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14
15 include "static_2/s_computation/fqus_fqup.ma".
16 include "static_2/static/fdeq_fqup.ma".
17 include "basic_2/rt_computation/fpbs_fqus.ma".
18
19 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
20
21 (* Advanced properties ******************************************************)
22
23 lemma tdeq_fpbs_trans: ∀h,o,T1,T. T1 ≛[h, o] T →
24                        ∀G1,G2,L1,L2,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
25 /3 width=5 by fdeq_fpbs_trans, tdeq_fdeq/ qed-.
26
27 lemma fpbs_tdeq_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T⦄ →
28                        ∀T2. T ≛[h, o] T2 →  ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
29 /3 width=5 by fpbs_fdeq_trans, tdeq_fdeq/ qed-.
30
31 (* Properties with plus-iterated structural successor for closures **********)
32
33 lemma fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
34                  ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
35 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 
36 /4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
37 qed.
38
39 lemma fpbs_fqup_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
40                        ∀G2,L2,T2. ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
41 /3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
42
43 lemma fqup_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
44                        ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
45 /3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-.