Data.Ring

class (Semiring a) <= Ring a  where

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)

Members

• sub :: a -> a -> a

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)

negate :: forall a. Ring a => a -> a

negate x can be used as a shorthand for zero - x.

Operator alias for Data.Ring.sub (left-associative / precedence 6)

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

Members

• subRecord :: forall rlproxy. rlproxy rowlist -> Record row -> Record row -> Record 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)

class SemiringRecord rowlist row subrow | rowlist -> subrow

Instances

• SemiringRecord Nil row ()
• (IsSymbol key, Cons key focus subrowTail subrow, SemiringRecord rowlistTail row subrowTail, Semiring focus) => SemiringRecord (Cons key focus rowlistTail) row subrow

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

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