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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 (* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
16  * Initial invocation: - Patience on me to gain peace and perfection! -
17  *)
18
19 include "ground/lib/relations.ma".
20 include "static_2/notation/functions/item0_0.ma".
21 include "static_2/notation/functions/snitem2_2.ma".
22
23 (* ATOMIC ARITY *************************************************************)
24
25 inductive aarity: Type[0] ≝
26   | AAtom: aarity                   (* atomic aarity construction *)
27   | APair: aarity → aarity → aarity (* binary aarity construction *)
28 .
29
30 interpretation "atomic arity construction (atomic)"
31    'Item0 = AAtom.
32
33 interpretation "atomic arity construction (binary)"
34    'SnItem2 A1 A2 = (APair A1 A2).
35
36 (* Basic inversion lemmas ***************************************************)
37
38 fact destruct_apair_apair_aux: ∀A1,A2,B1,B2. ②B1.A1 = ②B2.A2 → B1 = B2 ∧ A1 = A2.
39 #A1 #A2 #B1 #B2 #H destruct /2 width=1 by conj/
40 qed-.
41
42 lemma discr_apair_xy_x: ∀A,B. ②B.A = B → ⊥.
43 #A #B elim B -B
44 [ #H destruct
45 | #Y #X #IHY #_ #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
46 ]
47 qed-.
48
49 (* Basic_2A1: was: discr_tpair_xy_y *)
50 lemma discr_apair_xy_y: ∀B,A. ②B. A = A → ⊥.
51 #B #A elim A -A
52 [ #H destruct
53 | #Y #X #_ #IHX #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
54 ]
55 qed-.
56
57 (* Basic properties *********************************************************)
58
59 lemma eq_aarity_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
60 #A1 elim A1 -A1
61 [ #A2 elim A2 -A2 /2 width=1 by or_introl/
62   #B2 #A2 #_ #_ @or_intror #H destruct
63 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
64   [ -IHB1 -IHA1 @or_intror #H destruct
65   | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
66     [ #H destruct elim (IHA1 A2) -IHA1
67       [ #H destruct /2 width=1 by or_introl/
68       | #HA12 @or_intror #H destruct /2 width=1 by/
69       ]
70     | -IHA1 #HB12 @or_intror #H destruct /2 width=1 by/
71     ]
72   ]
73 ]
74 qed-.
75
76 lemma is_apear_dec (B) (X): Decidable (∃A. ②B.A = X).
77 #B * [| #X #A ]
78 [| elim (eq_aarity_dec X B) #HX ]
79 [| /3 width=2 by ex_intro, or_introl/ ]
80 @or_intror * #A #H destruct
81 /2 width=1 by/
82 qed-.