-(**************************************************************************)
-(* ___ *)
-(* ||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 "preamble.ma".
-
-(* TRICHOTOMY OPERATOR ******************************************************)
-
-(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
-let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
- match n1 with
- [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
- | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
- ].
-
-lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
-#A #a1 #a2 #a3 #n2 elim n2 -n2
-[ #n1 #H elim (lt_zero_false … H)
-| #n2 #IH #n1 elim n1 -n1 // /3 width=1/
-]
-qed.
-
-lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
-#A #a1 #a2 #a3 #n elim n -n normalize //
-qed.
-
-lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
-#A #a1 #a2 #a3 #n1 elim n1 -n1
-[ #n2 #H elim (lt_zero_false … H)
-| #n1 #IH #n2 elim n2 -n2 // /3 width=1/
-]
-qed.