+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_closed.ma".
+include "delayed_updating/syntax/path_guard.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+axiom list_ind_pippo (A) (Q:predicate …):
+ (∀l. (∀m. (∃p. p⨁{A}m = l) → Q m) → Q l) → ∀l. Q l.
+
+(* Destructions with pgc ****************************************************)
+
+lemma path_closed_des_guard (x) (n):
+ x ϵ 𝐂❨Ⓕ,n❩ →
+ ∃∃p,q. p ϵ 𝐆 & q ϵ 𝐂❨Ⓕ,𝟎❩ & p●q = x.
+#x @(list_ind_pippo … x) -x
+#x #IH #n #H0 @(insert_eq_1 … x … H0) -H0
+#y * -y -n
+[|*: #y #n [ #k #_ ] #Hy ] #H0 destruct
+[ /2 width=5 by pgc_empty, pcc_empty, ex3_2_intro/
+| elim (pcc_false_inv_append_bi … Hy) -Hy #r #s #Hr #Hs #H0 destruct
+ elim (IH … Hr) -IH -Hr [| /2 width=2 by ex_intro/ ]
+ #p #q #Hp #Hq #H0 destruct
+ @(ex3_2_intro … Hp) -Hp [1,3: // ]
+ /3 width=2 by pcc_append_bi, pcc_false_d_dx/
+| elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
+ #p #q #Hp #Hq #H0 destruct
+ /3 width=5 by pcc_m_dx, ex3_2_intro/
+| @(ex3_2_intro … (y◖𝗟) (𝐞)) //
+| elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
+ #p #q #Hp #Hq #H0 destruct
+ /3 width=5 by pcc_A_dx, ex3_2_intro/
+| elim (IH … Hy) -IH -Hy [| /2 width=2 by ex_intro/ ]
+ #p #q #Hp #Hq #H0 destruct
+ /3 width=5 by pcc_S_dx, ex3_2_intro/
+]
+qed-.