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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/notation/relations/lrsubeq_4.ma".
16 include "basic_2/static/aaa.ma".
17 include "basic_2/computation/acp_cr.ma".
19 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
21 inductive lsubc (RP) (G): relation lenv ≝
22 | lsubc_atom: lsubc RP G (⋆) (⋆)
23 | lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
24 | lsubc_abbr: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
25 lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
29 "local environment refinement (abstract candidates of reducibility)"
30 'LRSubEq RP G L1 L2 = (lsubc RP G L1 L2).
32 (* Basic inversion lemmas ***************************************************)
34 fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⊑[RP] L2 → L1 = ⋆ → L2 = ⋆.
35 #RP #G #L1 #L2 * -L1 -L2
37 | #I #L1 #L2 #V #_ #H destruct
38 | #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
42 (* Basic_1: was just: csubc_gen_sort_r *)
43 lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⊑[RP] L2 → L2 = ⋆.
44 /2 width=5 by lsubc_inv_atom1_aux/ qed-.
46 fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⊑[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
47 (∃∃K2. G ⊢ K1 ⊑[RP] K2 & L2 = K2.ⓑ{I}X) ∨
48 ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
50 L2 = K2. ⓛW & X = ⓝW.V & I = Abbr.
51 #RP #G #L1 #L2 * -L1 -L2
52 [ #I #K1 #V #H destruct
53 | #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
54 | #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10/
58 (* Basic_1: was: csubc_gen_head_r *)
59 lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⊑[RP] L2 →
60 (∃∃K2. G ⊢ K1 ⊑[RP] K2 & L2 = K2.ⓑ{I}X) ∨
61 ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
63 L2 = K2.ⓛW & X = ⓝW.V & I = Abbr.
64 /2 width=3 by lsubc_inv_pair1_aux/ qed-.
66 fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⊑[RP] L2 → L2 = ⋆ → L1 = ⋆.
67 #RP #G #L1 #L2 * -L1 -L2
69 | #I #L1 #L2 #V #_ #H destruct
70 | #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
74 (* Basic_1: was just: csubc_gen_sort_l *)
75 lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⊑[RP] ⋆ → L1 = ⋆.
76 /2 width=5 by lsubc_inv_atom2_aux/ qed-.
78 fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⊑[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
79 (∃∃K1. G ⊢ K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨
80 ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
82 L1 = K1.ⓓⓝW.V & I = Abst.
83 #RP #G #L1 #L2 * -L1 -L2
84 [ #I #K2 #W #H destruct
85 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
86 | #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8/
90 (* Basic_1: was just: csubc_gen_head_l *)
91 lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⊑[RP] K2.ⓑ{I} W →
92 (∃∃K1. G ⊢ K1 ⊑[RP] K2 & L1 = K1.ⓑ{I} W) ∨
93 ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
95 L1 = K1.ⓓⓝW.V & I = Abst.
96 /2 width=3 by lsubc_inv_pair2_aux/ qed-.
98 (* Basic properties *********************************************************)
100 (* Basic_1: was just: csubc_refl *)
101 lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⊑[RP] L.
102 #RP #G #L elim L -L // /2 width=1/
105 (* Basic_1: removed theorems 3:
106 csubc_clear_conf csubc_getl_conf csubc_csuba