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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 include "basic_2/static/aaa.ma".
16 include "basic_2/computation/acp_cr.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
20 inductive lsubc (RP:lenv→predicate term): relation lenv ≝
21 | lsubc_atom: lsubc RP (⋆) (⋆)
22 | lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V)
23 | lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ÷ A →
24 lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW)
28 "local environment refinement (abstract candidates of reducibility)"
29 'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
31 (* Basic inversion lemmas ***************************************************)
33 fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
36 | #I #L1 #L2 #V #_ #H destruct
37 | #L1 #L2 #V #W #A #_ #_ #_ #H destruct
41 (* Basic_1: was: csubc_gen_sort_r *)
42 lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
45 fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
46 (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
47 ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
49 L2 = K2. ⓛW & I = Abbr.
51 [ #I #K1 #V #H destruct
52 | #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
53 | #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
57 (* Basic_1: was: csubc_gen_head_r *)
58 lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
59 (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
60 ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
62 L2 = K2. ⓛW & I = Abbr.
65 fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
68 | #I #L1 #L2 #V #_ #H destruct
69 | #L1 #L2 #V #W #A #_ #_ #_ #H destruct
73 (* Basic_1: was: csubc_gen_sort_l *)
74 lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
77 fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
78 (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
79 ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
81 L1 = K1. ⓓV & I = Abst.
83 [ #I #K2 #W #H destruct
84 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
85 | #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
89 (* Basic_1: was: csubc_gen_head_l *)
90 lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. ⓑ{I} W →
91 (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
92 ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
94 L1 = K1. ⓓV & I = Abst.
97 (* Basic properties *********************************************************)
99 (* Basic_1: was: csubc_refl *)
100 lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
101 #RP #L elim L -L // /2 width=1/
104 (* Basic_1: removed theorems 2: csubc_clear_conf csubc_getl_conf *)