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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_2A/static/aaa_lift.ma".
16 include "basic_2A/computation/lsubc.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC 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_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
25 ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
26 #RP #G #L1 #L2 #H elim H -L1 -L2
27 [ #X #s #m #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
28 | #I #L1 #L2 #V #_ #IHL12 #X #s #m #H
29 elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
30 [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
31 /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
32 | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
34 | #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #m #H
35 elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
36 [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
37 /3 width=8 by lsubc_beta, drop_pair, ex2_intro/
38 | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
43 (* Basic_1: was: csubc_drop_conf_rev *)
44 lemma drop_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
45 ∀G,L1,K1,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
46 ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, m] L2 ≡ K2.
47 #RR #RS #RP #Hgcp #G #L1 #K1 #l #m #H elim H -L1 -K1 -l -m
48 [ #l #m #Hm #X #H elim (lsubc_inv_atom1 … H) -H
49 >Hm /2 width=3 by ex2_intro/
51 elim (lsubc_inv_pair1 … H) -H *
52 [ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
53 | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
54 /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
56 | #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12
57 elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
58 | #I #L1 #K1 #V1 #V2 #l #m #HLK1 #HV21 #IHLK1 #X #H
59 elim (lsubc_inv_pair1 … H) -H *
60 [ #K2 #HK12 #H destruct
61 elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
62 | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
63 elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
64 elim (IHLK1 … HK12) #K #HL1K #HK2
65 lapply (gcr_lift … Hgcp … HV2 … HLK1 … HV3) -HV2
66 lapply (gcr_lift … Hgcp … H1W2 … HLK1 … HW23) -H1W2
67 /4 width=11 by lsubc_beta, aaa_lift, drop_skip, ex2_intro/