{-# OPTIONS --no-exact-split --safe #-}
module Cubical.Data.Nat.Base where

open import Agda.Builtin.Nat public
  using (zero; suc; _+_)
  renaming (Nat to ; _-_ to _∸_; _*_ to _·_)

open import Cubical.Data.Nat.Literals public
open import Cubical.Data.Bool.Base
open import Cubical.Data.Sum.Base hiding (elim)
open import Cubical.Data.Empty.Base hiding (elim)
open import Cubical.Data.Unit.Base
open import Cubical.Data.Sigma.Base

predℕ :   
predℕ zero = zero
predℕ (suc n) = n

caseNat :  {}  {A : Type }  (a0 aS : A)    A
caseNat a0 aS zero    = a0
caseNat a0 aS (suc n) = aS

doubleℕ :   
doubleℕ zero = zero
doubleℕ (suc x) = suc (suc (doubleℕ x))

-- doublesℕ n m = 2^n · m
doublesℕ :     
doublesℕ zero m = m
doublesℕ (suc n) m = doublesℕ n (doubleℕ m)

-- iterate
iter :  {} {A : Type }    (A  A)  A  A
iter zero f z    = z
iter (suc n) f z = f (iter n f z)

elim :  {} {A :   Type }
   A zero
   ((n : )  A n  A (suc n))
   (n : )  A n
elim a₀ _ zero = a₀
elim a₀ f (suc n) = f n (elim a₀ f n)

elim+2 :  {} {A :   Type }  A 0  A 1
           ((n : )  (A (suc n)  A (suc (suc n))))
           (n : )  A n
elim+2 a0 a1 ind zero = a0
elim+2 a0 a1 ind (suc zero) = a1
elim+2 {A = A} a0 a1 ind (suc (suc n)) =
  ind n (elim+2 {A = A} a0 a1 ind (suc n))

isEven isOdd :   Bool
isEven zero = true
isEven (suc n) = isOdd n
isOdd zero = false
isOdd (suc n) = isEven n

--Typed version
private
  toType : Bool  Type
  toType false = 
  toType true = Unit

isEvenT :   Type
isEvenT n = toType (isEven n)

isOddT :   Type
isOddT n = isEvenT (suc n)

isZero :   Bool
isZero zero = true
isZero (suc n) = false

-- exponential

_^_ :     
m ^ 0 = 1
m ^ (suc n) = m · m ^ n


-- Iterated product
_ˣ_ :  {} (A :   Type ) (n : )  Type 
A ˣ zero = A zero
A ˣ suc n = (A ˣ n) × A (suc n)

 :  {} (A :   Type ) (0A : (n : )  A n)  (n : )  A ˣ n
 A 0A zero = 0A zero
 A 0A (suc n) = ( A 0A n) , (0A (suc n))