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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_ist.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_ist.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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:
+      ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f2❫ → 𝐓❪f1❫ → 𝐓❪f❫.
+#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:
+      ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f1❫.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i2 #Hf elim (gr_after_pat_des … Hf … H) -f /2 width=2 by ex_intro/
+qed-.
+
+(*** after_fwd_istot_sn *)
+lemma gr_after_des_ist_sn:
+      ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f2❫.
+#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
+#i #Hf elim (gr_after_pat_des … Hf … H) -f
+#i2 #Hf1 #Hf2 lapply (gr_pat_increasing … Hf1) -f1
+#Hi12 elim (gr_pat_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
+qed-.
+
+(*** after_at1_fwd *)
+lemma gr_after_pat_sn_des:
+      ∀f1,i1,i2. @❪i1, f1❫ ≘ i2 → ∀f2. 𝐓❪f2❫ → ∀f. f2 ⊚ f1 ≘ f →
+      ∃∃i. @❪i2, f2❫ ≘ i & @❪i1, f❫ ≘ 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:
+      ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → ∧∧ 𝐓❪f2❫ & 𝐓❪f1❫.
+/3 width=4 by gr_after_des_ist_sn, gr_after_des_ist_dx, conj/ qed-.