Coq.Relations.Relation_Definitions

# Module Coq.Relations.Relation_Definitions

``Section` Relation_Definition.

`Variable` A: Set.
`Definition` relation := A -> A -> Prop.

`Variable` R: relation.

`Section` General_Properties_of_Relations.

`Definition` reflexive : Prop := (x: A) (R x x).
`Definition` transitive : Prop := (x,y,z: A) (R x y) -> (R y z) -> (R x z).
`Definition` symmetric : Prop := (x,y: A) (R x y) -> (R y x).
`Definition` antisymmetric : Prop := (x,y: A) (R x y) -> (R y x) -> x=y.

`Definition` equiv := reflexive /\ transitive /\ symmetric.

`End` General_Properties_of_Relations.

`Section` Sets_of_Relations.

`Record` preorder : Prop := {
preord_refl : reflexive;
preord_trans : transitive }.

`Record` order : Prop := {
ord_refl : reflexive;
ord_trans : transitive;
ord_antisym : antisymmetric }.

`Record` equivalence : Prop := {
equiv_refl : reflexive;
equiv_trans : transitive;
equiv_sym : symmetric }.
`Record` PER : Prop := {
per_sym : symmetric;
per_trans : transitive }.

`End` Sets_of_Relations.

`Section` Relations_of_Relations.

`Definition` inclusion : relation -> relation -> Prop :=
[R1,R2: relation] (x,y:A) (R1 x y) -> (R2 x y).

`Definition` same_relation : relation -> relation -> Prop :=
[R1,R2: relation] (inclusion R1 R2) /\ (inclusion R2 R1).
`Definition` commut : relation -> relation -> Prop :=
[R1,R2:relation] (x,y:A) (R1 y x) -> (z:A) (R2 z y)
-> (EX y':A |(R2 y' x) & (R1 z y')).

`End` Relations_of_Relations.

`End` Relation_Definition.

`Hints` Unfold reflexive transitive antisymmetric symmetric : sets v62.

`Hints` Resolve Build_preorder Build_order Build_equivalence
Build_PER preord_refl preord_trans
ord_refl ord_trans ord_antisym
equiv_refl equiv_trans equiv_sym
per_sym per_trans : sets v62.

`Hints` Unfold inclusion same_relation commut : sets v62.

`

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