milestone update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / gr_nat_basic.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_basic.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_basic.ma
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+++ b/matita/matita/contribs/lambdadelta/ground/relocation/gr_nat_basic.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/relocation/gr_basic.ma".
+include "ground/relocation/gr_nat_uni.ma".
+
+(* NON-NEGATIVE APPLICATION FOR GENERIC RELOCATION MAPS *****************************)
+
+(* Properties with gr_basic **********************************************)
+
+lemma gr_nat_basic_lt (m) (n) (l):
+      l < m → @↑❪l, 𝐛❨m,n❩❫ ≘ l.
+#m @(nat_ind_succ … m) -m
+[ #n #i #H elim (nlt_inv_zero_dx … H)
+| #m #IH #n #l @(nat_ind_succ … l) -l
+  [ #_ /2 width=2 by refl, gr_pat_refl/
+  | #l #_ #H
+    lapply (nlt_inv_succ_bi … H) -H #Hlm
+    /3 width=7 by refl, gr_pat_push/
+  ]
+]
+qed.
+
+lemma gr_nat_basic_ge (m) (n) (l):
+      m ≤ l → @↑❪l, 𝐛❨m,n❩❫ ≘ l+n.
+#m @(nat_ind_succ … m) -m //
+#m #IH #n #l #H
+elim (nle_inv_succ_sn … H) -H #Hml #H >H -H
+/3 width=7 by gr_nat_push/
+qed.
+
+(* Inversion lemmas with gr_basic ****************************************)
+
+lemma gr_nat_basic_inv_lt (m) (n) (l) (k):
+      l < m → @↑❪l, 𝐛❨m,n❩❫ ≘ k → l = k.
+/3 width=4 by gr_nat_basic_lt, gr_nat_mono/ qed-.
+
+lemma gr_nat_basic_inv_ge (m) (n) (l) (k):
+      m ≤ l → @↑❪l, 𝐛❨m,n❩❫ ≘ k → l+n = k.
+/3 width=4 by gr_nat_basic_ge, gr_nat_mono/ qed-.