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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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(*                               three_gap.v                                *)
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(*                 THE THREE GAP THER0EM                 *)
(*                                                       *)
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Require Export preuve2.
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Section gap.
Variable N n : nat.
Hypothesis alpha_irr : forall n p : Z, (alpha * IZR p)%R <> IZR n.
Hypothesis prop_alpha : (0 < alpha)%R /\ (alpha < 1)%R.
Hypothesis prop_N : forall N : nat, N >= 2.
Hypothesis Hn : 0 < n /\ n < N.
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Definition succes := after N n.
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Definition num1 := n + first N.
Definition num2 := n + first N - last N.
Definition num3 := n - last N.
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(*                 Theorem                               *)
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Theorem three_gap : succes = num1 \/ succes = num2 \/ succes = num3.
generalize (inser2_trans_lt 0 n (last N) (N - first N) N Hn).