(** A [PartialOrder] is compatible with its underlying equivalence. *)
#[global]
Instance PartialOrder_proper `(PartialOrder A eqA R) :
Proper (eqA==>eqA==>iff) R.
(** From a [PartialOrder] to the corresponding [StrictOrder]: [lt = le /\ ~eq]. If the order is total, we could also say [gt = ~le]. *)

Lemma PartialOrder_StrictOrder `(PartialOrder A eqA R) :
StrictOrder (relation_conjunction R (complement eqA)).