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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/fr2_minus.ma b/matita/matita/contribs/lambdadelta/ground/relocation/fr2_minus.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/xoa/ex_3_1.ma".
+include "ground/notation/relations/rminus_3.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_minus.ma".
+include "ground/arith/nat_lt.ma".
+include "ground/relocation/fr2_map.ma".
+
+(* RELATIONAL SUBTRACTION FOR FINITE RELOCATION MAPS WITH PAIRS *******************************************)
+
+(*** minuss *)
+inductive fr2_minus: nat → relation fr2_map ≝
+(*** minuss_nil *)
+| fr2_minus_nil (i):
+  fr2_minus i (◊) (◊)
+(*** minuss_lt *)
+| fr2_minus_lt (f1) (f2) (d) (h) (i):
+  i < d → fr2_minus i f1 f2 → fr2_minus i (❨d,h❩;f1) (❨d-i,h❩;f2)
+(*** minuss_ge *)
+| fr2_minus_ge (f1) (f2) (d) (h) (i):
+  d ≤ i → fr2_minus (h+i) f1 f2 → fr2_minus i (❨d,h❩;f1) f2
+.
+
+interpretation
+  "minus (finite relocation maps with pairs)"
+  'RMinus f1 i f2 = (fr2_minus i f1 f2).
+
+(* Basic inversion lemmas ***************************************************)
+
+(*** minuss_inv_nil1 *)
+lemma fr2_minus_inv_nil_sn (f2) (i):
+      ◊ ▭ i ≘ f2 → f2 = ◊.
+#f2 #i @(insert_eq_1 … (◊))
+#f1 * -f1 -f2 -i
+[ //
+| #f1 #f2 #d #h #i #_ #_ #H destruct
+| #f1 #f2 #d #h #i #_ #_ #H destruct
+]
+qed-.
+
+(*** minuss_inv_cons1 *)
+lemma fr2_minus_inv_cons_sn (f1) (f2) (d) (h) (i):
+      ❨d, h❩;f1 ▭ i ≘ f2 →
+      ∨∨ ∧∧ d ≤ i & f1 ▭ h+i ≘ f2
+       | ∃∃f. i < d & f1 ▭ i ≘ f & f2 = ❨d-i,h❩;f.
+#g1 #f2 #d #h #i @(insert_eq_1 … (❨d,h❩;g1))
+#f1 * -f1 -f2 -i
+[ #i #H destruct
+| #f1 #f #d1 #h1 #i1 #Hid1 #Hf #H destruct /3 width=3 by ex3_intro, or_intror/
+| #f1 #f #d1 #h1 #i1 #Hdi1 #Hf #H destruct /3 width=1 by or_introl, conj/
+]
+qed-.
+
+(*** minuss_inv_cons1_ge *)
+lemma fr2_minus_inv_cons_sn_ge (f1) (f2) (d) (h) (i):
+      ❨d, h❩;f1 ▭ i ≘ f2 → d ≤ i → f1 ▭ h+i ≘ f2.
+#f1 #f2 #d #h #i #H
+elim (fr2_minus_inv_cons_sn … H) -H * // #f #Hid #_ #_ #Hdi
+elim (nlt_ge_false … Hid Hdi)
+qed-.
+
+(*** minuss_inv_cons1_lt *)
+lemma fr2_minus_inv_cons_sn_lt (f1) (f2) (d) (h) (i):
+      ❨d, h❩;f1 ▭ i ≘ f2 → i < d →
+      ∃∃f. f1 ▭ i ≘ f & f2 = ❨d-i,h❩;f.
+#f1 #f2 #d #h #i #H elim (fr2_minus_inv_cons_sn … H) -H *
+[ #Hdi #_ #Hid elim (nlt_ge_false … Hid Hdi)
+| /2 width=3 by ex2_intro/
+]
+qed-.