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(* 02110-1301 USA                                                     *)
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(*                                                                          *)
(*  Proof of the three gap theorem.                                         *)
(*                                                                          *)
(*  Micaela Mayero (INRIA-Rocquencourt)                                     *)
(*  September 1998                                                          *)
(*                                                                          *)
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(*                               prop_elem.v                                *)
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(*          Elementary properties first last             *)
(*                                                       *)
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Require Export tools.
Unset Standard Proposition Elimination Names.
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Lemma first_N : forall N : nat, N >= 2 -> first N < N.
intros; unfold first in |- *; case (N_classic N); intro.