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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 (b):
23 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂[b] ❨G2,L2,T2❩ →
25 ∃∃K2,T. ❨G1,K1❩ ⊢ T1 ⬈ T & ❨G1,K1,T❩ ⬂[b] ❨G2,K2,T2❩ & ❨G2,K2❩ ⊢ ⬈ L2.
26 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
28 elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
29 elim (lifts_total V (𝐔❨1❩)) #T #HVT
30 /3 width=5 by cpx_delta, fqu_drop, ex3_2_intro/
31 | /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
32 | /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
33 | /3 width=5 by lpx_bind_refl_dx, cpx_pair_sn, fqu_clear, ex3_2_intro/
34 | /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
35 | #I #G #K #T #U #HTU #K1 #H
36 elim (lpx_inv_bind_dx … H) -H #I0 #K0 #HK0 #HI0 #H destruct
37 /3 width=5 by fqu_drop, ex3_2_intro/
41 lemma fqu_lpx_trans (b):
42 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂[b] ❨G2,L2,T2❩ →
44 ∃∃K1,T. ❨G1,L1❩ ⊢ ⬈ K1 & ❨G1,L1❩ ⊢ T1 ⬈ T & ❨G1,K1,T❩ ⬂[b] ❨G2,K2,T2❩.
45 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
46 [ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
47 | /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
48 | #p #I #G2 #L2 #V2 #T2 #Hb #X #H
49 elim (lpx_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
50 /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
51 | #p #I #G2 #L2 #V2 #T2 #Hb #X #H
52 elim (lpx_inv_unit_sn … H) -H #K2 #HLK2 #H destruct
53 /3 width=5 by cpx_pair_sn, fqu_clear, ex3_2_intro/
54 | /3 width=5 by cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
55 | /3 width=5 by lpx_bind_refl_dx, fqu_drop, ex3_2_intro/
59 (* Properties with extended optional structural successor for closures ******)
61 lemma lpx_fquq_trans (b):
62 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂⸮[b] ❨G2,L2,T2❩ →
64 ∃∃K2,T. ❨G1,K1❩ ⊢ T1 ⬈ T & ❨G1,K1,T❩ ⬂⸮[b] ❨G2,K2,T2❩ & ❨G2,K2❩ ⊢ ⬈ L2.
65 #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
66 [ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
67 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
71 lemma fquq_lpx_trans (b):
72 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂⸮[b] ❨G2,L2,T2❩ →
74 ∃∃K1,T. ❨G1,L1❩ ⊢ ⬈ K1 & ❨G1,L1❩ ⊢ T1 ⬈ T & ❨G1,K1,T❩ ⬂⸮[b] ❨G2,K2,T2❩.
75 #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
76 [ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
77 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/