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.
unfold Strict_Rel_of at 1 in |- *.
red in |- *.
elim D; simpl in |- *.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
apply H'2 with (y:= y); tauto.
red in |- *; intro H'6.
elim H'4; intros H'7 H'8; apply H'8; clear H'4.
apply H'3; auto.
rewrite H'6; tauto.
Qed.
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.
unfold Strict_Rel_of at 1 in |- *.
red in |- *.
elim D; simpl in |- *.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
apply H'2 with (y:= y); tauto.
red in |- *; intro H'6.
elim H'5; intros H'7 H'8; apply H'8; clear H'5.
apply H'3; auto.
rewrite <- H'6; auto.
Qed.
Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).
red in |- *.
intros x y z H' H'0.
apply Strict_Rel_Transitive_with_Rel with (y:= y);
[ intuition | unfold Strict_Rel_of in H', H'0; intuition ].
Qed.
End Partial_order_facts.
Index
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