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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope R_scope.
Set Implicit Arguments.
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(** * Each bounded subset of N has a maximal element *)
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Definition Nbound (I:nat -> Prop) : Prop :=
exists n : nat, (forall i:nat, I i -> (i <= n)%nat).

(* Specifications: *)
Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}.
Lemma Nzorn :
forall I:nat -> Prop,
(exists n : nat, I n) ->
Nbound I -> { n:nat | I n /\ (forall i:nat, I i -> (i <= n)%nat) }.