+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+include "delayed_updating/substitution/fsubst_constructors.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/syntax/prototerm_constructors_eq.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(* Constructions with constructors for prototerm ****************************)
+
+lemma ibfr_abst_hd (t1) (t2) (r):
+ t1 ➡𝐢𝐛𝐟[r] t2 → 𝛌.t1 ➡𝐢𝐛𝐟[𝗟◗r] 𝛌.t2.
+#t1 #t2 #r *
+#p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (𝗟◗p) … Hb Hm Hn) -Hb -Hm -Hn
+[ -Ht2 //
+| -Ht2 /2 width=1 by in_comp_abst_hd/
+| @(subset_eq_trans … (abst_eq_repl … Ht2)) -Ht1 -Ht2
+ @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+ @fsubst_eq_repl // @lift_term_eq_repl_dx
+ >list_cons_shift @(subset_eq_canc_sn … (grafted_abst_hd … )) //
+]
+qed.
+
+lemma ibfr_appl_hd (v) (t1) (t2) (r):
+ t1 ➡𝐢𝐛𝐟[r] t2 → @v.t1 ➡𝐢𝐛𝐟[𝗔◗r] @v.t2.
+#v #t1 #t2 #r *
+#p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (𝗔◗p) … Hb Hm Hn) -Hb -Hm -Hn
+[ -Ht2 //
+| -Ht2 /2 width=1 by in_comp_appl_hd/
+| @(subset_eq_trans … (appl_eq_repl … Ht2)) -Ht1 -Ht2 [|*: // ]
+ @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+ @fsubst_eq_repl // @lift_term_eq_repl_dx
+ >list_cons_shift @(subset_eq_canc_sn … (grafted_appl_hd … )) //
+]
+qed.
+
+lemma ibfr_appl_sd (v1) (v2) (t) (r):
+ v1 ➡𝐢𝐛𝐟[r] v2 → @v1.t ➡𝐢𝐛𝐟[𝗦◗r] @v2.t.
+#v1 #v2 #t #r *
+#p #b #q #m #n #Hr #Hb #Hm #Hn #Hv1 #Hv2 destruct
+@(ex6_5_intro … (𝗦◗p) … Hb Hm Hn) -Hb -Hm -Hn
+[ -Hv2 //
+| -Hv2 /2 width=1 by in_comp_appl_sd/
+| @(subset_eq_trans ????? (appl_eq_repl …)) [3: @Hv2 |2,4: // |5: skip ]
+ @(subset_eq_canc_sn … (fsubst_appl_sd …)) @appl_eq_repl [| // ]
+ @fsubst_eq_repl // @lift_term_eq_repl_dx
+ >list_cons_shift @(subset_eq_canc_sn … (grafted_appl_sd … )) //
+]
+qed.
+
+lemma ibfr_beta_0 (v) (t) (q) (n):
+ q ϵ 𝐂❨Ⓕ,n❩ → q◖𝗱↑n ϵ t →
+ @v.𝛌.t ➡𝐢𝐛𝐟[𝗔◗𝗟◗q] @v.𝛌.(t[⋔q←🠡[𝐮❨↑n❩]v]).
+#v #t #q #n #Hn #Ht
+@(ex6_5_intro … (𝐞) (𝐞) q (𝟎) … Hn) -Hn
+[ //
+| //
+| //
+| /3 width=1 by in_comp_appl_hd, in_comp_abst_hd/
+| @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+ @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+ @fsubst_eq_repl // <nplus_zero_sn @lift_term_eq_repl_dx
+ >list_cons_comm @(subset_eq_canc_sn … (grafted_appl_sd … ))
+ @(subset_eq_canc_sn … (grafted_empty_dx … )) //
+]
+qed.
+
+lemma ibfr_beta_1 (v) (v1) (t) (q) (n):
+ q ϵ 𝐂❨Ⓕ,n❩ → q◖𝗱↑n ϵ t →
+ @v.@v1.𝛌.𝛌.t ➡𝐢𝐛𝐟[𝗔◗𝗔◗𝗟◗𝗟◗q] @v.@v1.𝛌.𝛌.(t[⋔q←🠡[𝐮❨↑↑n❩]v]).
+#v #v1 #t #q #n #Hn #Ht
+@(ex6_5_intro … (𝐞) (𝗔◗𝗟◗𝐞) q (𝟏) … Hn) -Hn
+[ //
+| /2 width=1 by pbc_empty, pbc_redex/
+| /3 width=1 by pcc_A_sn, pcc_L_sn, pcc_empty/
+| /5 width=1 by in_comp_appl_hd, in_comp_abst_hd/
+| @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+ @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+ @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+ @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+ @fsubst_eq_repl // <nplus_unit_sn @lift_term_eq_repl_dx
+ >list_cons_comm @(subset_eq_canc_sn … (grafted_appl_sd … ))
+ @(subset_eq_canc_sn … (grafted_empty_dx … )) //
+]
+qed.