End ErasureCorollaries.
(* Stratified Semantics with Separation -- this is where put everything together. *)
Class StratifiedSemanticsWithSeparation
{W V G S' U F world_erase VU GF}
`{StratifiedSemantics W V G S' U F world_erase VU GF}
{A}
{JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}
(* NOTE: Perhaps we also want   CA: Canc_alg A   *)
(projA : W -> A)
(* Projection function mapping worlds to elements of A *)
(H : pfunc A val -> Prop)
(* Set of primitive expressions on A -- we need these in order to use [assert_expr]-style predicates in our separation logic. *)
(HG : pfunc A val -> pfunc W val -> Prop)
(* Acceptable primitive expression pairs on A and W *)
: Type := {
sss_HG_wellformed : forall h g,
HG h g
-> H h /\ G g;
sss_primexpr_erasure : forall w a g h b1 b2,
HG h g
-> projA w = a
-> exprEval H (EPrim h) a b1
-> exprEval G (EPrim g) w b2
-> b1=b2;
sss_primexpr_safety : forall w a g h b1,
HG h g
-> projA w = a
-> exprEval H (EPrim h) a b1
-> exists b2, exprEval G (EPrim g) w b2
}.
Section UnpackStratifiedSemanticsWithSeparation.
Context
{W V G S' U F world_erase VU GF A projA H HG}
`{SSS : StratifiedSemanticsWithSeparation W V G S' U F world_erase VU GF A projA H HG}.

(* Specifications: *)
Lemma HG_wellformed : forall h g,
HG h g
-> H h /\ G g.
Lemma sep_expr_safety : forall e e' w b1,
expr_erase HG e e'
-> exprEval H e (projA w) b1
-> exists b2, exprEval G e' w b2.