matita/matita/contribs/lambda/trichotomy.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "preamble.ma".
  16
  17 (* TRICHOTOMY OPERATOR ******************************************************)
  18
  19 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
  20 let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
  21   match n1 with
  22   [ O    ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
  23   | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
  24   ].
  25
  26 lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
  27 #A #a1 #a2 #a3 #n2 elim n2 -n2
  28 [ #n1 #H elim (lt_zero_false … H)
  29 | #n2 #IH #n1 elim n1 -n1 // /3 width=1/
  30 ]
  31 qed.
  32
  33 lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
  34 #A #a1 #a2 #a3 #n elim n -n normalize //
  35 qed.
  36
  37 lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
  38 #A #a1 #a2 #a3 #n1 elim n1 -n1
  39 [ #n2 #H elim (lt_zero_false … H)
  40 | #n1 #IH #n2 elim n2 -n2 // /3 width=1/
  41 ]
  42 qed.