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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
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11 (*        v         GNU General Public License Version 2                  *)
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13 (**************************************************************************)
14
15 include "ground_2/lib/bool.ma".
16 include "ground_2/lib/arith.ma".
17
18 (* ITEMS ********************************************************************)
19
20 (* atomic items *)
21 inductive item0: Type[0] ≝
22    | Sort: nat → item0 (* sort: starting at 0 *)
23    | LRef: nat → item0 (* reference by index: starting at 0 *)
24    | GRef: nat → item0 (* reference by position: starting at 0 *)
25 .
26
27 (* binary binding items *)
28 inductive bind2: Type[0] ≝
29   | Abbr: bind2 (* abbreviation *)
30   | Abst: bind2 (* abstraction *)
31 .
32
33 (* binary non-binding items *)
34 inductive flat2: Type[0] ≝
35   | Appl: flat2 (* application *)
36   | Cast: flat2 (* explicit type annotation *)
37 .
38
39 (* binary items *)
40 inductive item2: Type[0] ≝
41   | Bind2: bool → bind2 → item2 (* polarized binding item *)
42   | Flat2: flat2 → item2        (* non-binding item *)
43 .
44
45 (* Basic inversion lemmas ***************************************************)
46
47 fact destruct_sort_sort_aux: ∀s1,s2. Sort s1 = Sort s2 → s1 = s2.
48 #s1 #s2 #H destruct //
49 qed-.
50
51 (* Basic properties *********************************************************)
52
53 lemma eq_item0_dec: ∀I1,I2:item0. Decidable (I1 = I2).
54 * #i1 * #i2 [2,3,4,6,7,8: @or_intror #H destruct ]
55 [2: elim (eq_nat_dec i1 i2) |1,3: elim (eq_nat_dec i1 i2) ] /2 width=1 by or_introl/
56 #Hni12 @or_intror #H destruct /2 width=1 by/
57 qed-.
58
59 (* Basic_1: was: bind_dec *)
60 lemma eq_bind2_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
61 * * /2 width=1 by or_introl/
62 @or_intror #H destruct
63 qed-.
64
65 (* Basic_1: was: flat_dec *)
66 lemma eq_flat2_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
67 * * /2 width=1 by or_introl/
68 @or_intror #H destruct
69 qed-.
70
71 (* Basic_1: was: kind_dec *)
72 lemma eq_item2_dec: ∀I1,I2:item2. Decidable (I1 = I2).
73 * [ #p1 ] #I1 * [1,3: #p2 ] #I2
74 [2,3: @or_intror #H destruct
75 | elim (eq_bool_dec p1 p2) #Hp
76   [ elim (eq_bind2_dec I1 I2) /2 width=1 by or_introl/ #HI ]
77   @or_intror #H destruct /2 width=1 by/
78 | elim (eq_flat2_dec I1 I2) /2 width=1 by or_introl/ #HI
79   @or_intror #H destruct /2 width=1 by/
80 ]
81 qed-.
82
83 (* Basic_1: removed theorems 21:
84             s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc
85             s_arith0 s_arith1
86             r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1
87             not_abbr_abst bind_dec_not
88 *)