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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 *)
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15 set "baseuri" "cic:/matita/RELATIONAL/NPlus/fwd".
18 include "NPlus/defs.ma".
20 (* primitive generation lemmas proved by elimination and inversion *)
22 theorem nplus_gen_zero_1: \forall q,r. (zero + q == r) \to q = r.
23 intros. elim H; clear H q r; intros;
29 theorem nplus_gen_succ_1: \forall p,q,r. ((succ p) + q == r) \to
30 \exists s. r = (succ s) \land p + q == s.
31 intros. elim H; clear H q r; intros;
35 rewrite > H1. clear H1 n2
36 ]; apply ex_intro; [| auto || auto ]. (**)
39 theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
40 intros. inversion H; clear H; intros;
43 lapply eq_gen_zero_succ to H2 as H0. apply H0
47 theorem nplus_gen_succ_2: \forall p,q,r. (p + (succ q) == r) \to
48 \exists s. r = (succ s) \land p + q == s.
49 intros. inversion H; clear H; intros;
50 [ lapply eq_gen_succ_zero to H as H0. apply H0
52 lapply linear eq_gen_succ_succ to H2 as H0.
53 rewrite > H0. clear H0 q.
54 apply ex_intro; [| auto ] (**)
58 theorem nplus_gen_zero_3: \forall p,q. (p + q == zero) \to
59 p = zero \land q = zero.
60 intros. inversion H; clear H; intros;
61 [ rewrite < H1. clear H1 p.
64 lapply eq_gen_zero_succ to H3 as H0. apply H0
68 theorem nplus_gen_succ_3: \forall p,q,r. (p + q == (succ r)) \to
69 \exists s. p = succ s \land (s + q == r) \lor
70 q = succ s \land p + s == r.
71 intros. inversion H; clear H; intros;
72 [ rewrite < H1. clear H1 p
74 lapply linear eq_gen_succ_succ to H3 as H0.
75 rewrite > H0. clear H0 r.
76 ]; apply ex_intro; [| auto || auto ] (**)
79 (* alternative proofs invoking nplus_gen_2 *)
81 variant nplus_gen_zero_3_alt: \forall p,q. (p + q == zero) \to
82 p = zero \land q = zero.
83 intros 2. elim q; clear q; intros;
84 [ lapply linear nplus_gen_zero_2 to H as H0.
85 rewrite > H0. clear H0 p.
88 lapply linear nplus_gen_succ_2 to H1 as H0.
90 lapply linear eq_gen_zero_succ to H1 as H0. apply H0
94 variant nplus_gen_succ_3_alt: \forall p,q,r. (p + q == (succ r)) \to
95 \exists s. p = succ s \land (s + q == r) \lor
96 q = succ s \land p + s == r.
97 intros 2. elim q; clear q; intros;
98 [ lapply linear nplus_gen_zero_2 to H as H0.
99 rewrite > H0. clear H0 p
101 lapply linear nplus_gen_succ_2 to H1 as H0.
103 lapply linear eq_gen_succ_succ to H1 as H0.
104 rewrite > H0. clear H0 r.
105 ]; apply ex_intro; [| auto || auto ]. (**)
108 (* other simplification lemmas *)
110 theorem nplus_gen_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
111 intros 2. elim q; clear q; intros;
112 [ lapply linear nplus_gen_zero_2 to H as H0.
113 rewrite > H0. clear H0 p
114 | lapply linear nplus_gen_succ_2 to H1 as H0.
116 lapply linear eq_gen_succ_succ to H2 as H0.
117 rewrite < H0 in H3. clear H0 a
121 theorem nplus_gen_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
122 intros 1. elim p; clear p; intros;
123 [ lapply linear nplus_gen_zero_1 to H as H0.
124 rewrite > H0. clear H0 q
125 | lapply linear nplus_gen_succ_1 to H1 as H0.
127 lapply linear eq_gen_succ_succ to H2 as H0.
128 rewrite < H0 in H3. clear H0 a