Definition exactly {A} (x: A) : pred A := fun w => w=x.

(* Specifications: *)
Lemma join_exactly {A} {JA: Join A}{PA: Perm_alg A}:
forall w1 w2 w3, join w1 w2 w3 -> exactly w1 * exactly w2 = exactly w3.
Lemma exists_and1 {A: Type} : forall {T: Type} (P: T -> pred A) (Q: pred A),
exp P && Q = EX x:T, P x && Q.
Lemma andp_comm {A: Type}: forall (P Q: pred A), P && Q = Q && P.
Lemma andp_assoc {A}: forall (P Q R: pred A),
((P && Q) && R = P && (Q && R)).
Lemma True_andp_eq {A}:
forall (P: Prop) (Q: pred A), P -> (!!P && Q)%pred = Q.