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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/computation/fpbc_fpns.ma".
16 include "basic_2/computation/fpbg.ma".
18 (* GENEARAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES *************)
20 (* Properties on parallel computation for "big tree" normal forms ***********)
22 lemma fpbg_fpns_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G, L, T⦄ →
23 ∀G2,L2,T2. ⦃G, L, T⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
24 #h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
25 [ /3 width=5 by fpbc_fpbg, fpbc_fpns_trans/
26 | /4 width=9 by fpbg_strap1, fpbc_fpns_trans/
30 lemma fpns_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >⋕[h, g] ⦃G2, L2, T2⦄ →
31 ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
32 #h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
33 [ /3 width=5 by fpbc_fpbg, fpns_fpbc_trans/
34 | /4 width=9 by fpbg_strap2, fpns_fpbc_trans/
38 lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
39 ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ ∨
40 ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
41 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
42 [ /2 width=1 by or_introl/
43 | #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
44 [ /3 width=5 by fpns_trans, or_introl/
45 | /5 width=5 by fpbc_fpbg, fpns_fpbc_trans, fpbu_fpbc, or_intror/
46 | /3 width=5 by fpbg_fpns_trans, or_intror/
47 | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
52 (* Advanced properties ******************************************************)
54 lemma fpbg_fpb_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
55 ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
56 ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
57 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
58 /3 width=5 by fpbg_fpns_trans, fpbg_strap1, fpbu_fpbc/
62 lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
63 ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >⋕[h, g] ⦃G2, L2, T2⦄ →
64 ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
65 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
66 /3 width=5 by fpns_fpbg_trans, fpbg_strap2, fpbu_fpbc/
69 lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
70 ∀G2,L2,T2. ⦃G, L, T⦄ >⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
71 #h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
74 (* Note: this is used in the closure proof *)
75 lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
76 ∀G1,L1,T1. ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄.
77 #h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/