// U60
// ---

U60.to_nat (x: U60) : Nat
U60.to_nat 0 = Nat.zero
U60.to_nat n  = (Nat.succ (U60.to_nat (- n 1)))

// Empty
// -----

Empty : Type

// Unit
// ----

Unit : Type
Unit.new : Unit

// Bool
// ----

type Bool {
  true : Bool
  false : Bool
}

Bool.not (a: Bool) : Bool
Bool.not Bool.true  = Bool.false
Bool.not Bool.false = Bool.true

Bool.and (a: Bool) (b: Bool) : Bool
Bool.and Bool.true  Bool.true  = Bool.true
Bool.and Bool.true  Bool.false = Bool.false
Bool.and Bool.false Bool.true  = Bool.false
Bool.and Bool.false Bool.false = Bool.false

Bool.if <r: Type> (b: Bool) (if_t: r) (if_f: r) : r
Bool.if r Bool.true  if_t if_f = if_t
Bool.if r Bool.false if_t if_f = if_f

Bool.not_not_theorem (a: Bool) : (Equal Bool a (Bool.not (Bool.not a)))
Bool.not_not_theorem Bool.true  = (Equal.refl Bool Bool.true)
Bool.not_not_theorem Bool.false = (Equal.refl Bool Bool.false)

Bool.true_not_false (e: (Equal Bool Bool.true Bool.false)) : Empty
Bool.true_not_false e = (Equal.rewrite e (x => (Bool.if Type x Unit Empty)) Unit.new)

// Nat
// ---

type Nat {
  zero : Nat
  succ (pred: Nat) : Nat
}

Nat.double (x: Nat) : Nat
Nat.double (Nat.succ x) = (Nat.succ (Nat.succ (Nat.double x)))
Nat.double (Nat.zero)   = (Nat.zero)

Nat.add (a: Nat) (b: Nat) : Nat
Nat.add (Nat.succ a) b = (Nat.succ (Nat.add a b))
Nat.add Nat.zero     b = b

Nat.comm.a (a: Nat) : (Equal Nat a (Nat.add a Nat.zero))
Nat.comm.a Nat.zero     = Equal.refl
Nat.comm.a (Nat.succ a) = (Equal.apply (x => (Nat.succ x)) (Nat.comm.a a))

Nat.comm.b (a: Nat) (b: Nat): (Equal Nat (Nat.add a (Nat.succ b)) (Nat.succ (Nat.add a b)))
Nat.comm.b Nat.zero     b = Equal.refl
Nat.comm.b (Nat.succ a) b = (Equal.apply (x => (Nat.succ x)) (Nat.comm.b a b))

Nat.comm (a: Nat) (b: Nat) : (Equal Nat (Nat.add a b) (Nat.add b a))
Nat.comm Nat.zero     b = (Nat.comm.a b)
Nat.comm (Nat.succ a) b =
  let e0 = (Equal.apply (x => (Nat.succ x)) (Nat.comm a b))
  let e1 = (Equal.mirror (Nat.comm.b b a))
  (Equal.chain e0 e1)

Nat.to_u60 (n: Nat) : U60
Nat.to_u60 Nat.zero     = 0
Nat.to_u60 (Nat.succ n) = (+ 1 (Nat.to_u60 n))
  
Nat.mul (a: Nat) (b: Nat) : Nat
Nat.mul (Nat.succ a) b = (Nat.add (Nat.mul a b) b) // (a + 1) * b = a*b + b
Nat.mul Nat.zero     b = Nat.zero                  // 0b = 0
 
Nat.mul.comm.a (x: Nat): (Equal (Nat.mul x Nat.zero) Nat.zero)
Nat.mul.comm.a Nat.zero = Equal.refl
Nat.mul.comm.a (Nat.succ x) =
  let e0 = (Nat.mul.comm.a x)
  let e1 = (Equal.apply (y => (Nat.add y Nat.zero)) e0)
  e1

// List
// ----

List (a: Type) : Type
List.nil <a> : (List a)
List.cons <a> (x: a) (xs: (List a)) : (List a)

List.negate (xs: (List Bool)) : (List Bool)
List.negate (List.cons Bool x xs) = (List.cons Bool (Bool.not x) (List.negate xs))
List.negate (List.nil Bool)       = (List.nil Bool)

List.tail <a> (xs: (List a)) : (List a)
List.tail a (List.cons t x xs) = xs

List.map <a> <b> (x: (List a)) (f: (x: a) -> b) : (List b)
List.map a b (List.nil t)       f = List.nil
List.map a b (List.cons t x xs) f = (List.cons (f x) (List.map xs f))

List.concat <a> (xs: (List a)) (ys: (List a)) : (List a)
List.concat a (List.cons u x xs) ys = (List.cons u x (List.concat a xs ys))
List.concat a (List.nil u)       ys = ys

List.flatten <a> (xs: (List (List a))) : (List a)
List.flatten a (List.cons u x xs) = (List.concat x (List.flatten xs))
List.flatten a (List.nil u)       = List.nil

List.bind <a: Type> <b: Type> (xs: (List a)) (f: a -> (List b)) : (List b)
List.bind a b xs f = (List.flatten b (List.map xs f))

List.pure <t: Type> (x: t) : (List t)
List.pure t x = (List.cons t x (List.nil t))

List.range.go (lim: Nat) (res: (List Nat)) : (List Nat)
List.range.go Nat.zero     res = res
List.range.go (Nat.succ n) res = (List.range.go n (List.cons n res))

List.sum (xs: (List Nat)) : Nat
List.sum (List.nil t)       = Nat.zero
List.sum (List.cons t x xs) = (Nat.add x (List.sum xs))

// Equal
// -----

Equal <t> (a: t) (b: t) : Type
Equal.refl <t> <a: t> : (Equal t a a)

Equal.mirror <t> <a: t> <b: t> (e: (Equal t a b)) : (Equal t b a)
Equal.mirror t a b (Equal.refl u k) = (Equal.refl u k)

Equal.apply <t> <u> <a: t> <b: t> (f: t -> t) (e: (Equal t a b)) : (Equal t (f a) (f b))
Equal.apply t u a b f (Equal.refl v k) = (Equal.refl v (f k))

Equal.rewrite <t> <a: t> <b: t> (e: (Equal t a b)) (p: t -> Type) (x: (p a)) : (p b)
Equal.rewrite t a b (Equal.refl u k) p x = (x :: (p k))

Equal.chain <t> <a: t> <b: t> <c: t> (e0: (Equal t a b)) (e1: (Equal t b c)) : (Equal t a c)
Equal.chain t a b c e0 (Equal.refl u x) = (e0 :: (Equal t a x))

// Monad
// -----

Monad (f: Type -> Type) : Type
Monad.new (f: Type -> Type)
  (pure: (a: Type) -> (x: a) -> (f a))
  (bind: (a: Type) -> (b: Type) -> (x: (f a)) -> (y: a -> (f b)) -> (f b))
  : (Monad f)

// Variadic
// --------

Variadic (r: Type) (n: Nat) : Type
Variadic r Nat.zero     = r
Variadic r (Nat.succ n) = r -> (Variadic r n)

// Examples
// --------

SNat : Type
SNat = (p: Type) -> ((SNat) -> p) -> p -> p

SZ : SNat
SZ = p => s => z => z