--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/dpredstar_6.ma".
+include "basic_2/unfold/lsstas.ma".
+include "basic_2/computation/cprs.ma".
+
+(* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
+
+definition cpds: ∀h. sd h → relation4 genv lenv term term ≝
+ λh,g,G,L,T1,T2.
+ ∃∃T,l1,l2. l2 ≤ l1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 & ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T & ⦃G, L⦄ ⊢ T ➡* T2.
+
+interpretation "decomposed extended parallel computation (term)"
+ 'DPRedStar h g G L T1 T2 = (cpds h g G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma ssta_cprs_cpds: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h, g] T →
+ ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
+/3 width=7/ qed.
+
+lemma lsstas_cpds: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 •*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
+/2 width=7/ qed.
+
+lemma cprs_cpds: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
+/2 width=7/ qed.
+
+lemma cpds_refl: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ T •*➡*[h, g] T.
+/2 width=2/ qed.
+
+lemma cpds_strap1: ∀h,g,G,L,T1,T,T2.
+ ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
+#h #g #G #L #T1 #T #T2 * /3 width=9/
+qed.