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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/dpredstar_7.ma".
16 include "basic_2/static/da.ma".
17 include "basic_2/unfold/lstas.ma".
18 include "basic_2/computation/cprs.ma".
20 (* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
22 definition cpds: ∀h. sd h → nat → relation4 genv lenv term term ≝
24 ∃∃T,l1. l2 ≤ l1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 & ⦃G, L⦄ ⊢ T1 •*[h, l2] T & ⦃G, L⦄ ⊢ T ➡* T2.
26 interpretation "decomposed extended parallel computation (term)"
27 'DPRedStar h g l G L T1 T2 = (cpds h g l G L T1 T2).
29 (* Basic properties *********************************************************)
31 lemma sta_cprs_cpds: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h] T →
32 ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 1] T2.
33 /3 width=6 by sta_lstas, ex4_2_intro/ qed.
35 lemma lstas_cpds: ∀h,g,G,L,T1,T2,l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
36 ∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l2] T2.
37 /2 width=6 by ex4_2_intro/ qed.
39 lemma cprs_cpds: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 ➡* T2 →
40 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 0] T2.
41 /2 width=6 by lstar_O, ex4_2_intro/ qed.
43 lemma cpds_refl: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ T •*➡*[h, g, 0] T.
44 /2 width=2 by cprs_cpds/ qed.
46 lemma cpds_strap1: ∀h,g,G,L,T1,T,T2,l.
47 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2.
48 #h #g #G #L #T1 #T #T2 #l * /3 width=8 by cprs_strap1, ex4_2_intro/
51 (* Basic forward lemmas *****************************************************)
53 lemma cpds_fwd_cprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 0] T2 →
56 #T #l #_ #_ #H lapply (lstas_inv_O … H) -l -H