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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/rt_transition/cpx_fqus.ma".
16 include "basic_2/rt_transition/lfpx_fquq.ma".
17 include "basic_2/rt_computation/cpxs_drops.ma".
18 include "basic_2/rt_computation/cpxs_cpxs.ma".
20 (* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
22 (* Properties with unbound parallel rt-transition on referred entries *******)
24 lemma lfpx_cpxs_conf: ∀h,G. s_r_confluent1 … (cpxs h G) (lfpx h G).
25 /3 width=5 by lfpx_cpx_conf, s_r_conf1_CTC1/ qed-.
27 lemma lfpx_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpx h G).
28 #h #G #L2 #T1 #T2 #H @(cpx_ind … H) -G -L2 -T1 -T2 //
29 [ #G #L2 #s1 #L1 #H elim (lfpx_inv_sort … H) -H * /2 width=1 by cpx_cpxs/
30 | #I #G #L2 #V #V2 #W2 #_ #IH #HVW2 #Y1 #H
31 elim (lfpx_inv_zero_pair_dx … H) -H #L1 #V1 #HL1 #HV1 #H destruct
32 /5 width=3 by lfpx_cpx_conf, cpxs_delta, cpxs_strap2/
33 | #I2 #G #L2 #V2 #W2 #i #_ #IH #HVW2 #Y1 #H
34 elim (lfpx_inv_lref_bind_dx … H) -H #I1 #L1 #HL1 #H destruct
35 /4 width=3 by cpxs_lref, cpxs_strap2/
36 | #p #I #G #L2 #V #V2 #T #T2 #_ #_ #IHV #IHT #L1 #H
37 elim (lfpx_inv_bind … H) -H /3 width=1 by cpxs_bind/
38 | #I #G #L2 #V #V2 #T #T2 #_ #_ #IHV #IHT #L1 #H
39 elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_flat/
40 | #G #L2 #V #T #T2 #T0 #_ #IH #HT02 #L1 #H
41 elim (lfpx_inv_bind … H) -H /3 width=3 by cpxs_zeta/
42 | #G #L2 #V #T #T2 #_ #IH #L1 #H
43 elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_eps/
44 | #G #L2 #V #V2 #T #_ #IH #L1 #H
45 elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_ee/
46 | #p #G #L2 #V #V2 #W #W2 #T #T2 #_ #_ #_ #IHV #IHW #IHT #L1 #H
47 elim (lfpx_inv_flat … H) -H #HV #H
48 elim (lfpx_inv_bind … H) -H /3 width=1 by cpxs_beta/
49 | #p #G #L2 #V #V2 #V0 #W #W2 #T #T2 #_ #_ #_ #IHV #IHW #IHT #HV20 #L1 #H
50 elim (lfpx_inv_flat … H) -H #HV #H
51 elim (lfpx_inv_bind … H) -H /3 width=3 by cpxs_theta/
55 lemma lfpx_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpx h G).
56 /3 width=6 by lfpx_cpx_conf, lfpx_cpx_trans, s_r_trans_CTC1/
59 (* Properties with plus-iterated structural successor for closures **********)
61 lemma lfpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
62 ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
63 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
64 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
65 [ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lfpx_fqu_trans … H12 … HKL1) -L1
66 /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
67 | #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
68 #L0 #T0 #HT10 #HT0 #HL0 elim (lfpx_fqu_trans … H2 … HL0) -L
69 #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
70 /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
74 (* Properties with star-iterated structural successor for closures **********)
76 lemma lfpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
77 ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
78 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
79 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H
80 [ #H12 elim (lfpx_fqup_trans … H12 … HKL1) -L1 /3 width=5 by fqup_fqus, ex3_2_intro/
81 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/