--- /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/unwind_k/unwind2_prototerm_eq.ma".
+include "delayed_updating/unwind_k/unwind2_path_append.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* TAILED UNWIND FOR PRETERM ************************************************)
+
+(* Constructions with fsubst and pic ****************************************)
+
+lemma unwind2_term_fsubst_pic_sn (f) (t) (u) (p): p Ļµ š ā
+ (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
+#f #t #u #p #Hp #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind2_path_append_pic_sn
+ /4 width=3 by in_comp_unwind2_path_term, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro ā¦ H1) @or_intror @conj // *
+ [ <list_append_empty_sn #H2 destruct
+ elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ /3 width=2 by unwind2_path_append_pic_sn/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic_dx (f) (t) (u) (p): p Ļµ š ā p Ļµ āµt ā t Ļµ š ā
+ ā¼[f](t[āpāu]) ā (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u].
+#f #t #u #p #Hp #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ /5 width=3 by unwind2_path_append_pic_sn, ex2_intro, or_introl/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
+ [ <list_append_empty_sn #Hr @(H0 ā¦ (š)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind2_path_inv_append_ppc_dx ā¦ Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p ā¦ Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic (f) (t) (u) (p): p Ļµ š ā p Ļµ āµt ā t Ļµ š ā
+ (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
+/4 width=3 by unwind2_term_fsubst_pic_sn, conj, unwind2_term_fsubst_pic_dx/ qed.
+
+(* Constructions with fsubst and ppc ****************************************)
+
+lemma unwind2_term_fsubst_ppc_sn (f) (t) (u) (p): u Ļµ š ā
+ (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
+#f #t #u #p #Hu #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind2_path_append_ppc_dx
+ /4 width=5 by in_comp_unwind2_path_term, in_comp_tpc_trans, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro ā¦ H1) @or_intror @conj // *
+ [ <list_append_empty_sn #H2 destruct
+ elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ @H0 -H0 [| <unwind2_path_append_ppc_dx /2 width=3 by ppc_rcons/ ]
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_ppc_dx (f) (t) (u) (p): u Ļµ š ā p Ļµ āµt ā t Ļµ š ā
+ ā¼[f](t[āpāu]) ā (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u].
+#f #t #u #p #Hu #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ @or_introl @ex2_intro
+ [|| <unwind2_path_append_ppc_dx /2 width=5 by in_comp_tpc_trans/ ]
+ /2 width=3 by ex2_intro/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
+ [ <list_append_empty_sn #Hr @(H0 ā¦ (š)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind2_path_inv_append_ppc_dx ā¦ Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p ā¦ Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_ppc (f) (t) (u) (p): u Ļµ š ā p Ļµ āµt ā t Ļµ š ā
+ (ā¼[f]t)[ā(āp)āā¼[ā¶[f]p]u] ā ā¼[f](t[āpāu]).
+/4 width=3 by unwind2_term_fsubst_ppc_sn, conj, unwind2_term_fsubst_ppc_dx/ qed.