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14
15 include "basic_2/rt_transition/fpb_fqup.ma".
16 include "basic_2/rt_computation/fpbs.ma".
17
18 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
19
20 (* Advanced eliminators *****************************************************)
21
22 lemma fpbs_ind:
23       ∀G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
24       (∀G,G2,L,L2,T,T2. ❨G1,L1,T1❩ ≥ ❨G,L,T❩ → ❨G,L,T❩ ≽ ❨G2,L2,T2❩ → Q G L T → Q G2 L2 T2) →
25       ∀G2,L2,T2. ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩ → Q G2 L2 T2.
26 /3 width=8 by tri_TC_star_ind/ qed-.
27
28 lemma fpbs_ind_dx:
29       ∀G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
30       (∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≽ ❨G,L,T❩ → ❨G,L,T❩ ≥ ❨G2,L2,T2❩ → Q G L T → Q G1 L1 T1) →
31       ∀G1,L1,T1. ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩ → Q G1 L1 T1.
32 /3 width=8 by tri_TC_star_ind_dx/ qed-.
33
34 (* Advanced properties ******************************************************)
35
36 lemma fpbs_refl:
37       tri_reflexive … fpbs.
38 /2 width=1 by tri_inj/ qed.
39
40 (* Properties with plus-iterated structural successor for closures **********)
41
42 lemma fqup_fpbs:
43       ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂+ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
44 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
45 /4 width=5 by fqu_fquq, fquq_fpb, tri_step/
46 qed.
47
48 lemma fpbs_fqup_trans:
49       ∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≥ ❨G,L,T❩ →
50       ∀G2,L2,T2. ❨G,L,T❩ ⬂+ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
51 #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
52 /3 width=5 by fpbs_strap1, fqu_fpb/
53 qed-.
54
55 lemma fqup_fpbs_trans:
56       ∀G,G2,L,L2,T,T2. ❨G,L,T❩ ≥ ❨G2,L2,T2❩ →
57       ∀G1,L1,T1. ❨G1,L1,T1❩ ⬂+ ❨G,L,T❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
58 #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -T1
59 /3 width=9 by fpbs_strap2, fqu_fpb/
60 qed-.