inductive nat : Type := zero : nat, succ : nat → nat namespace nat end nat open nat inductive list (A : Type) : Type := nil {} : list A, cons : A → list A → list A definition nil := @list.nil definition cons := @list.cons check nil check nil.{1} check @nil.{1} nat check @nil nat check cons nat.zero nil inductive vector (A : Type) : nat → Type := vnil {} : vector A zero, vcons : forall {n : nat}, A → vector A n → vector A (succ n) namespace vector end vector open vector check vcons zero vnil constant n : nat check vcons n vnil check vector.rec definition vector_to_list {A : Type} {n : nat} (v : vector A n) : list A := vector.rec (@nil A) (fun (n : nat) (a : A) (v : vector A n) (l : list A), cons a l) v coercion vector_to_list constant f : forall {A : Type}, list A → nat check f (cons zero nil) check f (vcons zero vnil)