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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 Ensembles.
Require Export Relations_1.
Require Export Relations_1_facts.
Require Export Partial_Order.
Require Export Cpo.
Section The_power_set_partial_order.
Variable U : Type.
Inductive Power_set (A:Ensemble U) : Ensemble (Ensemble U) :=
Definition_of_Power_set :
forall X:Ensemble U, Included U X A -> In (Ensemble U) (Power_set A) X.
Hint Resolve Definition_of_Power_set.
Theorem Empty_set_minimal : forall X:Ensemble U, Included U (Empty_set U) X.
intro X; red.
intros x H'; elim H'.
Qed.
Hint Resolve Empty_set_minimal.
Theorem Power_set_Inhabited :
forall X:Ensemble U, Inhabited (Ensemble U) (Power_set X).
intro X.
apply Inhabited_intro with (Empty_set U); auto with sets.
Qed.
Hint Resolve Power_set_Inhabited.
Theorem Inclusion_is_an_order : Order (Ensemble U) (Included U).
auto 6 with sets.
Qed.
Hint Resolve Inclusion_is_an_order.
Theorem Inclusion_is_transitive : Transitive (Ensemble U) (Included U).
elim Inclusion_is_an_order; auto with sets.
Qed.
Hint Resolve Inclusion_is_transitive.
Definition Power_set_PO : Ensemble U -> PO (Ensemble U).
intro A; try assumption.
apply Definition_of_PO with (Power_set A) (Included U); auto with sets.
Defined.
Hint Unfold Power_set_PO.
Theorem Strict_Rel_is_Strict_Included :
same_relation (Ensemble U) (Strict_Included U)
(Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U))).
auto with sets.
Qed.
Hint Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included.
Lemma Strict_inclusion_is_transitive_with_inclusion :
forall x y z:Ensemble U,
Strict_Included U x y -> Included U y z -> Strict_Included U x z.
intros x y z H' H'0; try assumption.
elim Strict_Rel_is_Strict_Included.
unfold contains.
intros H'1 H'2; try assumption.
apply H'1.
apply Strict_Rel_Transitive_with_Rel with (y := y); auto with sets.
Qed.
Lemma Strict_inclusion_is_transitive_with_inclusion_left :
forall x y z:Ensemble U,
Included U x y -> Strict_Included U y z -> Strict_Included U x z.
intros x y z H' H'0; try assumption.
elim Strict_Rel_is_Strict_Included.
unfold contains.
intros H'1 H'2; try assumption.
apply H'1.
apply Strict_Rel_Transitive_with_Rel_left with (y := y); auto with sets.
Qed.
Lemma Strict_inclusion_is_transitive :
Transitive (Ensemble U) (Strict_Included U).
apply cong_transitive_same_relation with
(R := Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U)));
auto with sets.
Qed.
Theorem Empty_set_is_Bottom :
forall A:Ensemble U, Bottom (Ensemble U) (Power_set_PO A) (Empty_set U).
intro A; apply Bottom_definition; simpl; auto with sets.
Qed.
Hint Resolve Empty_set_is_Bottom.
Theorem Union_minimal :
forall a b X:Ensemble U,
Included U a X -> Included U b X -> Included U (Union U a b) X.
intros a b X H' H'0; red.
intros x H'1; elim H'1; auto with sets.
Qed.
Hint Resolve Union_minimal.
Theorem Intersection_maximal :
forall a b X:Ensemble U,
Included U X a -> Included U X b -> Included U X (Intersection U a b).
auto with sets.
Qed.
Theorem Union_increases_l : forall a b:Ensemble U, Included U a (Union U a b).
auto with sets.
Qed.
Theorem Union_increases_r : forall a b:Ensemble U, Included U b (Union U a b).
auto with sets.
Qed.
Theorem Intersection_decreases_l :
forall a b:Ensemble U, Included U (Intersection U a b) a.
intros a b; red.
intros x H'; elim H'; auto with sets.
Qed.
Theorem Intersection_decreases_r :
forall a b:Ensemble U, Included U (Intersection U a b) b.
intros a b; red.
intros x H'; elim H'; auto with sets.
Qed.
Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l
Intersection_decreases_r.
Theorem Union_is_Lub :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b).
intros A a b H' H'0.
apply Lub_definition; simpl.
apply Upper_Bound_definition; simpl; auto with sets.
intros y H'1; elim H'1; auto with sets.
intros y H'1; elim H'1; simpl; auto with sets.
Qed.
Theorem Intersection_is_Glb :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Glb (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b)
(Intersection U a b).
intros A a b H' H'0.
apply Glb_definition; simpl.
apply Lower_Bound_definition; simpl; auto with sets.
apply Definition_of_Power_set.
generalize Inclusion_is_transitive; intro IT; red in IT; apply IT with a;
auto with sets.
intros y H'1; elim H'1; auto with sets.
intros y H'1; elim H'1; simpl; auto with sets.
Qed.
End The_power_set_partial_order.
Hint Resolve Empty_set_minimal: sets v62.
Hint Resolve Power_set_Inhabited: sets v62.
Hint Resolve Inclusion_is_an_order: sets v62.
Hint Resolve Inclusion_is_transitive: sets v62.
Hint Resolve Union_minimal: sets v62.
Hint Resolve Union_increases_l: sets v62.
Hint Resolve Union_increases_r: sets v62.
Hint Resolve Intersection_decreases_l: sets v62.
Hint Resolve Intersection_decreases_r: sets v62.
Hint Resolve Empty_set_is_Bottom: sets v62.
Hint Resolve Strict_inclusion_is_transitive: sets v62.
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