Local Open Scope pred.
Hint Rewrite @sepcon_emp @emp_sepcon @TT_and @andp_TT
@exp_sepcon1 @exp_sepcon2
@exp_andp1 @exp_andp2
@sepcon_andp_prop @sepcon_andp_prop'
: normalize.
Definition pure {A}{JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{agA: ageable A}{AgeA: Age_alg A}
(P: pred A) : Prop :=
P |-- emp.
Hint Resolve @pure_e : core.
Hint Resolve @pure_emp : core.
Hint Resolve @pure_existential : core.
Hint Rewrite @FF_sepcon @sepcon_FF : normalize.
Hint Rewrite @prop_true_andp using (solve [auto]) : normalize.
Hint Rewrite @true_eq using (solve [auto]) : normalize.
Hint Resolve @pure_con' : core.
Hint Resolve @pure_intersection1 @pure_intersection2 : core.
Hint Rewrite @FF_andp @andp_FF : normalize.
Hint Rewrite @andp_dup : normalize.

Lemma andp_emp_sepcon_TT {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{agA: ageable A}{AgeA: Age_alg A}:
forall (Q: pred A),
(forall w1 w2, core w1 = core w2 -> Q w1 -> Q w2) ->
(Q && emp * TT = Q).