{-# OPTIONS --warning=error --safe --without-K #-}

open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import WellFoundedInduction
open import Functions
open import Orders
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Addition
open import Numbers.Naturals.Order
open import Numbers.Naturals.Multiplication
open import Numbers.Naturals.Exponentiation
open import Semirings.Definition
open import Monoids.Definition
open import Maybe

module Numbers.Naturals.Subtraction where

_-N'_ : (a b : ℕ) → Maybe ℕ
zero -N' zero = yes 0
zero -N' succ b = no
succ a -N' zero = yes (succ a)
succ a -N' succ b = a -N' b

subtractZero : (a : ℕ) → a -N' 0 ≡ yes a
subtractZero zero = refl
subtractZero (succ a) = refl