OpenMath Content Dictionary: relation0

Canonical URL:
http://www.openmath.org/cd/relation0.ocd
CD Base:
http://www.openmath.org/cd
CD File:
relation0.ocd
CD as XML Encoded OpenMath:
relation0.omcd
Defines:
antisymmetric, equivalence, irreflexive, order, partial_equivalence, pre_order, reflexive, relation, strict_order, symmetric, transitive
Date:
2004-03-30
Version:
2 (Revision 1)
Review Date:
2017-12-31
Status:
experimental


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  Author: OpenMath Consortium
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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

Role:
application
Description:

Type constructor; returns the type of binary relations on a set.

Commented Mathematical property (CMP):
Is defined as "[A:Set] A -> A -> Prop"
Signatures:
sts


[Next: reflexive] [Last: pre_order] [Top]

reflexive

Role:
application
Description:

Proposition; the type of reflexive binary relations.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)](x:A)(R x x)
Signatures:
sts


[Next: irreflexive] [Previous: relation] [Top]

irreflexive

Role:
application
Description:

Proposition; the type of irreflexive binary relations.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)](x:A) ~(R x x)
Signatures:
sts


[Next: transitive] [Previous: reflexive] [Top]

transitive

Role:
application
Description:

Proposition; the type of transitive binary relations.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)](x,y,z:A)(R x y) -> (R y z) -> (R x z)
Signatures:
sts


[Next: symmetric] [Previous: irreflexive] [Top]

symmetric

Role:
application
Description:

Proposition; the type of symmetric binary relations.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)](x,y:A)(R x y) -> (R y x)
Signatures:
sts


[Next: antisymmetric] [Previous: transitive] [Top]

antisymmetric

Role:
application
Description:

Proposition; the type of antisymmetric binary relations.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)](x,y:A)(R x y) -> (R y x) -> (relation1::eq x y)
Signatures:
sts


[Next: partial_equivalence] [Previous: symmetric] [Top]

partial_equivalence

Role:
application
Description:

Proposition; the type of partial_equivalence relations, namely relations that are symmetric, and transitive.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)] (symmetric R) /\ (transitive R)
Signatures:
sts


[Next: equivalence] [Previous: antisymmetric] [Top]

equivalence

Role:
application
Description:

Proposition; the type of equivalence relations, namely relations that are reflexive, symmetric and transitive.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (symmetric R) /\ (transitive R)
Signatures:
sts


[Next: order] [Previous: partial_equivalence] [Top]

order

Role:
application
Description:

Proposition; the type of order relations, namely relations that are reflexive, antisymmetric and transitive.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (antisymmetric R) /\ (transitive R)
Signatures:
sts


[Next: strict_order] [Previous: equivalence] [Top]

strict_order

Role:
application
Description:

Proposition; the type of strict order relations, namely relations that are irreflexive, antisymmetric and transitive.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)] (irreflexive R) /\ (antisymmetric R) /\ (transitive R)
Signatures:
sts


[Next: pre_order] [Previous: order] [Top]

pre_order

Role:
application
Description:

Proposition; the type of preorder relations, namely relations that are reflexive and transitive.

Commented Mathematical property (CMP):
Defined as [A:symtype][R: (relation A)] (reflexive R) /\ (transitive R)
Signatures:
sts


[First: relation] [Previous: strict_order] [Top]