--- /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 "ground/relocation/pr_pushs.ma".
+include "ground/relocation/pr_tls.ma".
+include "ground/relocation/pr_nat.ma".
+include "ground/relocation/pr_coafter.ma".
+
+(* RELATIONAL CO-COMPOSITION FOR PARTIAL RELOCATION MAPS ********************)
+
+(* Destructions with pr_nat and pr_tls and pr_pushs *************************)
+
+(*** coafter_fwd_pushs *)
+lemma pr_coafter_des_pushs_dx (n) (m):
+ ∀g2,f1,g. g2 ~⊚ ⫯*[m]f1 ≘ g → @↑❪m, g2❫ ≘ n →
+ ∃∃f. ⫰*[n]g2 ~⊚ f1 ≘ f & ⫯*[n] f = g.
+#n @(nat_ind_succ … n) -n
+[ #m #g2 #f1 #g #Hg #H
+ elim (pr_nat_inv_zero_dx … H) -H [|*: // ] #f2 #H1 #H2 destruct
+ /2 width=3 by ex2_intro/
+| #n #IH * [| #m ] #g2 #f1 #g #Hg #H
+ [ elim (pr_nat_inv_zero_succ … H) -H [|*: // ] #f2 #Hmn #H destruct
+ elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+ | elim (pr_nat_inv_succ_bi … H) -H [1,4: * |*: // ] #f2 #Hmn #H destruct
+ [ elim (pr_coafter_inv_push_bi … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+ | elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+ ]
+ ]
+]
+qed-.