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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 "basic_2/rt_transition/cpx_fqus.ma".
16 include "basic_2/rt_transition/lpx_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 rt-transition for full local environments ********)
24 lemma lpx_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (λ_.lpx h G).
25 #h #G #L2 #T1 #T2 #H @(cpx_ind … H) -G -L2 -T1 -T2
27 | /3 width=2 by cpx_cpxs, cpx_ess/
28 | #I #G #K2 #V2 #V4 #W4 #_ #IH #HVW4 #L1 #H
29 elim (lpx_inv_pair_dx … H) -H #K1 #V1 #HK12 #HV12 #H destruct
30 /4 width=3 by cpxs_delta, cpxs_strap2/
31 | #I2 #G #K2 #T #U #i #_ #IH #HTU #L1 #H
32 elim (lpx_inv_bind_dx … H) -H #I1 #K1 #HK12 #HI12 #H destruct
33 /4 width=3 by cpxs_lref, cpxs_strap2/
34 |5,10: /4 width=1 by cpxs_beta, cpxs_bind, lpx_bind_refl_dx/
35 |6,8,9: /3 width=1 by cpxs_flat, cpxs_ee, cpxs_eps/
36 | /4 width=3 by cpxs_zeta, lpx_bind_refl_dx/
37 | /4 width=3 by cpxs_theta, cpxs_strap1, lpx_bind_refl_dx/
41 lemma lpx_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (λ_.lpx h G).
42 #h #G @s_r_trans_CTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
45 (* Advanced properties ******************************************************)
47 lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
48 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
49 ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
50 /4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
52 lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
53 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
54 ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
55 /4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
57 (* Properties with plus-iterated structural successor for closures **********)
59 (* Basic_2A1: uses: lpx_fqup_trans *)
60 lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
61 ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
62 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
63 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
64 [ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
65 /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
66 | #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
67 #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
68 #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
69 /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
73 (* Properties with star-iterated structural successor for closures **********)
75 (* Basic_2A1: uses: lpx_fqus_trans *)
76 lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
77 ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
78 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
79 #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H
80 [ #H12 elim (lpx_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/