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.