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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/notation/functions/two_0.ma".
+include "ground/arith/nat_le_minus_plus.ma".
+include "ground/arith/nat_lt.ma".
+
+(* ARITHMETICAL PROPERTIES FOR λδ-2A ****************************************)
+
+interpretation
+  "zero (non-negative integers)"
+  'Two = (ninj (psucc punit)).
+
+(* Equalities ***************************************************************)
+
+lemma plus_n_2: ∀n. (n + 𝟐) = n + 𝟏 + 𝟏.
+// qed.
+
+lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
+#a #b #c1 #H >nminus_comm <nminus_assoc_comm_23 //
+qed-.
+
+lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
+#a #b #c1 #c2 #H
+>(nminus_plus_assoc ? c1 c2) >(nminus_plus_assoc ? c1 c2)
+/2 width=1 by arith_b1/
+qed-.
+
+lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
+#x #a #b #c1
+<nplus_plus_comm_23 //
+qed.
+
+lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
+                a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
+#a1 #a2 #b #c1 #H1 #H2
+>nminus_plus_comm_23
+/2 width=1 by arith_b1/
+qed-.
+
+lemma arith_i: ∀x,y,z. y < x → x+z-y-(𝟏) = x-y-(𝟏)+z.
+/2 width=1 by nminus_plus_comm_23/ qed-.
+
+(* Constructions ************************************************************)
+
+fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
+// qed-.
+
+fact le_repl_sn_trans_aux: ∀x,y,z:nat. x ≤ z → y = x → y ≤ z.
+// qed-.
+
+lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
+/3 width=1 by nle_minus_bi_dx/ qed.
+
+lemma arith_j: ∀x,y,z. x-y-(𝟏) ≤ x-(y-z)-𝟏.
+/3 width=1 by nle_minus_bi_dx, nle_minus_bi_sn/ qed.
+
+lemma arith_k_sn: ∀z,x,y,n. z < x → x+n ≤ y → x-z-(𝟏)+n ≤ y-z-𝟏.
+#z #x #y #n #Hzx #Hxny
+>nminus_plus_comm_23 [2: /2 width=1 by nle_minus_bi_sn/ ]
+>nminus_plus_comm_23 [2: /2 width=1 by nlt_des_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+lemma arith_k_dx: ∀z,x,y,n. z < x → y ≤ x+n → y-z-(𝟏) ≤ x-z-(𝟏)+n.
+#z #x #y #n #Hzx #Hyxn
+>nminus_plus_comm_23 [2: /2 width=1 by nle_minus_bi_sn/ ]
+>nminus_plus_comm_23 [2: /2 width=1 by nlt_des_le/ ]
+/2 width=1 by monotonic_le_minus_l2/
+qed.
+
+(* Inversions ***************************************************************)
+
+lemma lt_plus_SO_to_le: ∀x,y. x < y + (𝟏) → x ≤ y.
+/2 width=1 by nle_inv_succ_bi/ qed-.
+
+(* Iterators ****************************************************************)
+
+lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+𝟏) b = f (f^l b).
+#B #f #b #l
+<niter_succ //
+qed.