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.
Lemma TT_i {A} : forall w: A, TT w.
Lemma TT_and {A}: forall (Q: pred A), TT && Q = Q.
Lemma andp_TT {A}: forall (P: pred A), P && TT = P.
Lemma TT_sepcon_TT {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: TT * TT = TT.