arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_minus_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_plus.ma".
+include "ground/arith/nat_minus.ma".
+
+(* SUBTRACTION FOR NON-NEGATIVE INTEGERS ************************************)
+
+(* Rewrites with nplus ******************************************************)
+
+(*** minus_plus_m_m *)
+lemma nminus_plus_sn_refl_sn (m) (n): m = m + n - n.
+#m #n elim n -n //
+#n #IH <nplus_succ_dx <nminus_succ_bi //
+qed.
+
+lemma nminus_plus_sn_refl_dx (m) (n): m = n + m - n.
+#m #n <nplus_comm //
+qed.
+
+(*** minus_plus *)
+theorem nminus_assoc (o) (m) (n): o-m-n = o-(m+n).
+#o #m #n elim n -n //
+#n #IH <nplus_succ_dx <nminus_succ_dx <nminus_succ_dx //
+qed-.
+
+(*** minus_plus_plus_l *)
+lemma nminus_plus_dx_bi (m) (n) (o): m - n = (m + o) - (n + o).
+#m #n #o <nminus_assoc <nminus_minus_comm //
+qed.
+
+(*** plus_minus_plus_plus_l *) (**)
+lemma plus_minus_plus_plus_l: ∀z,x,y,h. z + (x + h) - (y + h) = z + x - y.
+// qed-.
+
+(* Helper constructions with nplus ******************************************)
+
+(*** plus_to_minus *)
+lemma nminus_plus_dx (o) (m) (n): o = m+n → n = o-m.
+#o #m #n #H destruct //
+qed-.
+
+lemma nminus_plus_sn (o) (m) (n): o = m+n → m = o-n.
+#o #m #n #H destruct //
+qed-.
+
+(* Inversions with nplus ****************************************************)
+
+(*** discr_plus_xy_minus_xz *)
+lemma eq_inv_plus_nminus_refl_sn (m) (n) (o):
+      m + o = m - n →
+      ∨∨ ∧∧ 𝟎 = m & 𝟎 = o
+       | ∧∧ 𝟎 = n & 𝟎 = o.
+#m elim m -m
+[ /3 width=1 by or_introl, conj/
+| #m #IH #n @(nat_ind … n) -n
+  [ #o #Ho
+    lapply (eq_inv_nplus_bi_sn … (𝟎) Ho) -Ho
+    /3 width=1 by or_intror, conj/
+  | #n #_ #o
+    <nminus_succ_bi >nplus_succ_shift #Ho
+    elim (IH … Ho) -IH -Ho * #_ #H
+    elim (eq_inv_nzero_succ … H)
+  ]
+]
+qed-.
+
+(*** discr_minus_x_xy *)
+lemma eq_inv_nminus_refl_sn (m) (n): m = m - n → ∨∨ 𝟎 = m | 𝟎 = n.
+#m #n #Hmn
+elim (eq_inv_plus_nminus_refl_sn … (𝟎) Hmn) -Hmn * #H1 #H2
+/2 width=1 by or_introl, or_intror/
+qed-.