Library Coq.Relations.Relation_Definitions

Section Relation_Definition.

  Variable A : Type.

  Definition relation := A -> A -> Prop.

  Variable R : relation.

  Section General_Properties_of_Relations.

    Definition reflexive : Prop := forall x:A, R x x.
    Definition transitive : Prop := forall x y z:A, R x y -> R y z -> R x z.
    Definition symmetric : Prop := forall x y:A, R x y -> R y x.
    Definition antisymmetric : Prop := forall 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 (R1 R2:relation) : Prop :=
      forall x y:A, R1 x y -> R2 x y.

    Definition same_relation (R1 R2:relation) : Prop :=
      inclusion R1 R2 /\ inclusion R2 R1.

    Definition commut (R1 R2:relation) : Prop :=
      forall x y:A,
        R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'.

  End Relations_of_Relations.

End Relation_Definition.

Hint Unfold reflexive transitive antisymmetric symmetric: sets.

Hint 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.

Hint Unfold inclusion same_relation commut: sets.