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