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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/relocation/lleq_lleq.ma".
16 include "basic_2/computation/cpxs_lleq.ma".
17 include "basic_2/computation/lpxs_lleq.ma".
18 include "basic_2/computation/lpxs_lpxs.ma".
19 include "basic_2/computation/fpns.ma".
20 include "basic_2/computation/fpbs_alt.ma".
21 include "basic_2/computation/fpbu.ma".
23 (* UNITARY "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **************)
25 (* Properties on parallel computation for "big tree" normal forms ***********)
27 lemma fpns_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ⊢ ⋕➡*[h, g] ⦃F2, K2, T2⦄ →
28 ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
29 ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, U2⦄.
30 #h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
31 #K2 #HK12 #HT1 #G2 #L2 #U2 * -G2 -L2 -U2
32 [ #G2 #L2 #U2 #H12 elim (lpxs_lleq_fqup_trans … H12 … HK12 HT1) -K2
33 /3 width=5 by fpbu_fqup, fpns_intro, ex2_3_intro/
34 | /4 width=9 by fpbu_cpxs, fpns_intro, lpxs_cpxs_trans, lleq_cpxs_conf_dx, ex2_3_intro/
35 | /5 width=5 by fpbu_lpxs, lpxs_trans, lleq_canc_sn, ex2_3_intro/
39 lemma fpb_fpbu: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
40 ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ ∨
41 ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
42 #h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
43 [ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
44 [ /4 width=1 by fpbu_fqup, fqu_fqup, or_intror/
45 | * #H1 #H2 #H3 destruct /2 width=1 by or_introl/
47 | #T2 #HT12 elim (eq_term_dec T1 T2)
48 #HnT12 destruct /4 width=1 by fpbu_cpxs, cpx_cpxs, or_intror, or_introl/
49 | #L2 #HL12 elim (lleq_dec … T1 L1 L2 0)
50 /4 width=3 by fpbu_lpxs, fpns_intro, lpx_lpxs, or_intror, or_introl/
54 lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
55 ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ ∨
56 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
58 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
59 [ /2 width=1 by or_introl/
60 | #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
61 elim (fpb_fpbu … H1) -H1 #H1
64 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_fpbsa … H) -H
65 #L #T #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
66 [ -HT1 elim (fqus_inv_gen … HT2) -HT2
68 /5 width=9 by fpbsa_inv_fpbs, fpbu_fqup, ex3_2_intro, ex2_3_intro, or_intror/
69 | * #HG #HL #HT destruct elim (lleq_dec T2 L L2 0) #H
70 [ /3 width=1 by fpns_intro, or_introl/
71 | /5 width=5 by fpbu_lpxs, ex2_3_intro, or_intror/
74 | elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
75 /5 width=9 by fpbsa_inv_fpbs, fpbu_cpxs, cpx_cpxs, ex3_2_intro, ex2_3_intro, or_intror/