Ltac find_in_list A L :=
 match L with
  | A :: _ => constr:(O)
  | _ :: ?Y => let n := find_in_list A Y in constr:(S n)
  | nil => fail
  end.Ltac length_of R :=
 match R with
  |  nil => constr:(O)
  |  _:: ?R1 => let n := length_of R1 in constr:(S n)
 end.Fixpoint insert {A} (n: nat) (x: A) (ys: list A) {struct n} : list A :=
 match n with
 | O => x::ys
 | S n' => match ys with nil => x::ys | y::ys' => y::insert n' x ys' end
end.(* Note: in the grab_indexes function,
  it's important that the {struct} induction NOT be on xs, because
  that list might not be concrete all the way to the end, where the ns list will be concrete.
  Thus we do it this particular way.  *)
Fixpoint  grab_indexes' {A} (ns: list (option nat)) (xs: list A) {struct ns} : list A * list A :=
match ns, xs with
| nil, xs => (nil, xs)
| _, nil => (nil,nil)
| Some n::ns', x::xs' => let (al,bl) := grab_indexes' ns' xs'
                               in (insert n x al, bl)
| None :: ns', x::xs' => let (al,bl) := grab_indexes' ns' xs'
                                  in (al, x::bl)
end.Fixpoint grab_calc' (k: Z) (z: nat) (ns: list (option nat)): list (option nat) :=
match z, ns with
| O, _::ns' => Some (nat_of_Z k) :: ns'
| S z', None::ns' => None :: grab_calc' k z' ns'
| S z', Some n :: ns => Some n :: grab_calc' (k-1) z' ns
| O, nil => Some O :: nil
| S z', nil => None :: grab_calc' k z' nil
end.Fixpoint grab_calc (k: Z) (zs: list Z) (ns: list (option nat)) : list (option nat) :=
match zs with
| nil => ns
| z::zs' => grab_calc (k+1) zs' (grab_calc' k (nat_of_Z z) ns)
end.(* Eval compute in grab_calc 0 (3::1::5::nil) nil. *)

(* Define app_alt, just like app, so we have better control
  over which things get unfolded *)

Definition app_alt {A: Type} :=
fix app (l m : list A) : list A :=
  match l with
  | nil => m
  | a :: l1 => a :: app l1 m
  end.Definition grab_indexes {A} (ns: list Z) (xs: list A) : list A :=
    let (al,bl) := grab_indexes' (grab_calc 0 ns nil) xs in app_alt al bl.

(* Specifications: *)
Lemma fold_right_nil: forall {A B} (f: A -> B -> B) (z: B), fold_right f z nil = z.
Lemma fold_right_cons: forall {A B} (f: A -> B -> B) (z: B) x y, fold_right f z (x::y) = f x (fold_right f z y).
Lemma fold_right_and_app: forall (Q1 Q2: list (environ -> Prop)) rho, fold_right `(and) `(True) (Q1 ++ Q2) rho = (fold_right `(and) `(True) Q1 rho /\ fold_right `(and) `(True) Q2 rho).
Lemma fold_right_sepcon_app : forall P Q, fold_right_sepcon (P++Q) = fold_right_sepcon P * fold_right_sepcon Q.