Module KSa := KnotFullSa(TyFSA)(K)(KL).
Definition rmap := K.knot.
Instance Join_rmap: Join rmap := KSa.Join_knot.
Instance Perm_rmap : Perm_alg rmap:= KSa.Perm_knot.
Instance Sep_rmap : Sep_alg rmap:= KSa.Sep_knot.
Instance ag_rmap : ageable rmap := K.ageable_knot.
Instance Age_rmap: Age_alg rmap := KSa.asa_knot.
Inductive preds : Type :=
SomeP : forall A : TypeTree,
(forall ts: list Type, dependent_type_functor_rec ts A (pred rmap)) -> preds.
Definition NoneP := SomeP (ConstType unit) (fun _ => tt).
Inductive resource : Type :=
| NO: forall sh: Share.t, ~ readable_share sh -> resource
| YES: forall sh: Share.t, readable_share sh -> kind -> preds -> resource
| PURE: kind -> preds -> resource.
Definition resource2res (r: resource): res (pred rmap) :=
match r with
| NO sh nsh => NO' (pred rmap) sh nsh
| YES sh rsh k (SomeP A l) => YES' (pred rmap) sh rsh k (existT _ A l)
| PURE k (SomeP A l) => PURE' (pred rmap) k (existT _ A l)
end.
Definition res2resource (r: res (pred rmap)) : resource :=
match r with
| NO' sh nsh => NO sh nsh
| YES' sh rsh k (existT A l) => YES sh rsh k (SomeP A l)
| PURE' k (existT A l) => PURE k (SomeP A l)
end.
Definition ghost : Type := list (option ({g: Ghost & {a: @G g | ghost.valid a}} * preds)%type).
Definition pred2p (p: preds) : fpreds (pred rmap) :=
match p with SomeP A P => existT _ A P end.
Definition p2pred (p: fpreds (pred rmap)) : preds :=
match p with existT A P => SomeP A P end.
Definition ghost2g (r: ghost): SM.ghost (pred rmap) :=
map (option_map (fun '(a, b) => (a, pred2p b))) r.
Definition g2ghost (r: SM.ghost (pred rmap)) : ghost :=
map (option_map (fun '(a, b) => (a, p2pred b))) r.
Inductive res_join : resource -> resource -> resource -> Prop :=
| res_join_NO1 : forall sh1 nsh1 sh2 nsh2 sh3 nsh3
(RJ: join sh1 sh2 sh3),
res_join (NO sh1 nsh1) (NO sh2 nsh2) (NO sh3 nsh3)
| res_join_NO2 : forall sh1 rsh1 sh2 nsh2 sh3 rsh3 k p
(RJ: join sh1 sh2 sh3),
res_join (YES sh1 rsh1 k p) (NO sh2 nsh2) (YES sh3 rsh3 k p)
| res_join_NO3 : forall sh1 nsh1 sh2 rsh2 sh3 rsh3 k p
(RJ: join sh1 sh2 sh3),
res_join (NO sh1 nsh1) (YES sh2 rsh2 k p) (YES sh3 rsh3 k p)
| res_join_YES : forall sh1 rsh1 sh2 rsh2 sh3 rsh3 k p
(RJ: join sh1 sh2 sh3),
res_join (YES sh1 rsh1 k p) (YES sh2 rsh2 k p) (YES sh3 rsh3 k p)
| res_join_PURE : forall k p, res_join (PURE k p) (PURE k p) (PURE k p).
Instance Join_resource: Join resource := res_join.
Instance Perm_resource: Perm_alg resource.
Instance Sep_resource: Sep_alg resource.

Lemma ghost_join_inv: forall m1 m2 m3, ghost_join m1 m2 m3 ->
match m1, m2 with
| nil, _ => m3 = m2
| _, nil => m3 = m1
| a1 :: m1, a2 :: m2 => match m3 with nil => False
| a3 :: m3 => join a1 a2 a3 /\ ghost_join m1 m2 m3 end
end.