# Library Coq.Sets.Partial_Order

Require Export Ensembles.
Require Export Relations_1.
Section Partial_orders.
Variable U : Type.
Definition Carrier := Ensemble U.
Definition Rel := Relation U.
Record PO : Type := Definition_of_PO
{ Carrier_of : Ensemble U;
Rel_of : Relation U;
PO_cond1 : Inhabited U Carrier_of;
PO_cond2 : Order U Rel_of }.
Variable p : PO.
Definition Strict_Rel_of : Rel := fun x y:U => Rel_of p x y /\ x <> y.
Inductive covers (y x:U) : Prop :=
Definition_of_covers :
Strict_Rel_of x y ->
~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
covers y x.
End Partial_orders.
Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets v62.
Hint Resolve Definition_of_covers: sets v62.
Section Partial_order_facts.
Variable U : Type.
Variable D : PO U.
Lemma Strict_Rel_Transitive_with_Rel :
forall x y z:U,
Strict_Rel_of U D x y -> Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive_with_Rel_left :
forall x y z:U,
Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).
End Partial_order_facts.