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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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(*                               preuve1.v                                  *)
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(*           Intermediate proof 1                        *)
(*                                                       *)
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Require Export prop_fl.
Unset Standard Proposition Elimination Names.
Section particular.
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.
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Definition M (N : nat) := first N + last N.

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Lemma inter31a :
forall N n : nat, 0 < n -> n < last (M N) -> after (M N) n = n + first (M N).