Section LIST2.
Context {cs: compspecs}.
Context  {list_structid: ident} {list_link: ident} {list_token: share -> val -> mpred}.
Fixpoint lseg (ls: listspec list_structid list_link list_token) (dsh psh: share)
(contents: list (val * elemtype ls)) (x z: val) : mpred :=
match contents with
| (p,h)::hs => !! (p=x /\ ~ptr_eq x z) &&
EX y:val,  !! is_pointer_or_null y &&
list_token dsh x * list_cell ls dsh h x
* field_at psh list_struct (StructField list_link ::nil)
(valinject (nested_field_type list_struct (StructField list_link :: nil)) y) x
* lseg ls dsh psh hs y z
| nil => !! (ptr_eq x z) && emp
end.
Definition lseg_cons (ls: listspec list_structid list_link list_token) dsh psh (l: list (val * elemtype ls)) (x z: val) : mpred :=
!! (~ ptr_eq x z) &&
EX h:(elemtype ls), EX r:list (val * elemtype ls), EX y:val,
!!(l=(x,h)::r  /\ is_pointer_or_null y) &&
list_token dsh x * list_cell ls dsh h x *
field_at psh list_struct (StructField list_link :: nil) (valinject (nested_field_type list_struct (StructField list_link :: nil)) y) x *
lseg ls dsh psh r y z.
intros. rewrite lseg_unroll.
apply pred_ext. apply orp_left; auto.
rewrite andp_assoc.
do 2 (apply derives_extract_prop; intro).
congruence.
apply orp_right2. auto.
Qed.
Definition lseg_cons_right (ls: listspec list_structid list_link list_token)
dsh psh (l: list (val * elemtype ls)) (x z: val) : mpred :=
!! (~ ptr_eq x z) &&
EX h:(elemtype ls), EX r:list (val * elemtype ls), EX y:val,
!!(l=r++(y,h)::nil /\ is_pointer_or_null y)  &&
list_token dsh y * list_cell ls dsh h y *
field_at psh list_struct (StructField list_link :: nil) (valinject (nested_field_type list_struct (StructField list_link :: nil)) z) y *
lseg ls dsh psh r x y.

Lemma lseg_cons_right_null (ls: listspec list_structid list_link list_token): forall dsh psh l x h y,
list_token dsh y * list_cell ls dsh h y * field_at psh list_struct (StructField list_link :: nil) (valinject (nested_field_type list_struct (StructField list_link :: nil)) nullval) y *
lseg ls dsh psh l x y
|--   lseg ls dsh psh (l++(y,h)::nil) x nullval.