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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/static/aaa_lift.ma".
16 include "basic_2/computation/lsubc.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
20 (* Properties concerning basic local environment slicing ********************)
22 (* Basic_1: was: csubc_drop_conf_O *)
23 (* Note: the constant 0 can not be generalized *)
24 lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
25 ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ⊑[RP] K2.
26 #RP #L1 #L2 #H elim H -L1 -L2
28 >(ldrop_inv_atom1 … H) -H /2 width=3/
29 | #I #L1 #L2 #V #_ #IHL12 #X #e #H
30 elim (ldrop_inv_O1 … H) -H * #He #H destruct
31 [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/
32 | elim (IHL12 … H) -L2 /3 width=3/
34 | #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H
35 elim (ldrop_inv_O1 … H) -H * #He #H destruct
36 [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/
37 | elim (IHL12 … H) -L2 /3 width=3/
41 (* Basic_1: was: csubc_drop_conf_rev *)
42 lemma ldrop_lsubc_trans: ∀RR,RS,RP.
43 acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
44 ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 →
45 ∃∃L2. L1 ⊑[RP] L2 & ⇩[d, e] L2 ≡ K2.
46 #RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
48 >(lsubc_inv_atom1 … H) -H /2 width=3/
50 elim (lsubc_inv_pair1 … H) -H *
51 [ #K1 #HLK1 #H destruct /3 width=3/
52 | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/
54 | #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
55 elim (IHLK1 … HK12) -K1 /3 width=5/
56 | #L1 #K1 #I #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H
57 elim (lsubc_inv_pair1 … H) -H *
58 [ #K2 #HK12 #H destruct
59 elim (IHLK1 … HK12) -K1 /3 width=5/
60 | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct
61 elim (IHLK1 … HK12) #K #HL1K #HK2
62 lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA
63 lapply (s7 … HA … HV2 … HLK1 HV21) -HV2
64 elim (lift_total W2 d e) /4 width=9/