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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/btpredstarproper_8.ma".
16 include "basic_2/dynamic/ysc.ma".
17 include "basic_2/dynamic/yprs.ma".
19 (* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
21 inductive ygt (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
22 | ygt_inj : ∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
23 ygt h g G1 L1 T1 G2 L2 T2
24 | ygt_step: ∀G,L,L2,T. ygt h g G1 L1 T1 G L T → ⦃G, L⦄ ⊢ ➡ L2 → ygt h g G1 L1 T1 G L2 T
27 interpretation "'big tree' proper parallel computation (closure)"
28 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (ygt h g G1 L1 T1 G2 L2 T2).
30 (* Basic forvard lemmas *****************************************************)
32 lemma ygt_fwd_yprs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
33 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
34 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
35 /3 width=5 by yprs_strap1, ysc_ypr, ypr_lpr/
38 (* Basic properties *********************************************************)
40 lemma ysc_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
41 ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
44 lemma ygt_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
45 ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
46 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
47 lapply (ygt_fwd_yprs … H1) #H0
48 elim (ypr_inv_ysc … H2) -H2 [| * #HG2 #HL2 #HT2 destruct ] /2 width=5/
51 lemma ygt_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
52 ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
53 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -G2 -L2 -T2
54 [ /3 width=5 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
57 lemma ygt_yprs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
58 ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
59 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -G2 -L2 -T2 //
60 /2 width=5 by ygt_strap1/
63 lemma yprs_ygt_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
64 ∀G2,L2,T2. ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
65 #h #g #G1 #G #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -G -L -T //
66 /3 width=5 by ygt_strap2/
69 lemma fsupp_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
70 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2 /3 width=1/ /3 width=5/
73 lemma cprs_ygt: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
74 ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
75 #h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2
77 | #T #T2 #_ #HT2 #IHT1 #HT12
78 elim (term_eq_dec T1 T) #H destruct
80 | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
81 @(ygt_strap1 … HT1) -HT1 /2 width=1/
86 lemma lsubsv_ygt: ∀h,g,G,L1,L2,T. G ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) →
87 ⦃G, L1, T⦄ >[h, g] ⦃G, L2, T⦄.