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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_fqus.ma".
16 include "static_2/static/feqx_feqx.ma".
17 include "basic_2/rt_computation/cpxs_fqus.ma".
18 include "basic_2/rt_computation/cpxs_feqx.ma".
19 include "basic_2/rt_computation/lpxs_feqx.ma".
20 include "basic_2/rt_computation/fpbs_cpxs.ma".
22 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
24 (* Properties with unbound rt-computation on full local environments *******)
26 lemma lpxs_fpbs: ∀h,G,L1,L2,T. ❪G,L1❫ ⊢ ⬈*[h] L2 → ❪G,L1,T❫ ≥[h] ❪G,L2,T❫.
27 #h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
28 /3 width=5 by fpbq_lpx, fpbs_strap1/
31 lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L,T2❫ →
32 ∀L2. ❪G2,L❫ ⊢ ⬈*[h] L2 → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
33 #h #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
34 /3 width=5 by fpbs_strap1, fpbq_lpx/
37 lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ❪G1,L,T1❫ ≥[h] ❪G2,L2,T2❫ →
38 ∀L1. ❪G1,L1❫ ⊢ ⬈*[h] L → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
39 #h #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
40 /3 width=5 by fpbs_strap2, fpbq_lpx/
43 (* Basic_2A1: uses: lpxs_lleq_fpbs *)
44 lemma lpxs_feqx_fpbs: ∀h,G1,L1,L,T1. ❪G1,L1❫ ⊢ ⬈*[h] L →
45 ∀G2,L2,T2. ❪G1,L,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
46 /3 width=3 by lpxs_fpbs_trans, feqx_fpbs/ qed.
48 lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L,T2❫ →
49 ∀L2. ❪G2,L❫ ⊢ ⬈[h] L2 → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
50 /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
52 (* Properties with star-iterated structural successor for closures **********)
54 lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ⬂* ❪G2,L,T2❫ →
55 ∀L2. ❪G2,L❫ ⊢ ⬈*[h] L2 → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
56 /3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
58 (* Properties with unbound context-sensitive parallel rt-computation ********)
60 lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈*[h] T →
61 ∀G2,L,T2. ❪G1,L1,T❫ ⬂* ❪G2,L,T2❫ →
62 ∀L2.❪G2,L❫ ⊢ ⬈*[h] L2 → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
63 /3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
65 lemma fpbs_cpxs_teqx_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ❪G1,L1,T1❫ ≥ [h] ❪G3,L3,T3❫ →
66 ∀T4. ❪G3,L3❫ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 →
67 ∀G2,L4,T2. ❪G3,L3,T5❫ ⬂+ ❪G2,L4,T2❫ →
68 ∀L2. ❪G2,L4❫ ⊢ ⬈[h] L2 → ❪G1,L1,T1❫ ≥ [h] ❪G2,L2,T2❫.
69 #h #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
70 @(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
71 @(fpbs_fqup_trans … H34) -G2 -L4 -T2
72 /3 width=3 by fpbs_cpxs_trans, fpbs_teqx_trans/
75 (* Advanced properties ******************************************************)
77 (* Basic_2A1: uses: fpbs_intro_alt *)
78 lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈*[h] T →
79 ∀G,L,T0. ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ →
80 ∀L0. ❪G,L❫ ⊢ ⬈*[h] L0 →
81 ∀G2,L2,T2. ❪G,L0,T0❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ .
82 /3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_feqx/ qed.
84 (* Advanced inversion lemmas *************************************************)
86 (* Basic_2A1: uses: fpbs_inv_alt *)
87 lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ →
88 ∃∃G,L,L0,T,T0. ❪G1,L1❫ ⊢ T1 ⬈*[h] T & ❪G1,L1,T❫ ⬂* ❪G,L,T0❫
89 & ❪G,L❫ ⊢ ⬈*[h] L0 & ❪G,L0,T0❫ ≛ ❪G2,L2,T2❫.
90 #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
91 [ /2 width=9 by ex4_5_intro/
92 | #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
93 [ #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
94 elim (fquq_cpxs_trans … HT03 … H10) -T0
95 /3 width=9 by fqus_strap2, ex4_5_intro/
96 | #T0 #HT10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
97 /3 width=9 by cpxs_strap2, ex4_5_intro/
98 | #L0 #HL10 #_ * #G3 #L3 #L4 #T3 #T4 #HT13 #H34 #HL34 #H42
99 lapply (lpx_cpxs_trans … HT13 … HL10) -HT13 #HT13
100 elim (lpx_fqus_trans … H34 … HL10) -L0
101 /3 width=9 by lpxs_step_sn, cpxs_trans, ex4_5_intro/
102 | #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
103 elim (feqx_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
104 elim (feqx_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
105 elim (feqx_lpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
106 /3 width=13 by feqx_trans, ex4_5_intro/