+(**************************************************************************)
+(* ___ *)
+(* ||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/dbfr_constructors.ma".
+include "delayed_updating/reduction/ibfr_constructors.ma".
+include "delayed_updating/unwind/unwind2_prototerm_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_t: prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).⧣𝟏).
+
+definition l3_i1: prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.⧣𝟏).
+
+definition l3_i2: prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.⧣↑𝟐).
+
+definition l3_i2': prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.⧣↑↑𝟐).
+
+definition l3_d1: prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛕𝟏.(𝛌.⧣𝟏)).
+
+definition l3_d2: prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛕𝟏.𝛌.(⧣𝟐)).
+
+definition l3_d2': prototerm ≝
+ (𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛕𝟏.𝛌.𝛕𝟏.(⧣𝟐)).
+
+lemma l3_ti1:
+ l3_t ➡𝐢𝐛𝐟[𝗟◗𝗔◗𝗟◗𝗔◗𝐞] l3_i1.
+@ibfr_abst_hd
+@ibfr_eq_trans
+[ | @(ibfr_beta_0 … (𝟎))
+ [ /2 width=1 by pcc_A_sn, pcc_empty/
+ | >list_append_lcons_sn /2 width=1 by in_comp_appl_hd/
+ ]
+]
+@appl_eq_repl [ // ]
+@abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_appl_hd …))
+@appl_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_oref_pap … )) //
+qed.
+
+lemma l3_i12:
+ l3_i1 ➡𝐢𝐛𝐟[𝗟◗𝗔◗𝗟◗𝗔◗𝗟◗𝐞] l3_i2.
+@ibfr_abst_hd
+@ibfr_appl_hd
+@ibfr_abst_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_oref_pap … )) //
+qed.
+
+lemma l3_td1:
+ l3_t ➡𝐝𝐛𝐟[𝗟◗𝗔◗𝗟◗𝗔◗𝐞] l3_d1.
+@dbfr_abst_hd
+@dbfr_eq_trans
+[ | @(dbfr_beta_0 … (𝟎))
+ [ /2 width=1 by pcc_A_sn, pcc_empty/
+ | >list_append_lcons_sn /2 width=1 by in_comp_appl_hd/
+ ]
+]
+@appl_eq_repl [ // ]
+@abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_appl_hd …))
+@appl_eq_repl [ // ]
+@(subset_eq_canc_sn … (fsubst_empty_rc …))
+@iref_eq_repl //
+qed.
+
+lemma l3_di2:
+ l3_i2 ⇔ ▼[𝐢]l3_d2.
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_appl …)) @appl_eq_repl
+[ @(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+ @(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+]
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_appl …)) @appl_eq_repl
+[ @(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+]
+@(subset_eq_canc_sn … (unwind2_term_iref …))
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+qed.
+
+lemma l3_di2':
+ l3_i2' ⇔ ▼[𝐢]l3_d2'.
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_appl …)) @appl_eq_repl
+[ @(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+ @(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+]
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_appl …)) @appl_eq_repl
+[ @(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+]
+@(subset_eq_canc_sn … (unwind2_term_iref …))
+@(subset_eq_canc_sn … (unwind2_term_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_iref …))
+@(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+qed.