(* Given a separation algebra on type [A], we can make a separation logic with predicates of type  [pred A].  But the type   [B -> pred A] is also a natural separation logic.  These typeclasses automagically extend the separation-logic operators to the pointwise-function case. At present, this is just the beginnings of an experiment with this idea; don't expect it to work well enough to be useful. Andrew Appel, August 2012, following an approach suggested by Rob Dockins *)
Class StarOp A := {  starOp : A -> A -> A }.
Instance baseStarOp {A}{agA: ageable A}{JA: Join A}{PA: Perm_alg A}{AgeA: Age_alg A}
: StarOp (pred A) := {| starOp := sepcon |}.
Instance funStarOp (B: Type)(A: Type)(StarA: StarOp A) : StarOp (B -> A) :=
{| starOp := fun (P Q : B -> A) (b : B) =>  starOp (P b) (Q b) |}.
Notation "P '*' Q" := (starOp P Q) : pred.
(* Opaque baseStarOp *)
Class DerivesOp A := {  derivesOp : A -> A -> Prop }.
Instance baseDerivesOp {A}{agA: ageable A}
: DerivesOp (pred A) := {| derivesOp := @derives A agA|}.
Instance funDerivesOp (B: Type)(A: Type)(DerivesA: DerivesOp A) : DerivesOp (B -> A)
:= {| derivesOp := fun (P Q : B -> A)  => forall b, derivesOp (P b) (Q b) |}.
Notation "P '|--' Q" := (derivesOp P%pred Q%pred).
(* Opaque baseDerivesOp. *)
Class WandOp A := {  wandOp : A -> A -> A }.
Instance baseWandOp {A}{agA: ageable A}{JA: Join A}{PA: Perm_alg A}{AgeA: Age_alg A}
: WandOp (pred A) := {| wandOp := wand |}.
Instance funWandOp (B: Type)(A: Type)(WandA: WandOp A) : WandOp (B -> A) :=
{| wandOp := fun (P Q : B -> A) (b : B) =>  wandOp (P b) (Q b) |}.
Notation "P '-*' Q" := (wandOp P Q) : pred.
(* Opaque baseWandOp *)
Class EmpOp A := { Emp: A}.
Instance baseEmpOp {A}{agA: ageable A}{JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{AgeA: Age_alg A}
: EmpOp (pred A) := {| Emp := @emp A JA PA SA agA AgeA |}.
Instance funEmpOp  (B: Type)(A: Type)(EmpA: EmpOp A) : EmpOp (B -> A) :=
{| Emp := fun (b : B) =>  Emp  |}.
Section Test.
Variable environ: Type.
Variables (rmap : Type)
(Join_rmap: Join rmap) (Perm_rmap: @Perm_alg rmap Join_rmap)
(Sep_rmap: @Sep_alg rmap Join_rmap)
(Canc_rmap: @Canc_alg rmap Join_rmap)
(Disj_rmap: @Disj_alg rmap Join_rmap)
(ag_rmap: ageable rmap)
(Age_rmap: @Age_alg rmap Join_rmap ag_rmap).
Existing Instance  Join_rmap.
Existing Instance  Perm_rmap.
Existing Instance  Sep_rmap.
Existing Instance  Canc_rmap.
Existing Instance  Disj_rmap.
Existing Instance  ag_rmap.
Existing Instance  Age_rmap.

Lemma test1: forall (P : environ -> pred rmap) (Q: pred rmap),
P * Emp |-- P.