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 v62. 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 v62. Hint Unfold inclusion same_relation commut: sets v62.```
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