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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \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. *)
(****************************************************************************)
(*i $Id: Powerset.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*)
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:
(X: (Ensemble U)) (Included U X A) -> (In (Ensemble U) (Power_set A) X).
Hints Resolve Definition_of_Power_set.
Theorem Empty_set_minimal: (X: (Ensemble U)) (Included U (Empty_set U) X).
Intro X; Red.
Intros x H'; Elim H'.
Qed.
Hints Resolve Empty_set_minimal.
Theorem Power_set_Inhabited:
(X: (Ensemble U)) (Inhabited (Ensemble U) (Power_set X)).
Intro X.
Apply Inhabited_intro with (Empty_set U); Auto with sets.
Qed.
Hints Resolve Power_set_Inhabited.
Theorem Inclusion_is_an_order: (Order (Ensemble U) (Included U)).
Auto 6 with sets.
Qed.
Hints Resolve Inclusion_is_an_order.
Theorem Inclusion_is_transitive: (Transitive (Ensemble U) (Included U)).
Elim Inclusion_is_an_order; Auto with sets.
Qed.
Hints 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.
Hints 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.
Hints Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included.
Lemma Strict_inclusion_is_transitive_with_inclusion:
(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:
(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:
(A: (Ensemble U)) (Bottom (Ensemble U) (Power_set_PO A) (Empty_set U)).
Intro A; Apply Bottom_definition; Simpl; Auto with sets.
Qed.
Hints Resolve Empty_set_is_Bottom.
Theorem Union_minimal:
(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.
Hints Resolve Union_minimal.
Theorem Intersection_maximal:
(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: (a, b: (Ensemble U)) (Included U a (Union U a b)).
Auto with sets.
Qed.
Theorem Union_increases_r: (a, b: (Ensemble U)) (Included U b (Union U a b)).
Auto with sets.
Qed.
Theorem Intersection_decreases_l:
(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:
(a, b: (Ensemble U)) (Included U (Intersection U a b) b).
Intros a b; Red.
Intros x H'; Elim H'; Auto with sets.
Qed.
Hints Resolve Union_increases_l Union_increases_r Intersection_decreases_l
Intersection_decreases_r.
Theorem Union_is_Lub:
(A: (Ensemble U)) (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:
(A: (Ensemble U)) (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.
Hints Resolve Empty_set_minimal : sets v62.
Hints Resolve Power_set_Inhabited : sets v62.
Hints Resolve Inclusion_is_an_order : sets v62.
Hints Resolve Inclusion_is_transitive : sets v62.
Hints Resolve Union_minimal : sets v62.
Hints Resolve Union_increases_l : sets v62.
Hints Resolve Union_increases_r : sets v62.
Hints Resolve Intersection_decreases_l : sets v62.
Hints Resolve Intersection_decreases_r : sets v62.
Hints Resolve Empty_set_is_Bottom : sets v62.
Hints Resolve Strict_inclusion_is_transitive : sets v62.
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