{-# OPTIONS --cubical --no-import-sorts --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

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)