--- /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/gr_pat_lt.ma".
+include "ground/relocation/gr_ist.ma".
+include "ground/relocation/gr_after_pat.ma".
+
+(* RELATIONAL COMPOSITION FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(* Forward lemmas on istot **************************************************)
+
+(*** after_istot_fwd *)
+lemma gr_after_ist_des:
+ โf2,f1,f. f2 โ f1 โ f โ ๐โชf2โซ โ ๐โชf1โซ โ ๐โชfโซ.
+#f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
+#i2 #Hf1 elim (Hf2 i2) -Hf2
+/3 width=7 by gr_after_des_pat, ex_intro/
+qed-.
+
+(*** after_fwd_istot_dx *)
+lemma gr_after_des_ist_dx:
+ โf2,f1,f. f2 โ f1 โ f โ ๐โชfโซ โ ๐โชf1โซ.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i2 #Hf elim (gr_after_pat_des โฆ Hf โฆ H) -f /2 width=2 by ex_intro/
+qed-.
+
+(*** after_fwd_istot_sn *)
+lemma gr_after_des_ist_sn:
+ โf2,f1,f. f2 โ f1 โ f โ ๐โชfโซ โ ๐โชf2โซ.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i #Hf elim (gr_after_pat_des โฆ Hf โฆ H) -f
+#i2 #Hf1 #Hf2 lapply (gr_pat_increasing โฆ Hf1) -f1
+#Hi12 elim (gr_pat_le_ex โฆ Hf2 โฆ Hi12) -i2 /2 width=2 by ex_intro/
+qed-.
+
+(*** after_at1_fwd *)
+lemma gr_after_pat_sn_des:
+ โf1,i1,i2. @โชi1, f1โซ โ i2 โ โf2. ๐โชf2โซ โ โf. f2 โ f1 โ f โ
+ โโi. @โชi2, f2โซ โ i & @โชi1, fโซ โ i.
+#f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
+/3 width=8 by gr_after_des_pat, ex2_intro/
+qed-.
+
+(* Inversions with gr_ist *)
+
+(*** after_inv_istot *)
+lemma gr_after_inv_ist:
+ โf2,f1,f. f2 โ f1 โ f โ ๐โชfโซ โ โงโง ๐โชf2โซ & ๐โชf1โซ.
+/3 width=4 by gr_after_des_ist_sn, gr_after_des_ist_dx, conj/ qed-.