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[helm.git] / matita / matita / contribs / lambda_delta / ground_2 / tri.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_2/arith.ma".
+
+(* TRICOTOMY FUNCTION *******************************************************)
+
+let rec tri (A:Type[0]) m n a b c on m : A ≝
+  match m with 
+  [ O   ⇒ match n with [ O ⇒ b | S n ⇒ a ]
+  | S m ⇒ match n with [ O ⇒ c | S n ⇒ tri A m n a b c ]
+  ]. 
+
+(* Basic properties *********************************************************)
+
+lemma tri_lt: ∀A,a,b,c,n,m. m < n → tri A m n a b c = a.
+#A #a #b #c #n elim n -n
+[ #m #H elim (lt_zero_false … H)
+| #n #IH #m elim m -m // /3 width=1/
+]
+qed.
+
+lemma tri_eq: ∀A,a,b,c,m. tri A m m a b c = b.
+#A #a #b #c #m elim m -m normalize //
+qed.
+
+lemma tri_gt: ∀A,a,b,c,m,n. n < m → tri A m n a b c = c.
+#A #a #b #c #m elim m -m
+[ #n #H elim (lt_zero_false … H)
+| #m #IH #n elim n -n // /3 width=1/
+]
+qed.
+