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new definition of lleq allows to complete the proof of lemma 1000
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14
15
16
17 include "NPlus/defs.ma".
18
19 (* Inversion lemmas *********************************************************)
20
21 theorem nplus_inv_zero_1: ∀q,r. zero ⊕ q ≍ r → q = r.
22  intros. elim H; clear H q r; autobatch.
23 qed.
24
25 theorem nplus_inv_succ_1: ∀p,q,r. succ p ⊕ q ≍ r → 
26                           ∃s. r = succ s ∧ p ⊕ q ≍ s.
27  intros. elim H; clear H q r; intros;
28  [ autobatch depth = 3
29  | clear H1; decompose; destruct; autobatch depth = 4
30  ]
31 qed.
32
33 theorem nplus_inv_zero_2: ∀p,r. p ⊕ zero ≍ r → p = r.
34  intros; inversion H; clear H; intros; destruct; autobatch.
35 qed.
36
37 theorem nplus_inv_succ_2: ∀p,q,r. p ⊕ succ q ≍ r → 
38                           ∃s. r = succ s ∧ p ⊕ q ≍ s.
39  intros; inversion H; clear H; intros; destruct.
40  autobatch depth = 3.
41 qed.
42
43 theorem nplus_inv_zero_3: ∀p,q. p ⊕ q ≍ zero → 
44                           p = zero ∧ q = zero.
45  intros; inversion H; clear H; intros; destruct; autobatch.
46 qed.
47
48 theorem nplus_inv_succ_3: ∀p,q,r. p ⊕ q ≍ succ r →
49                              ∃s. p = succ s ∧ s ⊕ q ≍ r ∨
50                                q = succ s ∧ p ⊕ s ≍ r.
51  intros; inversion H; clear H; intros; destruct;
52  autobatch depth = 4.
53 qed.
54
55 (* Corollaries to inversion lemmas ******************************************)
56
57 theorem nplus_inv_succ_2_3: ∀p,q,r.
58                             p ⊕ succ q ≍ succ r → p ⊕ q ≍ r.
59  intros;
60  lapply linear nplus_inv_succ_2 to H; decompose; destruct; autobatch.
61 qed.
62
63 theorem nplus_inv_succ_1_3: ∀p,q,r.
64                             succ p ⊕ q ≍ succ r → p ⊕ q ≍ r.
65  intros;
66  lapply linear nplus_inv_succ_1 to H; decompose; destruct; autobatch.
67 qed.
68
69 theorem nplus_inv_eq_2_3: ∀p,q. p ⊕ q ≍ q → 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: ∀p,q. p ⊕ q ≍ p → 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.