Require Export VST.msl.iter_sepcon.
Require Export VST.concurrency.semax_conc_pred.
Require Export VST.concurrency.semax_conc.
Require Export VST.floyd.proofauto.
Require Export VST.floyd.sublist.
(* general list lemmas *)
Notation vint z := (Vint (Int.repr z)).
Notation vptrofs z := (Vptrofs (Ptrofs.repr z)).
Hint Resolve repable_0.
Definition complete MAX l := l ++ repeat (vptrofs 0) (Z.to_nat MAX - length l).
Definition remove_at {A} i (l : list A) := firstn i l ++ skipn (S i) l.
Open Scope logic.
Definition rotate {A} (l : list A) n m := sublist (m - n) (Zlength l) l ++
sublist 0 (m - n) l.
(* Lemma Znth_indep : forall {A} i (l : list A) d d', 0 <= i < Zlength l -> Znth i l d = Znth i l d'. *)
Fixpoint upto n :=
match n with
| O => []
| S n' => 0 :: map Z.succ (upto n')
end.
Opaque Z.of_nat.
(* Lemma Znth_map' : forall {A B} i (f : A -> B) al b, Znth i (map f al) (f b) = f (Znth i al b). *)
Transparent Z.of_nat.
Hint Resolve incl_nil.
Definition nil_dec {A} (l : list A) : {l = []} + {l <> []}.
Arguments eq_dec _ _ _ _ : simpl never.
Fixpoint extend {A} (l : list A) ls :=
match l, ls with
| x :: xs, y :: ys => (x :: y) :: extend xs ys
| _, _ => ls
end.
Fixpoint extendr {A} (l : list A) ls :=
match l, ls with
| x :: xs, y :: ys => (y ++ [x]) :: extendr xs ys
| _, _ => ls
end.
(* We could also consider an alpha-renaming axiom, although this may be unnecessary. *)
Hint Resolve lock_inv_isptr cond_var_isptr : saturate_local.
Hint Resolve unreadable_bot.
Definition join_Bot := join_Bot.
(* It's often useful to split Tsh in half. *)
Definition gsh1 := fst (slice.cleave Tsh).
Definition gsh2 := snd (slice.cleave Tsh).
Hint Resolve readable_gsh1 readable_gsh2 gsh1_gsh2_join.
Hint Resolve gsh1_not_bot gsh2_not_bot.
(* Lemma data_at_Tsh_conflict : forall {cs : compspecs} sh t v v' p, sepalg.nonidentity sh -> 0 < sizeof t -> data_at Tsh t v p * data_at sh t v' p |-- FF. *)
Definition remove_Znth {A} i (al : list A) := sublist 0 i al ++ sublist (i + 1) (Zlength al) al.

Lemma iter_sepcon_Znth: forall {A} {d : Inhabitant A} f (l : list A) i, 0 <= i < Zlength l ->
iter_sepcon f l = f (Znth i l) * iter_sepcon f (remove_Znth i l).