------------------------------------------------------------
-- Core ITT (minus W)

data Empty : * => { }

absurd [A : *] [x : Empty] : A => (
  Empty-Elim x (\_ => A)
)

neg [A : *] : * => (A -> Empty)

record Unit : * => tt { }

record Prod : [A : *] [B : A -> *] -> * =>
  prod {fst : A, snd : B fst}

data Bool : * => { true : Bool | false : Bool }

-- The if_then_else_ is provided by Bool-Elim

------------------------------------------------------------
-- Examples →

data Nat : * => { zero : Nat | suc : Nat -> Nat }

gt [n : Nat] : Nat -> * => (
  Nat-Elim
    n
    (\_ => Nat -> *)
    (\_ => Empty)
    (\n f m => Nat-Elim m (\_ => *) Unit (\m' _ => f m'))
)

data List : [A : *] -> * =>
  { nil : List A | cons : A -> List A -> List A }

length [A : *] [xs : List A] : Nat => (
    List-Elim xs (\_ => Nat) zero (\_ _ n => suc n)
)

head [A : *] [xs : List A] : gt (length A xs) zero -> A => (
    List-Elim
      xs
      (\xs => gt (length A xs) zero -> A)
      (\p => absurd A p)
      (\x _ _ _ => x)
)

------------------------------------------------------------
-- Examples ×