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.

Lemma incr_incr'
{worlds: Type}
{R: Relation worlds}
{po_R: PreOrder Krelation}
{J: Join worlds}:
forall m, increasing' m -> increasing m.