+(**************************************************************************)
+(* ___ *)
+(* ||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 un_t: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(⧣𝟐).𝛌.@(⧣𝟐).⧣𝟏.
+
+definition un_i1: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).⧣𝟏.
+
+lemma un_ti1:
+ un_t ➡𝐢𝐛𝐟[𝗔◗𝗟◗𝗟◗𝗦◗𝐞] un_i1.
+@(ibfr_eq_trans … (ibfr_beta_0 …))
+[ >list_append_lcons_sn
+ /3 width=1 by in_comp_appl_sd, in_comp_abst_hd/
+| /3 width=3 by pcc_S_sn, pcc_L_sn/
+| skip
+]
+@appl_eq_repl [ // ] @abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_appl_sd …)) @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.
+
+definition un_i2: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.⧣𝟏.
+
+lemma un_i12:
+ un_i1 ➡𝐢𝐛𝐟[𝗔◗𝗟◗𝗟◗𝗔◗𝗟◗𝗔◗𝐞] un_i2.
+@ibfr_appl_hd
+@ibfr_abst_hd
+@ibfr_abst_hd
+@(ibfr_eq_trans … (ibfr_beta_0 …))
+[ >list_append_lcons_sn
+ /2 width=1 by in_comp_appl_hd/
+| /2 width=1 by pcc_A_sn, pcc_empty/
+| skip
+]
+@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.
+
+definition un_i3: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.⧣↑𝟐.
+
+lemma un_i23:
+ un_i2 ➡𝐢𝐛𝐟[𝗔◗𝗟◗𝗟◗𝗔◗𝗟◗𝗔◗𝗟◗𝐞] un_i3.
+@ibfr_appl_hd
+@ibfr_abst_hd
+@ibfr_abst_hd
+@ibfr_appl_hd
+@ibfr_abst_hd
+@(ibfr_eq_trans … (ibfr_beta_0 …))
+[ // | // | skip ]
+@appl_eq_repl [ // ] @abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_empty_rc …))
+@(subset_eq_canc_sn … (lift_term_oref_pap … )) //
+qed.
+
+definition un_d1: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛕𝟐.𝛌.⧣𝟏).𝛌.@(⧣𝟐).⧣𝟏.
+
+lemma un_td1:
+ un_t ➡𝐝𝐛𝐟[𝗔◗𝗟◗𝗟◗𝗦◗𝐞] un_d1.
+@(dbfr_eq_trans … (dbfr_beta_0 …))
+[ >list_append_lcons_sn
+ /3 width=1 by in_comp_appl_sd, in_comp_abst_hd/
+| /3 width=3 by pcc_S_sn, pcc_L_sn/
+| skip
+]
+@appl_eq_repl [ // ] @abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_appl_sd …)) @appl_eq_repl [| // ]
+@(subset_eq_canc_sn … (fsubst_empty_rc …))
+@iref_eq_repl //
+qed.
+
+definition un_d2: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛕𝟐.𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛕𝟏.𝛕𝟐.𝛌.⧣𝟏.
+
+lemma un_d12:
+ un_d1 ➡𝐝𝐛𝐟[𝗔◗𝗟◗𝗟◗𝗔◗𝗟◗𝗔◗𝐞] un_d2.
+@dbfr_appl_hd
+@dbfr_abst_hd
+@dbfr_abst_hd
+@(dbfr_eq_trans … (dbfr_beta_0 …))
+[ >list_append_lcons_sn /2 width=1 by in_comp_appl_hd/
+| /2 width=1 by pcc_A_sn/
+| skip
+]
+@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 @iref_eq_repl //
+qed.
+
+definition un_d3: prototerm ≝
+ @(𝛌.⧣𝟏).𝛌.𝛌.@(𝛕𝟐.𝛌.⧣𝟏).𝛌.@(⧣𝟐).𝛌.𝛕𝟏.⧣𝟐.
+
+lemma un_di3:
+ un_i3 ⇔ ▼[𝐢] un_d3.
+@(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_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_appl …)) @appl_eq_repl
+[ @(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 …)) //
+]
+@(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_abst …)) @abst_eq_repl
+@(subset_eq_canc_sn … (unwind2_term_iref …))
+@(subset_eq_canc_sn … (unwind2_term_oref_pap …)) //
+qed.