--- /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 "ground/lib/subset.ma".
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/pitchfork_2.ma".
+include "delayed_updating/notation/functions/uptriangle_1.ma".
+
+(* PROTOTERM ****************************************************************)
+
+(* Note: a prototerm is a subset of complete paths *)
+definition prototerm: Type[0] ≝ 𝒫❨path❩.
+
+definition prototerm_grafted: path → prototerm → prototerm ≝
+ λp,t,q. p●q ϵ t.
+
+interpretation
+ "grafted (prototerm)"
+ 'Pitchfork t p = (prototerm_grafted p t).
+
+definition prototerm_root: prototerm → prototerm ≝
+ λt,q. ∃r. q●r ϵ t.
+
+interpretation
+ "root (prototerm)"
+ 'UpTriangle t = (prototerm_root t).
+
+(* Basic constructions ******************************************************)
+
+lemma prototerm_in_comp_root (p) (t):
+ p ϵ t → p ϵ ▵t.
+/2 width=2 by ex_intro/
+qed.