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 "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 extended rt-computation on full local environments ******)
27 ∀G,L1,L2,T. ❪G,L1❫ ⊢ ⬈* L2 → ❪G,L1,T❫ ≥ ❪G,L2,T❫.
28 #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
29 /3 width=5 by fpbq_lpx, fpbs_strap1/
32 lemma fpbs_lpxs_trans:
33 ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L,T2❫ →
34 ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
35 #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
36 /3 width=5 by fpbs_strap1, fpbq_lpx/
39 lemma lpxs_fpbs_trans:
40 ∀G1,G2,L,L2,T1,T2. ❪G1,L,T1❫ ≥ ❪G2,L2,T2❫ →
41 ∀L1. ❪G1,L1❫ ⊢ ⬈* L → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
42 #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
43 /3 width=5 by fpbs_strap2, fpbq_lpx/
46 (* Basic_2A1: uses: lpxs_lleq_fpbs *)
48 ∀G1,L1,L,T1. ❪G1,L1❫ ⊢ ⬈* L →
49 ∀G2,L2,T2. ❪G1,L,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
50 /3 width=3 by lpxs_fpbs_trans, feqx_fpbs/ qed.
53 ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L,T2❫ →
54 ∀L2. ❪G2,L❫ ⊢ ⬈ L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
55 /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
57 (* Properties with star-iterated structural successor for closures **********)
60 ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ⬂* ❪G2,L,T2❫ →
61 ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
62 /3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
64 (* Properties with extended context-sensitive parallel rt-computation *******)
66 lemma cpxs_fqus_lpxs_fpbs:
67 ∀G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈* T →
68 ∀G2,L,T2. ❪G1,L1,T❫ ⬂* ❪G2,L,T2❫ →
69 ∀L2.❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
70 /3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
72 lemma fpbs_cpxs_teqx_fqup_lpx_trans:
73 ∀G1,G3,L1,L3,T1,T3. ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ →
74 ∀T4. ❪G3,L3❫ ⊢ T3 ⬈* T4 → ∀T5. T4 ≛ T5 →
75 ∀G2,L4,T2. ❪G3,L3,T5❫ ⬂+ ❪G2,L4,T2❫ →
76 ∀L2. ❪G2,L4❫ ⊢ ⬈ L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
77 #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
78 @(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
79 @(fpbs_fqup_trans … H34) -G2 -L4 -T2
80 /3 width=3 by fpbs_cpxs_trans, fpbs_teqx_trans/
83 (* Advanced properties ******************************************************)
85 (* Basic_2A1: uses: fpbs_intro_alt *)
86 lemma fpbs_intro_star:
87 ∀G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈* T →
88 ∀G,L,T0. ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ →
90 ∀G2,L2,T2. ❪G,L0,T0❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ .
91 /3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_feqx/ qed.
93 (* Advanced inversion lemmas *************************************************)
95 (* Basic_2A1: uses: fpbs_inv_alt *)
97 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ →
98 ∃∃G,L,L0,T,T0. ❪G1,L1❫ ⊢ T1 ⬈* T & ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ & ❪G,L❫ ⊢ ⬈* L0 & ❪G,L0,T0❫ ≛ ❪G2,L2,T2❫.
99 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
100 [ /2 width=9 by ex4_5_intro/
101 | #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
102 [ #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
103 elim (fquq_cpxs_trans … HT03 … H10) -T0
104 /3 width=9 by fqus_strap2, ex4_5_intro/
105 | #T0 #HT10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
106 /3 width=9 by cpxs_strap2, ex4_5_intro/
107 | #L0 #HL10 #_ * #G3 #L3 #L4 #T3 #T4 #HT13 #H34 #HL34 #H42
108 lapply (lpx_cpxs_trans … HT13 … HL10) -HT13 #HT13
109 elim (lpx_fqus_trans … H34 … HL10) -L0
110 /3 width=9 by lpxs_step_sn, cpxs_trans, ex4_5_intro/
111 | #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
112 elim (feqx_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
113 elim (feqx_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
114 elim (feqx_lpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
115 /3 width=13 by feqx_trans, ex4_5_intro/