--- /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 "delayed_updating/reduction/ibfr_constructors.ma".
+include "delayed_updating/substitution/lift_prototerm_constructors.ma".
+include "ground/arith/pnat_two.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Example of unprotected balanced β-reduction ******************************)
+
+definition l3_t0: prototerm ≝
+ (𝛌.@(⧣𝟏).@(𝛌.@(⧣𝟐).⧣𝟏).𝛌.⧣𝟏).
+
+definition l3_t1: prototerm ≝
+ (𝛌.@(⧣𝟏).@(𝛌.@(⧣𝟐).⧣𝟏).𝛌.(𝛌.@(⧣↑𝟐).⧣𝟏)).
+
+lemma l3_t01:
+ l3_t0 ➡𝐢𝐛𝐟[𝗟◗𝗔◗𝗔◗𝗟◗𝐞] l3_t1.
+@ibfr_abst_hd
+@ibfr_appl_hd
+@ibfr_eq_trans [| @ibfr_beta_0 // ]
+@appl_eq_repl [ // ]
+@abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_empty_rc …))
+@(subset_eq_canc_sn … (lift_term_abst …))
+@abst_eq_repl
+@(subset_eq_canc_sn … (lift_term_appl … ))
+@appl_eq_repl
+@(subset_eq_canc_sn … (lift_term_oref_pap … )) //
+qed.