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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/notation/relations/btpredstar_8.ma".
16 include "basic_2/substitution/fsupp.ma".
17 include "basic_2/reduction/fpb.ma".
18 include "basic_2/computation/cprs.ma".
19 include "basic_2/computation/lprs.ma".
21 (* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
23 definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
24 λh,g. tri_TC … (fpb h g).
26 interpretation "'big tree' parallel computation (closure)"
27 'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
29 (* Basic eliminators ********************************************************)
31 lemma fpbs_ind: ∀h,g,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
32 (∀L,G2,G,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
33 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
34 /3 width=8 by tri_TC_star_ind/ qed-.
36 lemma fpbs_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
37 (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
38 ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
39 /3 width=8 by tri_TC_star_ind_dx/ qed-.
41 (* Basic properties *********************************************************)
43 lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
44 /2 width=1 by tri_inj/ qed.
46 lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
47 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
48 /2 width=1 by tri_inj/ qed.
50 lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
51 ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
52 /2 width=5 by tri_step/ qed-.
54 lemma fpbs_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
55 ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
56 /2 width=5 by tri_TC_strap/ qed-.
58 (* Note: this is a general property of bi_TC *)
59 lemma fsupp_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
60 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
61 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2
62 /3 width=5 by fpb_fsup, tri_step, fpb_fpbs/
65 lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
66 #h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2
67 /3 width=5 by fpb_cpr, fpbs_strap1/
70 lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
71 #h #g #G #L1 #L2 #T #H @(lprs_ind … H) -L2
72 /3 width=5 by fpb_lpr, fpbs_strap1/
75 lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
76 ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
77 /4 width=5 by fpbs_strap1, fpb_lpr, fpb_cpr/ qed.
79 lemma ssta_fpbs: ∀h,g,G,L,T,U,l.
80 ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h, g] U →
81 ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
82 /3 width=2 by fpb_fpbs, fpb_ssta/ qed.