably a delta
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14
15 set "baseuri" "cic:/matita/RELATIONAL/NPlus/inv".
16
17 include "NPlus/defs.ma".
18
19 (* Inversion lemmas *********************************************************)
20
21 theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
22  intros. elim H; clear H q r; autobatch.
23 qed.
24
25 theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to 
26                           \exists s. r = (succ s) \land p + q == s.
27  intros. elim H; clear H q r; intros;
28  [ autobatch depth = 4
29  | clear H1. decompose. subst. autobatch depth = 4
30  ]
31 qed.
32
33 theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
34  intros. inversion H; clear H; intros; subst. autobatch.
35 qed.
36
37 theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \to 
38                           \exists s. r = (succ s) \land p + q == s.
39  intros. inversion H; clear H; intros; subst.
40  autobatch depth = 4.
41 qed.
42
43 theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to 
44                           p = zero \land q = zero.
45  intros. inversion H; clear H; intros; subst. autobatch.
46 qed.
47
48 theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
49                           \exists s. p = succ s \land (s + q == r) \lor
50                                      q = succ s \land p + s == r.
51  intros. inversion H; clear H; intros; subst;
52  autobatch depth = 4.
53 qed.
54
55 (* Corollaries to inversion lemmas ******************************************)
56
57 theorem nplus_inv_succ_2_3: \forall p,q,r.
58                             (p + (succ q) == (succ r)) \to p + q == r.
59  intros. 
60  lapply linear nplus_inv_succ_2 to H. decompose. subst. autobatch.
61 qed.
62
63 theorem nplus_inv_succ_1_3: \forall p,q,r.
64                             ((succ p) + q == (succ r)) \to p + q == r.
65  intros. 
66  lapply linear nplus_inv_succ_1 to H. decompose. subst. autobatch.
67 qed.
68
69 theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
70  intros 2. elim q; clear q;
71  [ lapply linear nplus_inv_zero_2 to H
72  | lapply linear nplus_inv_succ_2_3 to H1
73  ]; autobatch.
74 qed.
75
76 theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
77  intros 1. elim p; clear p;
78  [ lapply linear nplus_inv_zero_1 to H
79  | lapply linear nplus_inv_succ_1_3 to H1.
80  ]; autobatch.
81 qed.