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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/static/feqx_feqx.ma".
16 include "basic_2/rt_transition/fpbq_fpb.ma".
17 include "basic_2/rt_computation/fpbs_fqup.ma".
18 include "basic_2/rt_computation/fpbg.ma".
20 (* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
22 (* Advanced forward lemmas **************************************************)
26 ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
27 #G1 #G2 #L1 #L2 #T1 #T2 *
28 /3 width=5 by fpbs_strap2, fpb_fpbq/
31 (* Advanced properties with sort-irrelevant equivalence on closures *********)
33 (* Basic_2A1: uses: fleq_fpbg_trans *)
34 lemma feqx_fpbg_trans:
35 ∀G,G2,L,L2,T,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ →
36 ∀G1,L1,T1. ❪G1,L1,T1❫ ≛ ❪G,L,T❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
37 #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
38 elim (feqx_fpb_trans … H1 … H0) -G -L -T
39 /4 width=9 by fpbs_strap2, fpbq_feqx, ex2_3_intro/
42 (* Properties with parallel proper rst-reduction on closures ****************)
45 ∀G1,G,G2,L1,L,L2,T1,T,T2.
46 ❪G1,L1,T1❫ ≻ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ →
47 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
48 /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
50 (* Properties with parallel rst-reduction on closures ***********************)
52 lemma fpbq_fpbg_trans:
53 ∀G1,G,G2,L1,L,L2,T1,T,T2.
54 ❪G1,L1,T1❫ ≽ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ →
55 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
56 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
57 elim (fpbq_inv_fpb … H1) -H1
58 /2 width=5 by feqx_fpbg_trans, fpb_fpbg_trans/
61 (* Properties with parallel rst-compuutation on closures ********************)
63 lemma fpbs_fpbg_trans:
64 ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ →
65 ∀G2,L2,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
66 #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
69 (* Advanced properties with plus-iterated structural successor for closures *)
71 lemma fqup_fpbg_trans:
72 ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ⬂+ ❪G,L,T❫ →
73 ∀G2,L2,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
74 /3 width=5 by fpbs_fpbg_trans, fqup_fpbs/ qed-.
76 (* Advanced inversion lemmas of parallel rst-computation on closures ********)
78 (* Basic_2A1: was: fpbs_fpbg *)
80 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ →
81 ∨∨ ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫
82 | ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
83 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
84 [ /2 width=1 by or_introl/
85 | #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
86 elim (fpbq_inv_fpb … H2) -H2 #H2
87 [ /3 width=5 by feqx_trans, or_introl/
88 | elim (feqx_fpb_trans … H1 … H2) -G -L -T
89 /4 width=5 by ex2_3_intro, or_intror, feqx_fpbs/
90 | /3 width=5 by fpbg_feqx_trans, or_intror/
91 | /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
96 (* Advanced properties of parallel rst-computation on closures **************)
99 ∀F1,F2,K1,K2,T1,T2. ❪F1,K1,T1❫ ≥ ❪F2,K2,T2❫ →
100 ∀G2,L2,U2. ❪F2,K2,T2❫ ≻ ❪G2,L2,U2❫ →
101 ∃∃G1,L1,U1. ❪F1,K1,T1❫ ≻ ❪G1,L1,U1❫ & ❪G1,L1,U1❫ ≥ ❪G2,L2,U2❫.
102 #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
103 [ #H12 #G2 #L2 #U2 #H2 elim (feqx_fpb_trans … H12 … H2) -F2 -K2 -T2
104 /3 width=5 by feqx_fpbs, ex2_3_intro/
105 | * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
106 @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/