Global Set Automatic Coercions Import.
Set Implicit Arguments.
(* Intended for use without Import. *)
Require Vector.
Require List.
Arguments Vector.cons [A] _ [n].
Arguments Vector.nil [A].
(* eliminators/inversion *)
Definition head A n (v: Vector.t A (S n)): A :=
match v with
| Vector.nil => I | Vector.cons h _ _ => h
end.
Definition tail n A (v: Vector.t A (S n)): Vector.t A n :=
match v with
| Vector.nil => I | Vector.cons _ _ t => t
end.
(* conversion to/from list *)
Fixpoint to_list X (n: nat) (v: Vector.t X n) {struct v}: List.list X :=
match v with
| Vector.nil => List.nil | Vector.cons x _ v' => List.cons x (to_list v')
end.
Fixpoint from_list A (l: List.list A): Vector.t A (List.length l) :=
match l return Vector.t A (List.length l) with
| List.nil => Vector.nil | List.cons h t => Vector.cons h (from_list t)
end.
Coercion to_list: Vector.t >-> List.list.
Coercion from_list: List.list >-> Vector.t.
Proof with try reflexivity. induction l... simpl. rewrite IHl... Qed.
Proof with auto.
revert f.
induction v...
intros.
simpl.
apply (IHv (fun (m: nat) (w: Vector.t X m) => f (S m) (Vector.cons h w))).
Qed.
Proof with auto.
induction n; intros.
rewrite (eq_nil x), (eq_nil y)...
rewrite (eq_cons x), (eq_cons y) in *...
inversion H.
rewrite (IHn (tail x) (tail y))...
Qed.
Proof with auto.
induction l; simpl; intros.
rewrite (eq_nil v)...
rewrite <- H...
simpl.
rewrite list_round_trip...
Qed.
(* simultaneous induction over two vectors *)

Lemma double_rect A B (P: forall n, Vector.t A n -> Vector.t B n -> Prop):
P 0 Vector.nil Vector.nil ->
(forall n (v: Vector.t A n) (w: Vector.t B n) (x: A) (y: B), P n v w -> P (S n) (Vector.cons x v) (Vector.cons y w)) ->
forall n (v: Vector.t A n) (w: Vector.t B n), P n v w.