Hint Resolve @fash_subp.

(* Specifications: *)
Lemma fash_allp {A} {agA:ageable A}:
forall  (B: Type) (F: B -> pred A),
# (allp F) = allp (fun z: B => # F z).
Lemma subp_i1 {A}{agA: ageable A}:
forall (P : pred nat) (Q R: pred A ), !P && Q |-- R -> P |-- Q >=> R.
Lemma eqp_nat: forall P Q: pred nat, (P <=> Q) = (P <--> Q).
Lemma prop_andp_subp {A}{agA : ageable A}:
forall (P: Prop) Q R w, (P -> app_pred (Q >=> R) w) -> app_pred ((!!P && Q) >=> R) w.
Lemma subp_e {A}{agA : ageable A}: forall P Q : pred A, TT |-- P >=> Q -> P |-- Q.