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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/notation/relations/btpredstarproper_8.ma".
16 include "basic_2/substitution/fqup.ma".
17 include "basic_2/reduction/fpbc.ma".
18 include "basic_2/computation/fpbs.ma".
20 (* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
22 inductive fpbg (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
23 | fpbg_inj : ∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
24 fpbg h g G1 L1 T1 G2 L2 T2
25 | fpbg_step: ∀G,L,L2,T. fpbg h g G1 L1 T1 G L T → ⦃G, L⦄ ⊢ ➡[h, g] L2 → fpbg h g G1 L1 T1 G L2 T
28 interpretation "'big tree' proper parallel computation (closure)"
29 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
31 (* Basic forvard lemmas *****************************************************)
33 lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
34 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
35 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
36 /3 width=5 by fpbs_strap1, fpbc_fwd_fpb, fpb_lpx/
39 (* Basic properties *********************************************************)
41 lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
42 ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
43 /3 width=5 by fpbg_inj, fpbg_step/ qed.
45 lemma fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
46 ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
47 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
48 lapply (fpbg_fwd_fpbs … H1) #H0
49 elim (fpb_fpbc_or_refl … H2) -H2 [| * #HG2 #HL2 #HT2 destruct ]
50 /2 width=5 by fpbg_inj, fpbg_step/
53 lemma fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
54 ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
55 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -G2 -L2 -T2
56 /3 width=5 by fpbg_step, fpbg_inj, fpbs_strap2/
59 lemma fpbg_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
60 ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
61 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(fpbs_ind … HT2) -G2 -L2 -T2
62 /2 width=5 by fpbg_strap1/
65 lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
66 ∀G2,L2,T2. ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
67 #h #g #G1 #G #L1 #L #T1 #T #HT1 @(fpbs_ind … HT1) -G -L -T
68 /3 width=5 by fpbg_strap2/
71 lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
72 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … L2 T2 H) -G2 -L2 -T2
73 /4 width=5 by fpbg_strap1, fpbc_fpbg, fpbc_fqu, fpb_fquq, fqu_fquq/
76 lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
77 ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
78 #h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
80 | #T #T2 #_ #HT2 #IHT1 #HT12
81 elim (term_eq_dec T1 T) #H destruct
83 | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
84 @(fpbg_strap1 … HT1) -HT1 /2 width=1 by fpb_cpx/
89 lemma cprs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
90 ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
91 /3 width=1 by cprs_cpxs, cpxs_fpbg/ qed.