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_subp {A}{agA: ageable A}:
forall (G: pred nat) (P Q R S: pred A),
G |-- (R >=> P) ->
G |-- (Q >=> S) ->
G |-- (P >=> Q) >=> (R >=> S).