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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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(*                               Nat_compl.v                                *)
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(*********************************************************)
(*             Complements for the natural numbers       *)
(*                                                       *)
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Global Set Asymmetric Patterns.
Require Export Arith.
Require Export Compare_dec.
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Definition decidable (P : Prop) := P \/ ~ P.
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(*       Technical lemmas                                 *)
(**********************************************************)
tauto.
Qed.
intros;
generalize (not_ex_all_not nat (fun n : nat => P n /\ Q n /\ R n /\ S n) H);
intro; generalize (not_symt_et (P k) (Q k) (R k) (S k) (H2 k));
intro; generalize (not_and_or (P k /\ Q k) (R k /\ S k) H3);
intro; generalize (or_to_imply (P k /\ Q k) (~ (R k /\ S k)) H4);
intro; cut (P k /\ Q k); auto with core arith.
intro; generalize (H5 H6); intro; apply (not_and_or (R k) (S k) H7).
Qed.
tauto.
Qed.

(**********)
Lemma princip_min :
forall P : nat -> Prop,
(forall n : nat, decidable (P n)) ->
forall n : nat,
P n -> exists m : nat, P m /\ (forall i : nat, P i -> i >= m).