arithmetics for λδ

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus_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/nat_pred_succ.ma".
+include "ground/arith/nat_plus.ma".
+
+(* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************)
+
+(* Inversions with npred ****************************************************)
+
+(*** plus_inv_S3_sn *)
+lemma eq_inv_succ_nplus_sn (o) (m) (n):
+      ↑o = m + n →
+      ∨∨ ∧∧ 𝟎 = m & n = ↑o
+       | ∧∧ m = ↑↓m & o = ↓m + n.
+#o #m @(nat_ind_succ … m) -m
+[ /3 width=1 by or_introl, conj/
+| #m #_ #n <nplus_succ_sn
+  /4 width=1 by eq_inv_nsucc_bi, or_intror, conj/
+]
+qed-.
+
+(*** plus_inv_S3_dx *)
+lemma eq_inv_succ_nplus_dx (o) (m) (n):
+      ↑o = m + n →
+      ∨∨ ∧∧ 𝟎 = n & m = ↑o
+       | ∧∧ n = ↑↓n & o = m + ↓n.
+#o #m #n @(nat_ind_succ … n) -n
+[ /3 width=1 by or_introl, conj/
+| #n #_ <nplus_succ_sn
+  /4 width=1 by eq_inv_nsucc_bi, or_intror, conj/
+]
+qed-.