Semiring solver (#50)

Numbers/Integers/Definition.agda

@@ -8,28 +8,3 @@ module Numbers.Integers.Definition where
 data ℤ : Set where
   nonneg : ℕ → ℤ
   negSucc : ℕ → ℤ
-
-data ℤSimple : Set where
-  negativeSucc : (a : ℕ) → ℤSimple
-  positiveSucc : (a : ℕ) → ℤSimple
-  zZero : ℤSimple
-
-convertZ : ℤ → ℤSimple
-convertZ (nonneg zero) = zZero
-convertZ (nonneg (succ x)) = positiveSucc x
-convertZ (negSucc x) = negativeSucc x
-
-convertZ' : ℤSimple → ℤ
-convertZ' (negativeSucc a) = negSucc a
-convertZ' (positiveSucc a) = nonneg (succ a)
-convertZ' zZero = nonneg 0
-
-zIsZ : (a : ℤ) → convertZ' (convertZ a) ≡ a
-zIsZ (nonneg zero) = refl
-zIsZ (nonneg (succ x)) = refl
-zIsZ (negSucc x) = refl
-
-zIsZ' : (a : ℤSimple) → convertZ (convertZ' a) ≡ a
-zIsZ' (negativeSucc a) = refl
-zIsZ' (positiveSucc a) = refl
-zIsZ' zZero = refl

Numbers/Naturals/Naturals.agda

@@ -10,6 +10,7 @@ open import Numbers.Naturals.Addition
 open import Numbers.Naturals.Order
 open import Numbers.Naturals.Multiplication
 open import Numbers.Naturals.Exponentiation
+open import Numbers.Naturals.Subtraction
 open import Semirings.Definition
 open import Monoids.Definition
 
@@ -20,6 +21,7 @@ module Numbers.Naturals.Naturals where
   open Numbers.Naturals.Multiplication using (_*N_ ; multiplicationNIsCommutative) public
   open Numbers.Naturals.Exponentiation using (_^N_) public
   open Numbers.Naturals.Order using (_<N_ ; le; succPreservesInequality; succIsPositive; addingIncreases; zeroNeverGreater; noIntegersBetweenXAndSuccX; a<SuccA; canRemoveSuccFrom<N) public
+  open Numbers.Naturals.Subtraction using (_-N'_) public
 
   ℕSemiring : Semiring 0 1 _+N_ _*N_
   Monoid.associative (Semiring.monoid ℕSemiring) a b c = equalityCommutative (additionNIsAssociative a b c)

Numbers/Naturals/Subtraction.agda

@@ -0,0 +1,27 @@
+{-# 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