milestone update in ground, partial commit

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/ynat_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/ynat_pred.ma
new file mode 100644 (file)
index 0000000..c0888b7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground/arith/ynat_pred.ma
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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.ma".
+include "ground/arith/ynat_nat.ma".
+
+(* PREDECESSOR FOR NON-NEGATIVE INTEGERS WITH INFINITY **********************)
+
+definition ypred_aux (n): ynat ≝
+           yinj_nat (↓n).
+
+(*** ypred *)
+definition ypred: ynat → ynat ≝
+           ynat_bind_nat ypred_aux (∞).
+
+interpretation
+  "successor (non-negative integers with infinity)"
+  'DownArrow x = (ypred x).
+
+(* Constructions ************************************************************)
+
+(*** ypred_O *)
+lemma ypred_inj (n): yinj_nat (↓n) = ↓(yinj_nat n).
+@(ynat_bind_nat_inj ypred_aux)
+qed.
+
+(*** ypred_Y *)
+lemma ypred_inf: ∞ = ↓∞.
+// qed.
+
+(* Inversions ***************************************************************)
+
+lemma eq_inv_ypred_inj (x) (n):
+      ↓x = yinj_nat n →
+      ∃∃m. x = yinj_nat m & n = ↓m.
+#x #n @(ynat_split_nat_inf … x) -x
+[ #m <ypred_inj #H <(eq_inv_yinj_nat_bi … H) -n
+  /2 width=3 by ex2_intro/
+| #H elim (eq_inv_inf_yinj_nat … H)
+]
+qed-.
+
+(*** ypred_inv_refl *)
+lemma ypred_inv_refl (x): x = ↓x → ∨∨ 𝟎 = x | ∞ = x.
+#x @(ynat_split_nat_inf … x) -x //
+#n <ypred_inj #H
+lapply (eq_inv_yinj_nat_bi … H) -H #H
+lapply (npred_inv_refl … H) -H #H
+/2 width=1 by or_introl/
+qed-.