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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/computation/lprs.ma".
16 include "basic_2/dynamic/ypr.ma".
18 (* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
20 definition yprs: ∀h. sd h → bi_relation lenv term ≝
21 λh,g. bi_TC … (ypr h g).
23 interpretation "'big tree' parallel computation (closure)"
24 'BTPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2).
26 (* Basic eliminators ********************************************************)
28 lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
29 (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
30 ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L2 T2.
31 /3 width=7 by bi_TC_star_ind/ qed-.
33 lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
34 (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
35 ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L1 T1.
36 /3 width=7 by bi_TC_star_ind_dx/ qed-.
38 (* Basic properties *********************************************************)
40 lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
43 lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
44 h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
47 lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
48 h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
51 lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
52 h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
55 lemma fw_yprs: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} →
56 h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
59 lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
60 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4 by ypr_cpr, yprs_strap1/
63 lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
64 #h #g #L1 #L2 #T #H @(lprs_ind … H) -L2 // /3 width=4 by ypr_lpr, yprs_strap1/
67 lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
68 h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
69 #h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by ypr_ssta, yprs_strap1/
72 lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
75 lemma lpr_cprs_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ ➡ L2 → L2 ⊢ T1 ➡* T2 →
76 h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
77 /3 width=4 by yprs_strap2, ypr_lpr, cprs_yprs/