Definition fashionable  (P: pred nat) := # P = P.
Notation "P '>=>' Q" := (# (P --> Q)) (at level 55, right associativity) : pred.
Notation "P '<=>' Q" := (# (P <--> Q)) (at level 57, no associativity) : pred.

Lemma subp_trans {A} `{ageable A} : forall G P Q R,
G |-- P >=> Q ->
G |-- Q >=> R ->
G |-- P >=> R.