Library Stdlib.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.

#[global]
Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets.
#[global]
Hint Resolve Definition_of_covers: sets.

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.