arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_lt_plus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.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/arith/nat_le_plus.ma".
+include "ground/arith/nat_lt.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+(* Basic constructions with plus ********************************************)
+
+(*** monotonic_lt_plus_l *)
+lemma nlt_plus_bi_dx (m) (n1) (n2): n1 < n2 → n1 + m < n2 + m.
+#m #n1 #n2 #H
+@nlt_i >nplus_succ_sn /2 width=1 by nle_plus_bi_dx/
+qed.
+
+(*** monotonic_lt_plus_r *)
+lemma nlt_plus_bi_sn (m) (n1) (n2): n1 < n2 → m + n1 < m + n2.
+#m #n1 #n2 #H
+@nlt_i >nplus_succ_dx /2 width=1 by nle_plus_bi_sn/
+qed.
+
+(*** lt_plus_Sn_r *) (**)
+lemma lt_plus_Sn_r: ∀a,x,n. a < a + x + ↑n.
+/2 width=1/ qed-.
+
+(* Basic inversions with plus ***********************************************)
+
+(*** lt_plus_to_lt_l *)
+lemma nlt_inv_plus_bi_dx (m) (n1) (n2): n1 + m < n2 + m → n1 < n2.
+/2 width=2 by nle_inv_plus_bi_dx/ qed-.
+
+(*** lt_plus_to_lt_r *)
+lemma nlt_inv_plus_bi_sn (m) (n1) (n2): m + n1 < m + n2 → n1 < n2.
+/2 width=2 by nle_inv_plus_bi_sn/ qed-.