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.