1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/dynamic/ysc.ma".
16 include "basic_2/dynamic/yprs.ma".
18 (* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
20 inductive ygt (h) (g) (L1) (T1): relation2 lenv term ≝
21 | ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ →
23 | ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → L ⊢ ➡ L2 → ygt h g L1 T1 L2 T
26 interpretation "'big tree' proper parallel computation (closure)"
27 'BTPRedStarProper h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
29 (* Basic forvard lemmas *****************************************************)
31 lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ →
32 h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
33 #h #g #L1 #L2 #T1 #T2 #H elim H -L2 -T2
34 /3 width=4 by yprs_strap1, ysc_ypr, ypr_lpr/
37 (* Basic properties *********************************************************)
39 lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
40 h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
43 lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
44 h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
45 #h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2
46 lapply (ygt_fwd_yprs … H1) #H0
47 elim (ypr_inv_ysc … H2) -H2 [| * #HL2 #H destruct ] /2 width=4/
50 lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
51 h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
52 #h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -L2 -T2
53 [ /3 width=4 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
56 lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
57 h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
58 #h #g #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -L2 -T2 //
59 /2 width=4 by ygt_strap1/
62 lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
63 ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
64 #h #g #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -L -T //
65 /3 width=4 by ygt_strap2/
68 lemma fsupp_ygt: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
69 #h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
72 lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
73 h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
74 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
76 | #T #T2 #_ #HT2 #IHT1 #HT12
77 elim (term_eq_dec T1 T) #H destruct
79 | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
80 @(ygt_strap1 … HT1) -HT1 /2 width=1/
85 lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → (T1 = T2 → ⊥) →
86 h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
87 #h #g #L #T1 #T2 #H @(sstas_ind … H) -T2
89 | #T #T2 #l #_ #HT2 #IHT1 #HT12 -HT12
90 elim (term_eq_dec T1 T) #H destruct
92 | lapply (IHT1 … H) -IHT1 -H #HT1
93 @(ygt_strap1 … HT1) -HT1 /2 width=2/
98 lemma lsubsv_ygt: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → (L1 = L2 → ⊥) →
99 h ⊢ ⦃L1, T⦄ >[g] ⦃L2, T⦄.