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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 set "baseuri" "cic:/matita/RELATIONAL/NPlus/inv".
17 include "NPlus/defs.ma".
19 (* Inversion lemmas *********************************************************)
21 theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
22 intros. elim H; clear H q r; auto.
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;
29 | clear H1. decompose. subst. auto depth = 4
33 theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
34 intros. inversion H; clear H; intros;
36 | clear H H1. destruct H2.
40 theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \to
41 \exists s. r = (succ s) \land p + q == s.
42 intros. inversion H; clear H; intros;
44 | clear H1 H3 r. destruct H2; clear H2. subst. auto depth = 4.
48 theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to
49 p = zero \land q = zero.
50 intros. inversion H; clear H; intros;
52 | clear H H1. destruct H3.
56 theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
57 \exists s. p = succ s \land (s + q == r) \lor
58 q = succ s \land p + s == r.
59 intros. inversion H; clear H; intros;
61 | clear H1. destruct H3. clear H3. subst.
65 (* Corollaries to inversion lemmas ******************************************)
67 theorem nplus_inv_succ_2_3: \forall p,q,r.
68 (p + (succ q) == (succ r)) \to p + q == r.
70 lapply linear nplus_inv_succ_2 to H. decompose. subst.
71 destruct H1. clear H1. subst. auto.
74 theorem nplus_inv_succ_1_3: \forall p,q,r.
75 ((succ p) + q == (succ r)) \to p + q == r.
77 lapply linear nplus_inv_succ_1 to H. decompose. subst.
78 destruct H1. clear H1. subst. auto.
81 theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
82 intros 2. elim q; clear q;
83 [ lapply linear nplus_inv_zero_2 to H
84 | lapply linear nplus_inv_succ_2_3 to H1
88 theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
89 intros 1. elim p; clear p;
90 [ lapply linear nplus_inv_zero_1 to H
91 | lapply linear nplus_inv_succ_1_3 to H1.