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 *)
13 (**************************************************************************)
17 include "NPlus/fun.ma".
19 (* Monoidal properties ******************************************************)
21 theorem nplus_zero_1: ∀q. zero ⊕ q ≍ q.
22 intros; elim q; clear q; autobatch.
25 theorem nplus_succ_1: ∀p,q,r. p ⊕ q ≍ r → succ p ⊕ q ≍ succ r.
26 intros; elim H; clear H q r; autobatch.
29 theorem nplus_comm: ∀p, q, x. p ⊕ q ≍ x → ∀y. q ⊕ p ≍ y → x = y.
30 intros 4; elim H; clear H q x;
31 [ lapply linear nplus_inv_zero_1 to H1
32 | lapply linear nplus_inv_succ_1 to H3. decompose
33 ]; destruct; autobatch.
36 theorem nplus_comm_rew: ∀p,q,r. p ⊕ q ≍ r → q ⊕ p ≍ r.
37 intros; elim H; clear H q r; autobatch.
40 theorem nplus_ass: ∀p1, p2, r1. p1 ⊕ p2 ≍ r1 → ∀p3, s1. r1 ⊕ p3 ≍ s1 →
41 ∀r3. p2 ⊕ p3 ≍ r3 → ∀s3. p1 ⊕ r3 ≍ s3 → s1 = s3.
42 intros 4; elim H; clear H p2 r1;
43 [ lapply linear nplus_inv_zero_1 to H2. destruct.
44 lapply nplus_mono to H1, H3. destruct. autobatch
45 | lapply linear nplus_inv_succ_1 to H3. decompose. destruct.
46 lapply linear nplus_inv_succ_1 to H4. decompose. destruct.
47 lapply linear nplus_inv_succ_2 to H5. decompose. destruct. autobatch
51 (* Corollaries of functional properties **************************************)
53 theorem nplus_inj_2: ∀p, q1, r. p ⊕ q1 ≍ r → ∀q2. p ⊕ q2 ≍ r → q1 = q2.
57 (* Corollaries of nonoidal properties ***************************************)
59 theorem nplus_comm_1: ∀p1, q, r1. p1 ⊕ q ≍ r1 → ∀p2, r2. p2 ⊕ q ≍ r2 →
60 ∀x. p2 ⊕ r1 ≍ x → ∀y. p1 ⊕ r2 ≍ y → x = y.
61 intros 4; elim H; clear H q r1;
62 [ lapply linear nplus_inv_zero_2 to H1
63 | lapply linear nplus_inv_succ_2 to H3.
64 lapply linear nplus_inv_succ_2 to H4. decompose. destruct.
65 lapply linear nplus_inv_succ_2 to H5. decompose
66 ]; destruct; autobatch.
69 theorem nplus_comm_1_rew: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ q ≍ r2 →
70 ∀s. p1 ⊕ r2 ≍ s → p2 ⊕ r1 ≍ s.
71 intros 4; elim H; clear H q r1;
72 [ lapply linear nplus_inv_zero_2 to H1. destruct
73 | lapply linear nplus_inv_succ_2 to H3. decompose. destruct.
74 lapply linear nplus_inv_succ_2 to H4. decompose. destruct
79 theorem nplus_shift_succ_sx: \forall p,q,r.
80 (p \oplus (succ q) \asymp r) \to (succ p) \oplus q \asymp r.
82 lapply linear nplus_inv_succ_2 to H as H0.
83 decompose. destruct. auto new timeout=100.
86 theorem nplus_shift_succ_dx: \forall p,q,r.
87 ((succ p) \oplus q \asymp r) \to p \oplus (succ q) \asymp r.
89 lapply linear nplus_inv_succ_1 to H as H0.
90 decompose. destruct. auto new timeout=100.
93 theorem nplus_trans_1: \forall p,q1,r1. (p \oplus q1 \asymp r1) \to
94 \forall q2,r2. (r1 \oplus q2 \asymp r2) \to
95 \exists q. (q1 \oplus q2 \asymp q) \land p \oplus q \asymp r2.
96 intros 2; elim q1; clear q1; intros;
97 [ lapply linear nplus_inv_zero_2 to H as H0.
99 | lapply linear nplus_inv_succ_2 to H1 as H0.
101 lapply linear nplus_inv_succ_1 to H2 as H0.
103 lapply linear H to H4, H3 as H0.
105 ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
108 theorem nplus_trans_2: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ r1 ≍ r2 →
109 ∃p. p1 ⊕ p2 ≍ p ∧ p ⊕ q ≍ r2.
110 intros 2; elim q; clear q; intros;
111 [ lapply linear nplus_inv_zero_2 to H as H0.
113 | lapply linear nplus_inv_succ_2 to H1 as H0.
115 lapply linear nplus_inv_succ_2 to H2 as H0.
117 lapply linear H to H4, H3 as H0.
119 ]; autobatch. apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)