module Simple;
  inductive T {
    tt : T;
  };

  someT : T;
  someT ≔ tt;

  inductive Bool  {
    false : Bool;
    true : Bool;
  };


  inductive Nat  {
     zero : Nat;
     suc : Nat → Nat;
  };

  infix 3 ==;
  == : Nat → Nat → Bool;
  == zero zero ≔ true;
  == (suc a) (suc b) ≔ a == b;
  == _ _ ≔ false;

  infixl 4 +;
  + : Nat → Nat → Nat;
  + zero b ≔ b;
  + (suc a) b ≔ suc (a + b);

  infixr 5 ∷;
  inductive List {
    nil : List;
    ∷ : Nat → List → List;
   };

  foldr : (Nat → Nat → Nat) → Nat → List → Nat;
  foldr _ v nil ≔ v;
  foldr f v (a ∷ as) ≔ f a (foldr f v as);

  sum : List → Nat;
  sum ≔ foldr (+) zero;

end;