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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 "ground/xoa/ex_4_2.ma".
16 include "basic_2A/notation/relations/dpredstar_7.ma".
17 include "basic_2A/static/da.ma".
18 include "basic_2A/computation/cprs.ma".
20 (* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
22 definition scpds: ∀h. sd h → nat → relation4 genv lenv term term ≝
24 ∃∃T,d1. d2 ≤ d1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 & ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T ➡* T2.
26 interpretation "stratified decomposed parallel computation (term)"
27 'DPRedStar h g d G L T1 T2 = (scpds h g d G L T1 T2).
29 (* Basic properties *********************************************************)
31 lemma sta_cprs_scpds: ∀h,g,G,L,T1,T,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T →
32 ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 1] T2.
33 /2 width=6 by ex4_2_intro/ qed.
35 lemma lstas_scpds: ∀h,g,G,L,T1,T2,d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 →
36 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d2] T2.
37 /2 width=6 by ex4_2_intro/ qed.
39 lemma scpds_strap1: ∀h,g,G,L,T1,T,T2,d.
40 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2.
41 #h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_strap1, ex4_2_intro/
44 (* Basic forward lemmas *****************************************************)
46 lemma scpds_fwd_cprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 0] T2 →
48 #h #g #G #L #T1 #T2 * /3 width=3 by cprs_strap2, lstas_cpr/