Definition dec_share_nonidentity (sh: Share.t) : {~identity sh}+{identity sh} :=
   (Sumbool.sumbool_not _ _ (dec_share_identity sh)).Definition perm_of_sh (sh: Share.t): option permission :=
  if writable0_share_dec sh
  then if eq_dec sh Share.top
            then Some Freeable
            else Some Writable
    else if readable_share_dec sh
         then Some Readable
         else if eq_dec sh Share.bot
                   then None
              else Some Nonempty.Definition contents_at (m: mem) (loc: address) : memval :=
  ZMap.get (snd loc) (PMap.get (fst loc) (mem_contents m)).Definition valshare (r: resource) : share :=
    match r with
      | YES sh rsh _ _ => Share.glb Share.Rsh sh
      | _ => Share.bot
    end.Definition res_retain' (r: resource) : Share.t :=
 match r with
  | NO sh _ => sh
  | YES sh _ _ _ => Share.glb Share.Lsh sh
  | PURE _ _ => Share.top
 end.Definition perm_of_res (r: resource) :=
  (*  perm_of_sh (res_retain' r) (valshare r). *)
 match r with
 | NO sh _ => if eq_dec sh Share.bot then None else Some Nonempty
 | PURE _ _ => Some Nonempty
 | YES sh rsh (VAL _) _ => perm_of_sh sh
 | YES sh rsh _ _ => Some Nonempty
 end.(*To do a case analysis over perm_of_res, use:
functional induction (perm_of_res_explicit r1) using perm_of_res_expl_ind 
We define the induction shceme bellow. *)
Definition perm_of_res_lock_explicit
             (r : compcert_rmaps.RML.R.resource):=
    match r with
    | compcert_rmaps.RML.R.NO _ _ => None
    | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.VAL _) _ => None
    | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.LK _ _) _ =>
      if writable0_share_dec (Share.glb Share.Rsh sh)
      then if eq_dec (Share.glb Share.Rsh sh) Share.top then Some Freeable else Some Writable
      else if readable_share_dec (Share.glb Share.Rsh sh) then Some Readable else
             if eq_dec  (Share.glb Share.Rsh sh) Share.bot then None else Some Nonempty
    | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.FUN _ _) _ => None
    | compcert_rmaps.RML.R.PURE _ _ => None
    end.Definition perm_of_res' (r: resource) :=
  (*  perm_of_sh (res_retain' r) (valshare r). *)
 match r with
 | NO sh _ => if eq_dec sh Share.bot then None else Some Nonempty
 | PURE _ _ => Some Nonempty
 | YES sh _ _ _ => perm_of_sh sh
 end.Definition perm_of_res_lock (r: resource) := 
  (*  perm_of_sh (res_retain' r) (valshare r). *)
 match r with
 | YES sh rsh (LK _ _) _ => perm_of_sh (Share.glb Share.Rsh sh)
 | _ => None 
 end.(*To do a case analysis over perm_of_res_lock, use:
functional induction (perm_of_res_lock_explicit r1) using perm_of_res_lock_expl_ind 
We define the induction shceme bellow. *)
Definition perm_of_res_explicit
               (r : compcert_rmaps.RML.R.resource):=
        match r with
        | compcert_rmaps.RML.R.NO sh _ => if eq_dec sh Share.bot then None else Some Nonempty
           | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.VAL _) _ =>
             if writable0_share_dec sh
             then if eq_dec sh Share.top then Some Freeable else Some Writable
             else
               if readable_share_dec sh
               then Some Readable
               else if eq_dec sh Share.bot then None else Some Nonempty
           | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.LK _ _) _ => Some Nonempty
           | compcert_rmaps.RML.R.YES sh _ (compcert_rmaps.FUN _ _) _ => Some Nonempty
           | compcert_rmaps.RML.R.PURE _ _ => Some Nonempty
        end.

Lemma perm_of_res_op2: forall r, perm_order'' (perm_of_res' r) (perm_of_res_lock r).