This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. relation0 http://www.openmath.org/cd http://www.openmath.org/cd/relation0.ocd 2006-03-30 experimental 2004-03-30 2 0 Binary relations properties, equivalence relation, orders, up to the definition of a setoid as a set with an equivalence relations defined on its elements. Initial version: O. Caprotti relation application Type constructor; returns the type of binary relations on a set. Is defined as "[A:Set] A -> A -> Prop" reflexive application Proposition; the type of reflexive binary relations. Defined as [A:symtype][R: (relation A)](x:A)(R x x) irreflexive application Proposition; the type of irreflexive binary relations. Defined as [A:symtype][R: (relation A)](x:A) ~(R x x) transitive application Proposition; the type of transitive binary relations. Defined as [A:symtype][R: (relation A)](x,y,z:A)(R x y) -> (R y z) -> (R x z) symmetric application Proposition; the type of symmetric binary relations. Defined as [A:symtype][R: (relation A)](x,y:A)(R x y) -> (R y x) antisymmetric application Proposition; the type of antisymmetric binary relations. Defined as [A:symtype][R: (relation A)](x,y:A)(R x y) -> (R y x) -> (relation1::eq x y) partial_equivalence application Proposition; the type of partial_equivalence relations, namely relations that are symmetric, and transitive. Defined as [A:symtype][R: (relation A)] (symmetric R) /\ (transitive R) equivalence application Proposition; the type of equivalence relations, namely relations that are reflexive, symmetric and transitive. Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (symmetric R) /\ (transitive R) order application Proposition; the type of order relations, namely relations that are reflexive, antisymmetric and transitive. Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (antisymmetric R) /\ (transitive R) strict_order application Proposition; the type of strict order relations, namely relations that are irreflexive, antisymmetric and transitive. Defined as [A:symtype][R: (relation A)] (irreflexive R) /\ (antisymmetric R) /\ (transitive R) pre_order application Proposition; the type of preorder relations, namely relations that are reflexive and transitive. Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (transitive R)