Definition mallocN_spec :=
DECLARE _mallocN
WITH n: Z
PRE [ 1%positive OF tint]
PROP (4 <= n <= Int.max_unsigned)
LOCAL (temp 1%positive (Vint (Int.repr n)))
SEP ()
POST [ tptr tvoid ]
EX v: val,
PROP (malloc_compatible n v)
LOCAL (temp ret_temp v)
SEP (memory_block Tsh n v).
Definition freeN_spec :=
DECLARE _freeN
WITH p : val , n : Z
PRE [ 1%positive OF tptr tvoid , 2%positive OF tint]
(* we should also require natural_align_compatible (eval_id 1) *)
PROP() LOCAL (temp 1%positive p; temp 2%positive (Vint (Int.repr n)))
SEP (memory_block Tsh n p)
POST [ tvoid ]
PROP () LOCAL () SEP ().
Definition insert_spec :=
DECLARE _insert
WITH p0: val, x: Z, v: val, t0: tree val
PRE  [ _p OF (tptr (tptr t_struct_tree)), _x OF tint,
_value OF (tptr Tvoid)   ]
PROP( Int.min_signed <= x <= Int.max_signed; is_pointer_or_null v)
LOCAL(temp _p p0; temp _x (Vint (Int.repr x)); temp _value v)
SEP (treebox_rep t0 p0)
POST [ Tvoid ]
PROP()
LOCAL()
SEP (treebox_rep (insert x v t0) p0).
Definition lookup_spec :=
DECLARE _lookup
WITH b: val, x: Z, v: val, t: tree val
PRE  [ _t OF (tptr (tptr t_struct_tree)), _x OF tint  ]
PROP( Int.min_signed <= x <= Int.max_signed)
LOCAL(temp _t b; temp _x (Vint (Int.repr x)))
SEP (treebox_rep t b)
POST [ tptr Tvoid ]
PROP()
LOCAL(temp ret_temp (lookup nullval x t))
SEP (treebox_rep t b).
(* Definition turn_left_spec := DECLARE _turn_left WITH ta: tree val, x: Z, vx: val, tb: tree val, y: Z, vy: val, tc: tree val, b: val, l: val, pa: val, r: val PRE  [ __l OF (tptr (tptr (Tstruct _tree noattr))), _l OF (tptr (Tstruct _tree noattr)), _r OF (tptr (Tstruct _tree noattr))] PROP(Int.min_signed <= x <= Int.max_signed; is_pointer_or_null vx) LOCAL(temp __l b; temp _l l; temp _r r) SEP (data_at Tsh (tptr t_struct_tree) l b; data_at Tsh t_struct_tree (Vint (Int.repr x), (vx, (pa, r))) l; tree_rep ta pa; tree_rep (T tb y vy tc) r) POST [ Tvoid ] EX pc: val, PROP(Int.min_signed <= y <= Int.max_signed; is_pointer_or_null vy) LOCAL() SEP (data_at Tsh (tptr t_struct_tree) r b; data_at Tsh t_struct_tree (Vint (Int.repr y), (vy, (l, pc))) r; tree_rep (T ta x vx tb) l; tree_rep tc pc).  Definition pushdown_left_spec := DECLARE _pushdown_left WITH ta: tree val, x: Z, v: val, tb: tree val, b: val, p: val PRE  [ _t OF (tptr (tptr (Tstruct _tree noattr)))] PROP(Int.min_signed <= x <= Int.max_signed; tc_val (tptr Tvoid) v) LOCAL(temp _t b) SEP (data_at Tsh (tptr t_struct_tree) p b; field_at Tsh t_struct_tree [StructField _key] (Vint (Int.repr x)) p; field_at Tsh t_struct_tree [StructField _value] v p; treebox_rep ta (field_address t_struct_tree [StructField _left] p); treebox_rep tb (field_address t_struct_tree [StructField _right] p)) POST [ Tvoid ] PROP() LOCAL() SEP (treebox_rep (pushdown_left ta tb) b).  Definition delete_spec := DECLARE _delete WITH b: val, x: Z, t: tree val PRE  [ _t OF (tptr (tptr t_struct_tree)), _x OF tint] PROP( Int.min_signed <= x <= Int.max_signed) LOCAL(temp _t b; temp _x (Vint (Int.repr x))) SEP (treebox_rep t b) POST [ Tvoid ] PROP() LOCAL() SEP (treebox_rep (delete x t) b). *)
Definition Gprog : funspecs :=
ltac:(with_library prog [
mallocN_spec; freeN_spec;
insert_spec; lookup_spec(*;
turn_left_spec; pushdown_left_spec; delete_spec*)
]).

Lemma tree_rep_saturate_local:
forall t p, tree_rep t p |-- !! is_pointer_or_null p.