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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_after.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_after.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_after_eq.ma".
+
+(* RELATIONAL COMPOSITION FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(* Main properties **********************************************************)
+
+(*** after_trans1 *)
+corec theorem gr_after_trans_sn:
+              ∀f0,f3,f4. f0 ⊚ f3 ≘ f4 →
+              ∀f1,f2. f1 ⊚ f2 ≘ f0 →
+              ∀f. f2 ⊚ f3 ≘ f → f1 ⊚ f ≘ f4.
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (gr_after_inv_push … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (gr_after_inv_push_bi … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by gr_after_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (gr_after_inv_push … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (gr_after_inv_push_next … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by gr_after_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (gr_after_inv_next … Hg0 … H0) -g0 *
+  [ #f1 #f2 #Hf0 #H1 #H2
+    cases (gr_after_inv_next_sn … Hg … H2) -g2
+    #f #Hf #H /3 width=7 by gr_after_push/
+  | #f1 #Hf0 #H1 /3 width=6 by gr_after_next/
+  ]
+]
+qed-.
+
+(*** after_trans2 *)
+corec theorem gr_after_trans_dx:
+              ∀f1,f0,f4. f1 ⊚ f0 ≘ f4 →
+              ∀f2, f3. f2 ⊚ f3 ≘ f0 →
+              ∀f. f1 ⊚ f2 ≘ f → f ⊚ f3 ≘ f4.
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (gr_after_inv_push … Hg0 … H0) -g0
+  #f2 #f3 #Hf0 #H2 #H3
+  cases (gr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
+  #f #Hf #H /3 width=7 by gr_after_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (gr_after_inv_next … Hg0 … H0) -g0 *
+  [ #f2 #f3 #Hf0 #H2 #H3
+    cases (gr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=7 by gr_after_push/
+  | #f2 #Hf0 #H2
+    cases (gr_after_inv_push_next … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=6 by gr_after_next/
+  ]
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+  cases (gr_after_inv_next_sn … Hg … H1) -g1
+  #f #Hg #H /3 width=6 by gr_after_next/
+]
+qed-.
+
+(* Main inversion lemmas ****************************************************)
+
+(*** after_mono *)
+corec theorem gr_after_mono:
+              ∀f1,f2,x,y. f1 ⊚ f2 ≘ x → f1 ⊚ f2 ≘ y → x ≡ y.
+#f1 #f2 #x #y * -f1 -f2 -x
+#f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
+[ cases (gr_after_inv_push_bi … Hy … H1 H2) -g1 -g2 /3 width=8 by gr_eq_push/
+| cases (gr_after_inv_push_next … Hy … H1 H2) -g1 -g2 /3 width=8 by gr_eq_next/
+| cases (gr_after_inv_next_sn … Hy … H1) -g1 /3 width=8 by gr_eq_next/
+]
+qed-.
+
+(*** after_mono_eq *)
+lemma gr_after_mono_eq:
+      ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
+      f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
+/4 width=4 by gr_after_mono, gr_after_eq_repl_back_dx, gr_after_eq_repl_back_sn/ qed-.