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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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(*                                                                          *)
(*                         Naive Set Theory in Coq                          *)
(*                                                                          *)
(*                     INRIA                        INRIA                   *)
(*              Rocquencourt                        Sophia-Antipolis        *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*									    *)
(*			         Gilles Kahn 				    *)
(*				 Gerard Huet				    *)
(*									    *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
(* to the Newton Institute for providing an exceptional work environment    *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
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Require Export Finite_sets.
Require Export Constructive_sets.
Require Export Classical.
Require Export Classical_sets.
Require Export Powerset.
Require Export Powerset_facts.
Require Export Powerset_Classical_facts.
Require Export Finite_sets_facts.
Require Export Image.
Section Approx.
Variable U : Type.
Inductive Approximant (A X:Ensemble U) : Prop :=
Defn_of_Approximant : Finite U X -> Included U X A -> Approximant A X.
End Approx.
#[global]
Hint Resolve Defn_of_Approximant : core.
Section Infinite_sets.
Variable U : Type.

(* Specifications: *)
Lemma make_new_approximant :
forall A X:Ensemble U,
~ Finite U A -> Approximant U A X -> Inhabited U (Setminus U A X).
Lemma approximants_grow :
forall A X:Ensemble U,
~ Finite U A ->
forall n:nat,
cardinal U X n ->
Included U X A ->  exists Y : _, cardinal U Y (S n) /\ Included U Y A.
Lemma approximants_grow' :
forall A X:Ensemble U,
~ Finite U A ->
forall n:nat,
cardinal U X n ->
Approximant U A X ->
exists Y : _, cardinal U Y (S n) /\ Approximant U A Y.
Lemma approximant_can_be_any_size :
forall A X:Ensemble U,
~ Finite U A ->
forall n:nat,  exists Y : _, cardinal U Y n /\ Approximant U A Y.