--- /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/dpconvstar_8.ma".
+include "basic_2/computation/scpds.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
+
+definition scpes: ∀h. sd h → nat → nat → relation4 genv lenv term term ≝
+ λh,o,d1,d2,G,L,T1,T2.
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] T & ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T.
+
+interpretation "stratified decomposed parallel equivalence (term)"
+ 'DPConvStar h o d1 d2 G L T1 T2 = (scpes h o d1 d2 G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma scpds_div: ∀h,o,G,L,T1,T2,T,d1,d2.
+ ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] T → ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T →
+ ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
+/2 width=3 by ex2_intro/ qed.
+
+lemma scpes_sym: ∀h,o,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
+ ⦃G, L⦄ ⊢ T2 •*⬌*[h, o, d2, d1] T1.
+#h #o #G #L #T1 #T2 #L1 #d2 * /2 width=3 by scpds_div/
+qed-.