propagating the arithmetics library, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat_lt_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat_lt_pred.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_pred.ma".
+include "ground/arith/pnat_lt.ma".
+
+(* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
+
+(* Destructions with ppred **************************************************)
+
+lemma plt_des_gen (p) (q): p < q → q = ↑↓q.
+#p #q elim q -q //
+#H elim (plt_inv_unit_dx … H)
+qed-.
+
+(* Inversions with ppred ****************************************************)
+
+lemma plt_inv_gen (p) (q): p < q → ∧∧ p ≤ ↓q & q = ↑↓q.
+/2 width=1 by ple_inv_succ_sn/ qed-.
+
+lemma plt_inv_succ_sn (p) (q): ↑p < q → ∧∧ p < ↓q & q = ↑↓q.
+/2 width=1 by ple_inv_succ_sn/ qed-.
+
+lemma plt_inv_pred_dx (p) (q): p < ↓q → ↑p < q.
+#p #q #H >(plt_des_gen (𝟏) q)
+[ /2 width=1 by plt_succ_bi/
+| /3 width=3 by ple_plt_trans, plt_ple_trans/
+]
+qed-.
+
+lemma plt_inv_pred_bi (p) (q):
+      ↓p < ↓q → p < q.
+/3 width=3 by plt_inv_pred_dx, ple_plt_trans/
+qed-.
+
+(* Constructions with ppred *************************************************)
+
+lemma plt_unit_sn (q): q = ↑↓q → 𝟏 < q.
+// qed.
+
+lemma plt_pred_bi (p) (q): 𝟏 < p → p < q → ↓p < ↓q.
+#p #q #Hp #Hpq
+@ple_inv_succ_bi
+<(plt_des_gen … Hp) -Hp
+<(plt_des_gen … Hpq) //
+qed.