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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! -
19 * term binders polarized to control ζ reduction.
20 * 2012 April 16 (anniversary milestone):
21 * context-sensitive subject equivalence for atomic arity assignment.
23 * context-sensitive strong normalization for simply typed terms.
25 * support for abstract candidates of reducibility.
27 * confluence for context-sensitive parallel reduction.
29 * confluence for context-free parallel reduction.
31 * specification starts.
34 include "ground_2/star.ma".
35 include "basic_2/notation.ma".
37 (* ATOMIC ARITY *************************************************************)
39 inductive aarity: Type[0] ≝
40 | AAtom: aarity (* atomic aarity construction *)
41 | APair: aarity → aarity → aarity (* binary aarity construction *)
44 interpretation "aarity construction (atomic)"
47 interpretation "aarity construction (binary)"
48 'SnItem2 A1 A2 = (APair A1 A2).
50 (* Basic inversion lemmas ***************************************************)
52 lemma discr_apair_xy_x: ∀A,B. ②B. A = B → ⊥.
55 | #Y #X #IHY #_ #H destruct
56 -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
61 lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → ⊥.
64 | #Y #X #_ #IHX #H destruct
65 -H (**) (* destruct: the destucted equality is not erased *)
70 (* Basic properties *********************************************************)
72 lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
74 [ #A2 elim A2 -A2 /2 width=1/
75 #B2 #A2 #_ #_ @or_intror #H destruct
76 | #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
77 [ -IHB1 -IHA1 @or_intror #H destruct
78 | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
79 [ #H destruct elim (IHA1 A2) -IHA1
80 [ #H destruct /2 width=1/
81 | #HA12 @or_intror #H destruct /2 width=1/
83 | -IHA1 #HB12 @or_intror #H destruct /2 width=1/