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3 (*      ||M||                                                             *)
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14
15 include "basic_2/s_transition/fquq.ma".
16 include "basic_2/rt_transition/cpm_drops.ma".
17 include "basic_2/rt_transition/cpr.ma".
18 include "basic_2/rt_transition/lfpr_fqup.ma".
19
20 (* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************)
21
22 (* Properties with supclosure ***********************************************)
23
24 lemma fqu_cpr_trans_dx: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
25                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
26                         ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
27 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
28 /3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
29 #I #G #L #V #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
30 /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
31 qed-.
32
33 lemma fqu_cpr_trans_sn: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
34                         ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
35                         ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
36 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
37 /3 width=5 by lfpr_pair, cpr_pair_sn, cpr_flat, cpm_bind, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex3_2_intro/
38 #I #G #L #V #U #T #HUT #U2 #HU2 elim (cpm_lifts_sn … HU2 (Ⓣ) … HUT) -U
39 /3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
40 qed-.
41
42 (* Properties with optional supclosure **************************************)
43
44 lemma fquq_cpr_trans_dx: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
45                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
46                          ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
47 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
48 [ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
49 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
50 ]
51 qed-.
52
53 lemma fquq_cpr_trans_sn: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
54                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
55                          ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
56 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
57 [ #HT12 #U2 #HTU2 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
58 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
59 ]
60 qed-.