(*Join juice from all locks*)
Fixpoint join_list' (ls: seq.seq (option res)) (r:option res):=
if ls is phi::ls' then exists (r':option res),
@join _ (@Join_lower res _) phi r' r /\ join_list' ls' r' else r=None.
Definition join_locks tp r:= join_list' (map snd (AMap.elements (lset tp))) r.
(*Join all the juices*)
Inductive join_all: thread_pool -> res -> Prop:=
AllJuice tp r0 r1 r:
join_threads tp r0 ->
join_locks tp r1 ->
join (Some r0) r1 (Some r) ->
join_all tp r.
Definition juicyLocks_in_lockSet (lset : lockMap) (juice: rmap):=
forall loc,
(forall i, 0 <= i < LKSIZE -> exists sh psh P, juice @ (fst loc, snd loc + i) = YES sh psh (LK LKSIZE i) P)  ->
AMap.find loc lset.
(* I removed the NO case for two reasons: * - To ensure that lset is "valid" (lr_valid), it needs inherit it from the rmap * - there was no real reason to have a NO other than speculation of the future. *)
Definition lockSet_in_juicyLocks (lset : lockMap) (juice: rmap):=
forall loc, AMap.find loc lset ->
(exists sh,
forall i, 0 <= i < LKSIZE -> exists sh' psh' P, join_sub sh sh' /\ juice @ (fst loc, snd loc + i) = YES sh' psh' (LK LKSIZE i) P).
Definition lockSet_in_juicyLocks' (lset : lockMap) (juice: rmap):=
forall loc, AMap.find loc lset ->
Mem.perm_order'' (Some Nonempty) (perm_of_res (juice @ loc)).

Lemma lockSet_in_juic_weak: forall lset juice,
lockSet_in_juicyLocks lset juice -> lockSet_in_juicyLocks' lset juice.