Local Open Scope kripke_model.
Definition increasing
{worlds: Type}
{R: Relation worlds}
{J: Join worlds}: worlds -> Prop :=
fun m => forall n n', join m n n' -> n <= n'.
Definition increasing'
{worlds: Type}
{R: Relation worlds}
{J: Join worlds}: worlds -> Prop :=
fun m => forall n, m <= n -> increasing n.

(* Specifications: *)
Lemma incr_incr'
{worlds: Type}
{R: Relation worlds}
{po_R: PreOrder Krelation}
{J: Join worlds}:
forall m, increasing' m -> increasing m.
(* * In a separation algebra with discrete order, * an element is increasing iff it is a unit.*)
Lemma disc_incr_unit
{worlds: Type}
{R: Relation worlds}
{po_R: PreOrder Krelation}
{J: Join worlds}:
IdentityKripkeIntuitionisticModel worlds ->
forall e, increasing e <-> unit_element e.