1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/static/ffdeq_fqus.ma".
16 include "basic_2/static/ffdeq_ffdeq.ma".
17 include "basic_2/rt_computation/cpxs_fqus.ma".
18 include "basic_2/rt_computation/cpxs_ffdeq.ma".
19 include "basic_2/rt_computation/cpxs_lfpx.ma".
20 include "basic_2/rt_computation/lfpxs_ffdeq.ma".
21 include "basic_2/rt_computation/fpbs_cpxs.ma".
23 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
25 (* Properties with unbound parallel rt-computation on referred entries ******)
27 (* Basic_2A1: uses: lpxs_fpbs *)
28 lemma lfpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
29 #h #o #G #L1 #L2 #T #H @(lfpxs_ind_sn … H) -L2
30 /3 width=5 by fpbq_lfpx, fpbs_strap1/
33 (* Basic_2A1: uses: fpbs_lpxs_trans *)
34 lemma fpbs_lfpxs_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
35 ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
36 #h #o #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lfpxs_ind_sn … H) -L2
37 /3 width=5 by fpbs_strap1, fpbq_lfpx/
40 (* Basic_2A1: uses: lpxs_fpbs_trans *)
41 lemma lfpxs_fpbs_trans: ∀h,o,G1,G2,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
42 ∀L1. ⦃G1, L1⦄ ⊢ ⬈*[h, T1] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
43 #h #o #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lfpxs_ind_dx … H) -L1
44 /3 width=5 by fpbs_strap2, fpbq_lfpx/
47 (* Basic_2A1: uses: lpxs_lleq_fpbs *)
48 lemma lfpxs_ffdeq_fpbs: ∀h,o,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, T1] L →
49 ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
50 /3 width=3 by lfpxs_fpbs_trans, ffdeq_fpbs/ qed.
52 lemma fpbs_lfpx_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
53 ∀L2. ⦃G2, L⦄ ⊢ ⬈[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
54 /3 width=3 by fpbs_lfpxs_trans, lfpx_lfpxs/ qed-.
56 (* Properties with star-iterated structural successor for closures **********)
58 (* Basic_2A1: uses: fqus_lpxs_fpbs *)
59 lemma fqus_lfpxs_fpbs: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
60 ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
61 /3 width=3 by fpbs_lfpxs_trans, fqus_fpbs/ qed.
63 (* Properties with unbound context-sensitive parallel rt-computation ********)
65 (* Basic_2A1: uses: cpxs_fqus_lpxs_fpbs *)
66 lemma cpxs_fqus_lfpxs_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
67 ∀G2,L,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ →
68 ∀L2.⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
69 /3 width=5 by cpxs_fqus_fpbs, fpbs_lfpxs_trans/ qed.
71 lemma fpbs_cpxs_tdeq_fqup_lfpx_trans: ∀h,o,G1,G3,L1,L3,T1,T3. ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G3, L3, T3⦄ →
72 ∀T4. ⦃G3, L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛[h, o] T5 →
73 ∀G2,L4,T2. ⦃G3, L3, T5⦄ ⊐+ ⦃G2, L4, T2⦄ →
74 ∀L2. ⦃G2, L4⦄ ⊢ ⬈[h, T2] L2 → ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G2, L2, T2⦄.
75 #h #o #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
76 @(fpbs_lfpx_trans … HL42) -L2 (**) (* full auto too slow *)
77 @(fpbs_fqup_trans … H34) -G2 -L4 -T2
78 /3 width=3 by fpbs_cpxs_trans, fpbs_tdeq_trans/
81 (* Advanced properties ******************************************************)
83 lemma fpbs_intro_fstar: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
84 ∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ →
85 ∀L0. ⦃G, L⦄ ⊢ ⬈*[h, T0] L0 →
86 ∀G2,L2,T2. ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
87 /3 width=5 by cpxs_fqus_lfpxs_fpbs, fpbs_strap1, fpbq_ffdeq/ qed.
89 (* Advanced inversion lemmas *************************************************)
91 lemma fpbs_inv_fstar: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
92 ∃∃G,L,L0,T,T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄
93 & ⦃G, L⦄ ⊢ ⬈*[h, T0] L0 & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
94 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
95 [ /2 width=9 by ex4_5_intro/
96 | #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
97 [ #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
98 elim (fquq_cpxs_trans … HT03 … H10) -T0
99 /3 width=9 by fqus_strap2, ex4_5_intro/
100 | #T0 #HT10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
101 /3 width=9 by cpxs_strap2, ex4_5_intro/
102 | #L0 #HL10 #_ * #G3 #L3 #L4 #T3 #T4 #HT13 #H34 #HL34 #H42
103 lapply (lfpx_cpxs_trans … HT13 … HL10) -HT13 #HT13
104 lapply (lfpx_cpxs_conf … HT13 … HL10) -HL10 #HL10
105 elim (lfpx_fqus_trans … H34 … HL10) -L0
106 /3 width=9 by lfpxs_step_sn, cpxs_trans, ex4_5_intro/
107 | #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
108 elim (ffdeq_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
109 elim (ffdeq_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
110 elim (ffdeq_lfpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
111 /3 width=13 by ffdeq_trans, ex4_5_intro/