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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/notation/relations/btpredstaralt_8.ma".
16 include "basic_2/computation/llpxs_cpxs.ma".
17 include "basic_2/computation/fpbs_fpbs.ma".
19 (* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
21 (* Note: alternative definition of fpbs *)
22 definition fpbsa: ∀h. sd h → tri_relation genv lenv term ≝
23 λh,g,G1,L1,T1,G2,L2,T2.
24 ∃∃L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
25 ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ &
26 ⦃G2, L⦄ ⊢ ➡*[h, g, T2, 0] L2.
28 interpretation "'big tree' parallel computation (closure) alternative"
29 'BTPRedStarAlt h g G1 L1 T1 G2 L2 T2 = (fpbsa h g G1 L1 T1 G2 L2 T2).
31 (* Basic properties *********************************************************)
33 lemma fpb_fpbsa_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
34 ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
35 #h #g #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 ]
36 #G2 #L2 #T2 * #L0 #T0 #HT0 #HG2 #HL02
37 [ elim (fquq_cpxs_trans … HT0 … HG1) -T
38 /3 width=7 by fqus_strap2, ex3_2_intro/
39 | /3 width=5 by cpxs_strap2, ex3_2_intro/
40 | lapply (llpx_cpxs_trans … HT0 … HL1) -HT0 #HT10
41 lapply (cpxs_llpx_conf … HT10 … HL1) -HL1 #HL1
42 elim (llpx_fqus_trans … HG2 … HL1) -L
43 /3 width=7 by llpxs_strap2, cpxs_trans, ex3_2_intro/
47 (* Main properties **********************************************************)
49 theorem fpbs_fpbsa: ∀h,g,G1,G2,L1,L2,T1,T2.
50 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
51 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
52 /2 width=5 by fpb_fpbsa_trans, ex3_2_intro/
55 (* Main inversion lemmas ****************************************************)
57 theorem fpbsa_inv_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
58 ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
59 #h #g #G1 #G2 #L1 #L2 #T1 #T2 *
60 /4 width=5 by fpbs_trans, fqus_fpbs, cpxs_fpbs, llpxs_fpbs/