Data.CommutativeRing

class (Ring a) <= CommutativeRing a 

The CommutativeRing class is for rings where multiplication is commutative.

Instances must satisfy the following law in addition to the Ring laws:

• Commutative multiplication: a * b = b * a

Instances

• CommutativeRing Int
• CommutativeRing Number
• CommutativeRing Unit
• (CommutativeRing b) => CommutativeRing (a -> b)
• (RowToList row list, CommutativeRingRecord list row row) => CommutativeRing (Record row)
• CommutativeRing (Proxy a)
• CommutativeRing (Proxy2 a)
• CommutativeRing (Proxy3 a)

class (RingRecord rowlist row subrow) <= CommutativeRingRecord rowlist row subrow | rowlist -> subrow

A class for records where all fields have CommutativeRing instances, used to implement the CommutativeRing instance for records.

Instances

• CommutativeRingRecord Nil row ()
• (IsSymbol key, Cons key focus subrowTail subrow, CommutativeRingRecord rowlistTail row subrowTail, CommutativeRing focus) => CommutativeRingRecord (Cons key focus rowlistTail) row subrow

class (Semiring a) <= Ring a 

The Ring class is for types that support addition, multiplication, and subtraction operations.

Instances must satisfy the following laws in addition to the Semiring laws:

• Additive inverse: a - a = zero
• Compatibility of sub and negate: a - b = a + (zero - b)

Instances

• Ring Int
• Ring Number
• Ring Unit
• (Ring b) => Ring (a -> b)
• Ring (Proxy a)
• Ring (Proxy2 a)
• Ring (Proxy3 a)
• (RowToList row list, RingRecord list row row) => Ring (Record row)

class (SemiringRecord rowlist row subrow) <= RingRecord rowlist row subrow | rowlist -> subrow

Instances

• RingRecord Nil row ()
• (IsSymbol key, Cons key focus subrowTail subrow, RingRecord rowlistTail row subrowTail, Ring focus) => RingRecord (Cons key focus rowlistTail) row subrow

class Semiring a  where

The Semiring class is for types that support an addition and multiplication operation.

Instances must satisfy the following laws:

• Commutative monoid under addition:
  • Associativity: (a + b) + c = a + (b + c)
  • Identity: zero + a = a + zero = a
  • Commutative: a + b = b + a
• Monoid under multiplication:
  • Associativity: (a * b) * c = a * (b * c)
  • Identity: one * a = a * one = a
• Multiplication distributes over addition:
  • Left distributivity: a * (b + c) = (a * b) + (a * c)
  • Right distributivity: (a + b) * c = (a * c) + (b * c)
• Annihilation: zero * a = a * zero = zero

Note: The Number and Int types are not fully law abiding members of this class hierarchy due to the potential for arithmetic overflows, and in the case of Number, the presence of NaN and Infinity values. The behaviour is unspecified in these cases.

Members

• add :: a -> a -> a
• zero :: a
• mul :: a -> a -> a
• one :: a

Instances

• Semiring Int
• Semiring Number
• (Semiring b) => Semiring (a -> b)
• Semiring Unit
• Semiring (Proxy a)
• Semiring (Proxy2 a)
• Semiring (Proxy3 a)
• (RowToList row list, SemiringRecord list row row) => Semiring (Record row)

Operator alias for Data.Semiring.add (left-associative / precedence 6)

Operator alias for Data.Semiring.mul (left-associative / precedence 7)