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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/prototerm.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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.