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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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19 (*#***********************************************************************)
21 (* v * The Coq Proof Assistant / The Coq Development Team *)
23 (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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27 (* // * This file is distributed under the terms of the *)
29 (* * GNU Lesser General Public License Version 2.1 *)
31 (*#***********************************************************************)
33 (*#***************************************************************************)
37 (* Naive Set Theory in Coq *)
43 (* Rocquencourt Sophia-Antipolis *)
61 (* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *)
63 (* to the Newton Institute for providing an exceptional work environment *)
65 (* in Summer 1995. Several developments by E. Ledinot were an inspiration. *)
67 (*#***************************************************************************)
69 (*i $Id: Powerset.v,v 1.5.2.1 2004/07/16 19:31:18 herbelin Exp $ i*)
71 include "Sets/Ensembles.ma".
73 include "Sets/Relations_1.ma".
75 include "Sets/Relations_1_facts.ma".
77 include "Sets/Partial_Order.ma".
79 include "Sets/Cpo.ma".
82 Section The_power_set_partial_order
86 cic:/Coq/Sets/Powerset/The_power_set_partial_order/U.var
89 inline procedural "cic:/Coq/Sets/Powerset/Power_set.ind".
92 Hint Resolve Definition_of_Power_set.
95 inline procedural "cic:/Coq/Sets/Powerset/Empty_set_minimal.con" as theorem.
98 Hint Resolve Empty_set_minimal.
101 inline procedural "cic:/Coq/Sets/Powerset/Power_set_Inhabited.con" as theorem.
104 Hint Resolve Power_set_Inhabited.
107 inline procedural "cic:/Coq/Sets/Powerset/Inclusion_is_an_order.con" as theorem.
110 Hint Resolve Inclusion_is_an_order.
113 inline procedural "cic:/Coq/Sets/Powerset/Inclusion_is_transitive.con" as theorem.
116 Hint Resolve Inclusion_is_transitive.
119 inline procedural "cic:/Coq/Sets/Powerset/Power_set_PO.con" as definition.
122 Hint Unfold Power_set_PO.
125 inline procedural "cic:/Coq/Sets/Powerset/Strict_Rel_is_Strict_Included.con" as theorem.
128 Hint Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included.
131 inline procedural "cic:/Coq/Sets/Powerset/Strict_inclusion_is_transitive_with_inclusion.con" as lemma.
133 inline procedural "cic:/Coq/Sets/Powerset/Strict_inclusion_is_transitive_with_inclusion_left.con" as lemma.
135 inline procedural "cic:/Coq/Sets/Powerset/Strict_inclusion_is_transitive.con" as lemma.
137 inline procedural "cic:/Coq/Sets/Powerset/Empty_set_is_Bottom.con" as theorem.
140 Hint Resolve Empty_set_is_Bottom.
143 inline procedural "cic:/Coq/Sets/Powerset/Union_minimal.con" as theorem.
146 Hint Resolve Union_minimal.
149 inline procedural "cic:/Coq/Sets/Powerset/Intersection_maximal.con" as theorem.
151 inline procedural "cic:/Coq/Sets/Powerset/Union_increases_l.con" as theorem.
153 inline procedural "cic:/Coq/Sets/Powerset/Union_increases_r.con" as theorem.
155 inline procedural "cic:/Coq/Sets/Powerset/Intersection_decreases_l.con" as theorem.
157 inline procedural "cic:/Coq/Sets/Powerset/Intersection_decreases_r.con" as theorem.
160 Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l
161 Intersection_decreases_r.
164 inline procedural "cic:/Coq/Sets/Powerset/Union_is_Lub.con" as theorem.
166 inline procedural "cic:/Coq/Sets/Powerset/Intersection_is_Glb.con" as theorem.
169 End The_power_set_partial_order
173 Hint Resolve Empty_set_minimal: sets v62.
177 Hint Resolve Power_set_Inhabited: sets v62.
181 Hint Resolve Inclusion_is_an_order: sets v62.
185 Hint Resolve Inclusion_is_transitive: sets v62.
189 Hint Resolve Union_minimal: sets v62.
193 Hint Resolve Union_increases_l: sets v62.
197 Hint Resolve Union_increases_r: sets v62.
201 Hint Resolve Intersection_decreases_l: sets v62.
205 Hint Resolve Intersection_decreases_r: sets v62.
209 Hint Resolve Empty_set_is_Bottom: sets v62.
213 Hint Resolve Strict_inclusion_is_transitive: sets v62.