--- /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/reducibility/trf.ma".
+
+(* CONTEXT-FREE IRREDUCIBLE TERMS *******************************************)
+
+definition tif: predicate term โ ฮปT. ๐[T] โ False.
+
+interpretation "context-free irreducibility (term)"
+ 'NotReducible T = (tif T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tif_inv_abbr: โV,T. ๐[โV.T] โ False.
+/2 width=1/ qed-.
+
+lemma tif_inv_abst: โV,T. ๐[โV.T] โ ๐[V] โง ๐[T].
+/4 width=1/ qed-.
+
+lemma tif_inv_appl: โV,T. ๐[โV.T] โ โงโง ๐[V] & ๐[T] & ๐[T].
+#V #T #HVT @and3_intro /3 width=1/
+generalize in match HVT; -HVT elim T -T //
+* // * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
+qed-.
+
+lemma tif_inv_cast: โV,T. ๐[โฃV.T] โ False.
+/2 width=1/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tif_atom: โI. ๐[โช{I}].
+/2 width=4/ qed.
+
+lemma tif_abst: โV,T. ๐[V] โ ๐[T] โ ๐[โV.T].
+#V #T #HV #HT #H
+elim (trf_inv_abst โฆ H) -H /2 width=1/
+qed.
+
+lemma tif_appl: โV,T. ๐[V] โ ๐[T] โ ๐[T] โ ๐[โV.T].
+#V #T #HV #HT #S #H
+elim (trf_inv_appl โฆ H) -H /2 width=1/
+qed.