load "nat.quox";

namespace list {

def0 Vec : 0.ℕ → 0.★ → ★ =
  λ n A ⇒
  caseω n return ★ of {
    zero           ⇒ {nil};
    succ _, 0.Tail ⇒ A × Tail
  };

def0 List : 0.★ → ★ =
  λ A ⇒ (len : ℕ) × Vec len A;

def nil : 0.(A : ★) → List A =
  λ A ⇒ (0, 'nil);

def cons : 0.(A : ★) → 1.A → 1.(List A) → List A =
  λ A x xs ⇒ case1 xs return List A of { (len, elems) ⇒ (succ len, x, elems) };

def foldr' : 0.(A B : ★) →
             1.B → ω.(1.A → 1.B → B) → 1.(n : ℕ) → 1.(Vec n A) → B =
  λ A B z c n ⇒
  case1 n return n' ⇒ 1.(Vec n' A) → B of {
    zero ⇒
      λ nil ⇒ case1 nil return B of { 'nil ⇒ z };
    succ n, 1.ih ⇒
      λ cons ⇒ case1 cons return B of { (first, rest) ⇒ c first (ih rest) }
  };

def foldr : 0.(A B : ★) → 1.B → ω.(1.A → 1.B → B) → 1.(List A) → B =
  λ A B z c xs ⇒
  case1 xs return B of { (len, elems) ⇒ foldr' A B z c len elems };

def sum : 1.(List ℕ) → ℕ = foldr ℕ ℕ 0 nat.plus;

def numbers : List ℕ = (5, (0, 1, 2, 3, 4, 'nil));

def number-sum : sum numbers ≡ 10 : ℕ = δ _ ⇒ 10;

}