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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 *)
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15 include "static_2/s_transition/fquq.ma".
16 include "basic_2/rt_transition/lpx.ma".
18 (* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
20 (* Properties with extended structural successor for closures ***************)
22 lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
23 ∀K1. ❪G1,K1❫ ⊢ ⬈[h] L1 →
24 ∃∃K2,T. ❪G1,K1❫ ⊢ T1 ⬈[h] T & ❪G1,K1,T❫ ⬂[b] ❪G2,K2,T2❫ & ❪G2,K2❫ ⊢ ⬈[h] L2.
25 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
27 elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
28 elim (lifts_total V (𝐔❨1❩)) #T #HVT
29 /3 width=5 by cpx_delta, fqu_drop, ex3_2_intro/
30 | /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
31 | /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
32 | /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_clear, ex3_2_intro/
33 | /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
34 | #I #G #K #T #U #HTU #K1 #H
35 elim (lpx_inv_bind_dx … H) -H #I0 #K0 #HK0 #HI0 #H destruct
36 /3 width=5 by fqu_drop, ex3_2_intro/
40 lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
41 ∀K2. ❪G2,L2❫ ⊢ ⬈[h] K2 →
42 ∃∃K1,T. ❪G1,L1❫ ⊢ ⬈[h] K1 & ❪G1,L1❫ ⊢ T1 ⬈[h] T & ❪G1,K1,T❫ ⬂[b] ❪G2,K2,T2❫.
43 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
44 [ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
45 | /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
46 | #p #I #G2 #L2 #V2 #T2 #Hb #X #H
47 elim (lpx_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
48 /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
49 | #p #I #G2 #L2 #V2 #T2 #Hb #X #H
50 elim (lpx_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
51 /3 width=5 by cpx_pair_sn, fqu_clear, ex3_2_intro/
52 | /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
53 | /3 width=5 by lpx_bind_refl_dx, fqu_drop, ex3_2_intro/
57 (* Properties with extended optional structural successor for closures ******)
59 lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
60 ∀K1. ❪G1,K1❫ ⊢ ⬈[h] L1 →
61 ∃∃K2,T. ❪G1,K1❫ ⊢ T1 ⬈[h] T & ❪G1,K1,T❫ ⬂⸮[b] ❪G2,K2,T2❫ & ❪G2,K2❫ ⊢ ⬈[h] L2.
62 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
63 [ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
64 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
68 lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
69 ∀K2. ❪G2,L2❫ ⊢ ⬈[h] K2 →
70 ∃∃K1,T. ❪G1,L1❫ ⊢ ⬈[h] K1 & ❪G1,L1❫ ⊢ T1 ⬈[h] T & ❪G1,K1,T❫ ⬂⸮[b] ❪G2,K2,T2❫.
71 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
72 [ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
73 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/