1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground_2/xoa/ex_1_3.ma".
16 include "static_2/notation/functions/item0_1.ma".
17 include "static_2/notation/functions/snitem2_3.ma".
18 include "static_2/notation/functions/snbind2_4.ma".
19 include "static_2/notation/functions/snbind2pos_3.ma".
20 include "static_2/notation/functions/snbind2neg_3.ma".
21 include "static_2/notation/functions/snflat2_3.ma".
22 include "static_2/notation/functions/star_1.ma".
23 include "static_2/notation/functions/lref_1.ma".
24 include "static_2/notation/functions/gref_1.ma".
25 include "static_2/notation/functions/snabbr_3.ma".
26 include "static_2/notation/functions/snabbrpos_2.ma".
27 include "static_2/notation/functions/snabbrneg_2.ma".
28 include "static_2/notation/functions/snabst_3.ma".
29 include "static_2/notation/functions/snabstpos_2.ma".
30 include "static_2/notation/functions/snabstneg_2.ma".
31 include "static_2/notation/functions/snappl_2.ma".
32 include "static_2/notation/functions/sncast_2.ma".
33 include "static_2/syntax/item.ma".
35 (* TERMS ********************************************************************)
38 inductive term: Type[0] ≝
39 | TAtom: item0 → term (* atomic item construction *)
40 | TPair: item2 → term → term → term (* binary item construction *)
43 interpretation "term construction (atomic)"
46 interpretation "term construction (binary)"
47 'SnItem2 I T1 T2 = (TPair I T1 T2).
49 interpretation "term binding construction (binary)"
50 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
52 interpretation "term positive binding construction (binary)"
53 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
55 interpretation "term negative binding construction (binary)"
56 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
58 interpretation "term flat construction (binary)"
59 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
61 interpretation "sort (term)"
62 'Star s = (TAtom (Sort s)).
64 interpretation "local reference (term)"
65 'LRef i = (TAtom (LRef i)).
67 interpretation "global reference (term)"
68 'GRef l = (TAtom (GRef l)).
70 interpretation "abbreviation (term)"
71 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
73 interpretation "positive abbreviation (term)"
74 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
76 interpretation "negative abbreviation (term)"
77 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
79 interpretation "abstraction (term)"
80 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
82 interpretation "positive abstraction (term)"
83 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
85 interpretation "negative abstraction (term)"
86 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
88 interpretation "application (term)"
89 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
91 interpretation "native type annotation (term)"
92 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
94 (* Basic properties *********************************************************)
96 lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ[p]W.T
97 | (∀p,W,T. X = ⓛ[p]W.T → ⊥).
98 * [ #I | * [ #p * | #I ] #V #T ]
99 [3: /3 width=4 by ex1_3_intro, or_introl/ ]
100 @or_intror #q #W #U #H destruct
103 (* Basic_1: was: term_dec *)
104 lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
105 #T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
106 [1,4: @or_intror #H destruct
107 | elim (eq_item2_dec I1 I2) #HI
108 [ elim (IHV1 V2) -IHV1 #HV
109 [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
111 @or_intror #H destruct /2 width=1 by/
112 | elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
113 @or_intror #H destruct /2 width=1 by/
117 (* Basic inversion lemmas ***************************************************)
119 fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪[I1] = ⓪[I2] → I1 = I2.
120 #I1 #I2 #H destruct //
123 fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②[I1]T1.V1 = ②[I2]T2.V2 →
124 ∧∧T1 = T2 & I1 = I2 & V1 = V2.
125 #I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
128 lemma discr_tpair_xy_x: ∀I,T,V. ②[I]V.T = V → ⊥.
131 | #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
135 (* Basic_1: was: thead_x_y_y *)
136 lemma discr_tpair_xy_y: ∀I,V,T. ②[I]V.T = T → ⊥.
139 | #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
143 lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
144 (②[I]V1.T1 = ②[I]V2.T2 → ⊥) →
145 (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
146 #I #V1 #T1 #V2 #T2 #H
147 elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
148 @or_intror @conj // #HT12 destruct /2 width=1 by/
151 lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
152 (②[I] V1. T1 = ②[I]V2.T2 → ⊥) →
153 (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
154 #I #V1 #T1 #V2 #T2 #H
155 elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
156 @or_intror @conj // #HT12 destruct /2 width=1 by/
159 (* Basic_1: removed theorems 3:
160 not_void_abst not_abbr_void not_abst_void