EuclideanDomain -> SemiEuclidean

src/ZkFold/Base/Algebra/Basic/Class.hs

@@ -269,16 +269,20 @@ intScale n a | n < 0     = naturalFromInteger (-n) `scale` negate a
 -}
 class (AdditiveMonoid a, MultiplicativeMonoid a, FromConstant Natural a) => Semiring a
 
-{- | A Euclidean domain @R@ is an integral domain which can be endowed
-with at least one function @f : R\{0} -> R+@ s.t.
-If @a@ and @b@ are in @R@ and @b@ is nonzero, then there exist @q@ and @r@ in @R@ such that
-@a = bq + r@ and either @r = 0@ or @f(r) < f(b)@.
+{- | A semi-Euclidean-domain @a@ is a semiring without zero divisors which can
+be endowed with at least one function @f : a\{0} -> R+@ s.t. if @x@ and @y@ are
+in @a@ and @y@ is nonzero, then there exist @q@ and @r@ in @a@ such that
+@x = qy + r@ and either @r = 0@ or @f(r) < f(y)@.
 
-@q@ and @r@ are called respectively a quotient and a remainder of the division (or Euclidean division) of @a@ by @b@.
+@q@ and @r@ are called respectively a quotient and a remainder of the division
+(or Euclidean division) of @x@ by @y@.
 
-The function @divMod@ associated with this class produces @q@ and @r@ given @a@ and @b@.
+The function @divMod@ associated with this class produces @q@ and @r@
+given @a@ and @b@.
+
+This is a generalization of a notion of Euclidean domains to semirings.
 -}
-class Semiring a => EuclideanDomain a where
+class Semiring a => SemiEuclidean a where
     {-# MINIMAL divMod | (div, mod) #-}
 
     divMod :: a -> a -> (a, a)
@@ -479,7 +483,7 @@ instance AdditiveMonoid Natural where
 
 instance Semiring Natural
 
-instance EuclideanDomain Natural where
+instance SemiEuclidean Natural where
     divMod = Haskell.divMod
 
 instance BinaryExpansion Natural where
@@ -517,7 +521,7 @@ instance FromConstant Natural Integer where
 
 instance Semiring Integer
 
-instance EuclideanDomain Integer where
+instance SemiEuclidean Integer where
     divMod = Haskell.divMod
 
 instance Ring Integer

src/ZkFold/Base/Algebra/Basic/Field.hs

@@ -95,7 +95,7 @@ instance KnownNat p => FromConstant Natural (Zp p) where
 
 instance KnownNat p => Semiring (Zp p)
 
-instance KnownNat p => EuclideanDomain (Zp p) where
+instance KnownNat p => SemiEuclidean (Zp p) where
     divMod a b = let (q, r) = Haskell.divMod (fromZp a) (fromZp b)
                   in (toZp . fromIntegral $ q, toZp . fromIntegral $ r)
 

src/ZkFold/Base/Algebra/Basic/Sources.hs

@@ -77,6 +77,6 @@ instance (Finite a, Ord i) => BinaryExpansion (Sources a i) where
   type Bits (Sources a i) = [Sources a i]
   binaryExpansion = replicate (numberOfBits @a)
 
-instance Ord i => EuclideanDomain (Sources a i) where
+instance Ord i => SemiEuclidean (Sources a i) where
   div = (<>)
   mod = (<>)

src/ZkFold/Symbolic/Algorithms/Hash/Blake2b.hs

@@ -15,8 +15,8 @@ import           Prelude                                           hiding (Num (
                                                                     replicate, splitAt, truncate, (!!), (&&), (^))
 
 import           ZkFold.Base.Algebra.Basic.Class                   (AdditiveGroup (..), AdditiveSemigroup (..),
-                                                                    EuclideanDomain (..), Exponent (..),
-                                                                    FromConstant (..), MultiplicativeSemigroup (..),
+                                                                    Exponent (..), FromConstant (..),
+                                                                    MultiplicativeSemigroup (..), SemiEuclidean (..),
                                                                     divMod, one, zero, (-!))
 import           ZkFold.Base.Algebra.Basic.Number
 import           ZkFold.Prelude                                    (length, replicate, splitAt, (!!))

src/ZkFold/Symbolic/Data/UInt.hs

@@ -102,7 +102,7 @@ cast n =
 eea
     :: forall n c r
     .  Symbolic c
-    => EuclideanDomain (UInt n r c)
+    => SemiEuclidean (UInt n r c)
     => KnownNat n
     => KnownNat (NumberOfRegisters (BaseField c) n r)
     => AdditiveGroup (UInt n r c)
@@ -200,7 +200,7 @@ instance
     , KnownRegisterSize rs
     , r ~ NumberOfRegisters (BaseField c) n rs
     , NFData (c (Vector r))
-    ) => EuclideanDomain (UInt n rs c) where
+    ) => SemiEuclidean (UInt n rs c) where
     divMod numerator d = bool @(Bool c) (q, r) (zero, zero) (d == zero)
         where
             (q, r) = Haskell.foldl longDivisionStep (zero, zero) [value @n -! 1, value @n -! 2 .. 0]

src/ZkFold/Symbolic/MonadCircuit.hs

@@ -18,7 +18,7 @@ import           ZkFold.Base.Algebra.Basic.Class
 -- | A @'WitnessField'@ should support all algebraic operations
 -- used inside an arithmetic circuit.
 type WitnessField n a = ( FiniteField a, ToConstant a, Const a ~ n
-                        , FromConstant n a, EuclideanDomain n)
+                        , FromConstant n a, SemiEuclidean n)
 
 -- | A type of witness builders. @i@ is a type of variables, @a@ is a base field.
 --