End POrderTheory.
Hint Resolve lexx le_refl lt_irreflexive.
Module Import ReversePOrder.
Section ReversePOrder.
Canonical reverse_eqType (T : eqType) := EqType T [eqMixin of T^r].
Canonical reverse_choiceType (T : choiceType) := [choiceType of T^r].
Context {display : unit}.
Local Notation porderType := (porderType display).
Variable T : porderType.
Definition reverse_le (x y : T) := (y <= x).
Definition reverse_lt (x y : T) := (y < x).

(* Specifications: *)
Lemma reverse_lt_neqAle (x y : T) : reverse_lt x y = (x != y) && (reverse_le x y).
Lemma leIx x y z : (y <= x) || (z <= x) -> y `&` z <= x.
Lemma leIr x y : y `&` x <= x.
Fact joinA : associative join.