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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 *)
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15 (* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
16 * Suggested invocation to start formal specifications with:
17 * - Patience on me to gain peace and perfection! -
18 * 2012 April 16 (anniversary milestone):
19 * context-sensitive subject equivalence for atomic arity assignment.
21 * context-sensitive strong normalization for simply typed terms.
23 * support for abstract candidates of reducibility.
25 * confluence for context-sensitive parallel reduction.
27 * confluence for context-free parallel reduction.
29 * specification starts.
32 include "ground_2/star.ma".
33 include "basic_2/notation.ma".
35 (* ATOMIC ARITY *************************************************************)
37 inductive aarity: Type[0] ≝
38 | AAtom: aarity (* atomic aarity construction *)
39 | APair: aarity → aarity → aarity (* binary aarity construction *)
42 interpretation "aarity construction (atomic)"
45 interpretation "aarity construction (binary)"
46 'SnItem2 A1 A2 = (APair A1 A2).
48 (* Basic inversion lemmas ***************************************************)
50 lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥.
53 | #Y #X #IHY #_ #H destruct
54 -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
59 lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥.
62 | #Y #X #_ #IHX #H destruct
63 -H (**) (* destruct: the destucted equality is not erased *)
68 (* Basic properties *********************************************************)
70 lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
72 [ #A2 elim A2 -A2 /2 width=1/
73 #B2 #A2 #_ #_ @or_intror #H destruct
74 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
75 [ -IHB1 -IHA1 @or_intror #H destruct
76 | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
77 [ #H destruct elim (IHA1 A2) -IHA1
78 [ #H destruct /2 width=1/
79 | #HA12 @or_intror #H destruct /2 width=1/
81 | -IHA1 #HB12 @or_intror #H destruct /2 width=1/