load "nat.quox";

namespace vec {

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

def elim : 0.(A : ★) → 0.(P : (n : ℕ) → Vec n A → ★) →
           P 0 'nil →
           ω.((x : A) → 0.(n : ℕ) → 0.(xs : Vec n A) →
                P n xs → P (succ n) (x, xs)) →
           (n : ℕ) → (xs : Vec n A) → P n xs =
  λ A P pn pc n ⇒
    case n return n' ⇒ (xs' : Vec n' A) → P n' xs' of {
      zero ⇒ λ n ⇒
        case n return n' ⇒ P 0 n' of { 'nil ⇒ pn };
      succ n, ih ⇒ λ c ⇒
        case c return c' ⇒ P (succ n) c' of {
          (first, rest) ⇒ pc first n rest (ih rest)
        }
    };

#[compile-scheme "(lambda% (n xs) xs)"]
def up : 0.(A : ★) → (n : ℕ) → Vec n A → Vec¹ n A =
  λ A n ⇒
    case n return n' ⇒ Vec n' A → Vec¹ n' A of {
      zero ⇒ λ xs ⇒
        case xs return Vec¹ 0 A of { 'nil ⇒ 'nil };
      succ n', f' ⇒ λ xs ⇒
        case xs return Vec¹ (succ n') A of {
          (first, rest) ⇒ (first, f' rest)
        }
    }

}

def0 Vec = vec.Vec;


namespace list {

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

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

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

def elim : 0.(A : ★) → 0.(P : List A → ★) →
           P (Nil A) →
           ω.((x : A) → 0.(xs : List A) → P xs → P (Cons A x xs)) →
           (xs : List A) → P xs =
  λ A P pn pc xs ⇒
    case xs return xs' ⇒ P xs' of { (len, elems) ⇒
      vec.elim A (λ n xs ⇒ P (n, xs))
        pn (λ x n xs ih ⇒ pc x (n, xs) ih)
        len elems
    };

-- [fixme] List A <: List¹ A should be automatic, imo
#[compile-scheme "(lambda (xs) xs)"]
def up : 0.(A : ★) → List A → List¹ A =
  λ A xs ⇒
  case xs return List¹ A of { (len, elems) ⇒
    case nat.dup! len return List¹ A of { [p] ⇒
      caseω p return List¹ A of { (lenω, eq0) ⇒
        case eq0 return List¹ A of { [eq] ⇒
          (lenω, vec.up A lenω (coe (𝑖 ⇒ Vec (eq @𝑖) A) @1 @0 elems))
        }
      }
    }
  };

def foldr : 0.(A B : ★) → B → ω.(A → B → B) → List A → B =
  λ A B z f xs ⇒ elim A (λ _ ⇒ B) z (λ x _ y ⇒ f x y) xs;

def map : 0.(A B : ★) → ω.(A → B) → List A → List B =
  λ A B f ⇒ foldr A (List B) (Nil B) (λ x ys ⇒ Cons B (f x) ys);

def0 All : (A : ★) → (P : A → ★) → List A → ★ =
  λ A P xs ⇒ foldr¹ A ★ True (λ x ps ⇒ P x × ps) (up A xs);

}

def0 List = list.List;