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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(****************************************************************************)
(* *)
(* 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. *)
(****************************************************************************)
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 Gt.
Require Export Lt.
Require Export Le.
Require Export Finite_sets_facts.
Require Export Image.
Require Export Infinite_sets.
Require Export Compare_dec.
Require Export Relations_1.
Require Export Partial_Order.
Require Export Cpo.
Section Integers_sect.
Inductive Integers : Ensemble nat :=
Integers_defn : forall x:nat, In nat Integers x.
Lemma le_reflexive : Reflexive nat le.
Proof.
red; auto with arith.
Qed.
Lemma le_antisym : Antisymmetric nat le.
Proof.
red; intros x y H H'; rewrite (le_antisym x y); auto.
Qed.
Lemma le_trans : Transitive nat le.
Proof.
red; intros; apply le_trans with y; auto.
Qed.
Lemma le_Order : Order nat le.
Proof.
split; [exact le_reflexive | exact le_trans | exact le_antisym].
Qed.
Lemma triv_nat : forall n:nat, In nat Integers n.
Proof.
exact Integers_defn.
Qed.
Definition nat_po : PO nat.
apply Definition_of_PO with (Carrier_of := Integers) (Rel_of := le);
auto with sets arith.
apply Inhabited_intro with (x := 0).
apply Integers_defn.
exact le_Order.
Defined.
Lemma le_total_order : Totally_ordered nat nat_po Integers.
Proof.
apply Totally_ordered_definition.
simpl.
intros H' x y H'0.
elim le_or_lt with (n := x) (m := y).
intro H'1; left; auto with sets arith.
intro H'1; right.
cut (y <= x); auto with sets arith.
Qed.
Lemma Finite_subset_has_lub :
forall X:Ensemble nat,
Finite nat X -> exists m : nat, Upper_Bound nat nat_po X m.
Proof.
intros X H'; elim H'.
exists 0.
apply Upper_Bound_definition.
unfold nat_po. simpl. apply triv_nat.
intros y H'0; elim H'0; auto with sets arith.
intros A H'0 H'1 x H'2; try assumption.
elim H'1; intros x0 H'3; clear H'1.
elim le_total_order.
simpl.
intro H'1; try assumption.
lapply H'1; [ intro H'4; idtac | try assumption ]; auto with sets arith.
generalize (H'4 x0 x).
clear H'4.
clear H'1.
intro H'1; lapply H'1;
[ intro H'4; elim H'4;
[ intro H'5; try exact H'5; clear H'4 H'1 | intro H'5; clear H'4 H'1 ]
| clear H'1 ].
exists x.
apply Upper_Bound_definition. simpl. apply triv_nat.
intros y H'1; elim H'1.
generalize le_trans.
intro H'4; red in H'4.
intros x1 H'6; try assumption.
apply H'4 with (y := x0). elim H'3; simpl; auto with sets arith. trivial.
intros x1 H'4; elim H'4. unfold nat_po; simpl; trivial.
exists x0.
apply Upper_Bound_definition.
unfold nat_po. simpl. apply triv_nat.
intros y H'1; elim H'1.
intros x1 H'4; try assumption.
elim H'3; simpl; auto with sets arith.
intros x1 H'4; elim H'4; auto with sets arith.
red.
intros x1 H'1; elim H'1; apply triv_nat.
Qed.
Lemma Integers_has_no_ub :
~ (exists m : nat, Upper_Bound nat nat_po Integers m).
Proof.
red; intro H'; elim H'.
intros x H'0.
elim H'0; intros H'1 H'2.
cut (In nat Integers (S x)).
intro H'3.
specialize H'2 with (y := S x); lapply H'2;
[ intro H'5; clear H'2 | try assumption; clear H'2 ].
simpl in H'5.
absurd (S x <= x); auto with arith.
apply triv_nat.
Qed.
Lemma Integers_infinite : ~ Finite nat Integers.
Proof.
generalize Integers_has_no_ub.
intro H'; red; intro H'0; try exact H'0.
apply H'.
apply Finite_subset_has_lub; auto with sets arith.
Qed.
End Integers_sect.
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