Hint Resolve @boxy_FF.

(* Specifications: *)
Lemma TT_i  {A} `{ageable A}: forall w: A,  app_pred TT w.
Hint Resolve @TT_i.
Lemma prop_andp_left {A}{agA: ageable A}: forall (P: Prop) Q R, (P -> Q |-- R) -> !!P && Q |-- R.
Lemma prop_andp_right {A}{agA: ageable A}: forall (P: Prop) Q R, P -> Q |-- R -> Q |-- !!P && R.
Lemma prop_true_andp:
forall (P: Prop) A `{ageable A} (Q: pred A), P -> (!! P && Q = Q).
Lemma prop_false_andp:
forall (P: Prop) A `{ageable A} (Q: pred A),
~P -> !! P && Q = FF.
Lemma prop_andp_e {A} `{ageable A}:  forall P Q (w:A), (!! P && Q) w -> P /\ Q w.
Lemma prop_andp_i {A} `{ageable A}:  forall P Q (w:A), P /\ app_pred Q w -> (!! P && Q) w.
Lemma later_derives {A} `{agA : ageable A}: forall {P Q}, (P |-- Q) -> (|> P |-- |> Q).
Lemma boxy_allp {A} `{agA : ageable A}:
forall (M: modality) (B: Type) F,
reflexive _ (app_mode M) ->
(forall (x:B), boxy M (F x)) -> boxy M (allp F).