partial update in static_2

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat_lt_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat_lt_plus.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/ground/arith/pnat_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/pnat_le_plus.ma".
+include "ground/arith/pnat_lt.ma".
+
+(* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
+
+(* Constructions with pplus *************************************************)
+
+lemma plt_plus_bi_dx (p) (q1) (q2): q1 < q2 → q1 + p < q2 + p.
+#p #q1 #q2 #H
+@plt_i >pplus_succ_sn /2 width=1 by ple_plus_bi_dx/
+qed.
+
+lemma plt_plus_bi_sn (p) (q1) (q2): q1 < q2 → p + q1 < p + q2.
+#p #q1 #q2 #H
+@plt_i >pplus_succ_dx /2 width=1 by ple_plus_bi_sn/
+qed.
+
+lemma plt_plus_dx_dx_refl (p) (q): p < p + q.
+/2 width=1 by ple_plus_bi_sn/ qed.
+
+lemma plt_plus_dx_sn_refl (p) (q): p < q + p.
+/2 width=1 by ple_plus_bi_dx/ qed.
+
+lemma plt_plus_dx_sn (r) (p) (q): q ≤ p → q < r + p.
+/2 width=3 by ple_plt_trans/ qed.
+
+lemma plt_plus_dx_dx (r) (p) (q): q ≤ p → q < p + r.
+/2 width=3 by ple_plt_trans/ qed.