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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 *)
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15 include "basic_2/relocation/lex.ma".
16 include "basic_2/relocation/drops_cext2.ma".
17 include "basic_2/relocation/drops_lexs.ma".
19 (* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
21 definition dedropable_sn: predicate … ≝
22 λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
23 ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
25 definition dropable_sn: predicate … ≝
26 λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
27 ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
29 definition dropable_dx: predicate … ≝
30 λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
31 ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
33 (* Properties with generic extension ****************************************)
35 (* Basic_2A1: was: lpx_sn_liftable_dedropable *)
36 lemma lex_liftable_dedropable_sn (R): c_reflexive … R →
37 d_liftable2_sn … lifts R → dedropable_sn R.
38 #R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 * #f1 #Hf1 #HK12
39 elim (lexs_liftable_co_dedropable_sn … HLK1 … HK12) -K1
40 /3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, coafter_isid_dx, ex3_intro, ex2_intro/
43 (* Inversion lemmas with generic extension **********************************)
45 (* Basic_2A1: was: lpx_sn_deliftable_dropable *)
46 lemma lex_dropable_sn (R): dropable_sn R.
47 #R #b #f #L1 #K1 #HLK1 #H1f #L2 * #f2 #Hf2 #HL12
48 elim (lexs_co_dropable_sn … HLK1 … HL12) -L1
49 /3 width=3 by coafter_isid_dx, ex2_intro/
52 (* Basic_2A1: was: lpx_sn_dropable *)
53 lemma lex_dropable_dx (R): dropable_dx R.
54 #R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #Hf
55 elim (lexs_co_dropable_dx … HL12 … HLK2) -L2
56 /3 width=5 by coafter_isid_dx, ex2_intro/
59 (* Basic_2A1: includes: lpx_sn_drop_conf *)
60 lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
61 ∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
62 ∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
63 #R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf
64 elim (lexs_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
65 [ #Z2 #K2 #HLK2 #HK12 #H
66 elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
67 /3 width=5 by ex3_2_intro, ex2_intro/
68 | /3 width=3 by coafter_isid_dx, isid_push/
72 (* Basic_2A1: includes: lpx_sn_drop_trans *)
73 lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
74 ∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
75 ∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
76 #R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf
77 elim (lexs_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
78 [ #Z1 #K1 #HLK1 #HK12 #H
79 elim (ext2_inv_pair_dx … H) -H #V1 #HV12 #H destruct
80 /3 width=5 by ex3_2_intro, ex2_intro/
81 | /3 width=3 by coafter_isid_dx, isid_push/