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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/dynamic/lsubsv.ma".
17 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
19 (* Properties concerning basic local environment slicing ********************)
21 (* Note: the constant 0 cannot be generalized *)
22 lemma lsubsv_ldrop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
23 ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
24 ∃∃K2. G ⊢ K1 ¡⫃[h, g] K2 & ⇩[s, 0, e] L2 ≡ K2.
25 #h #g #G #L1 #L2 #H elim H -L1 -L2
26 [ /2 width=3 by ex2_intro/
27 | #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
28 elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
30 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
31 <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, ldrop_pair, ex2_intro/
32 | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
34 | #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K1 #s #e #H
35 elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
37 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
38 <(ldrop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, ldrop_pair, ex2_intro/
39 | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
44 (* Note: the constant 0 cannot be generalized *)
45 lemma lsubsv_ldrop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⫃[h, g] L2 →
46 ∀K2,s, e. ⇩[s, 0, e] L2 ≡ K2 →
47 ∃∃K1. G ⊢ K1 ¡⫃[h, g] K2 & ⇩[s, 0, e] L1 ≡ K1.
48 #h #g #G #L1 #L2 #H elim H -L1 -L2
49 [ /2 width=3 by ex2_intro/
50 | #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
51 elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
53 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
54 <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, ldrop_pair, ex2_intro/
55 | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
57 | #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K2 #s #e #H
58 elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
60 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
61 <(ldrop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_abbr, ldrop_pair, ex2_intro/
62 | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/