Data.BooleanAlgebra

class (HeytingAlgebra a) <= BooleanAlgebra a 

The BooleanAlgebra type class represents types that behave like boolean values.

Instances should satisfy the following laws in addition to the HeytingAlgebra law:

• Excluded middle:
  • a || not a = tt

Instances

• BooleanAlgebra Boolean
• BooleanAlgebra Unit
• (BooleanAlgebra b) => BooleanAlgebra (a -> b)
• (RowToList row list, BooleanAlgebraRecord list row row) => BooleanAlgebra (Record row)

class (HeytingAlgebraRecord rowlist row subrow) <= BooleanAlgebraRecord rowlist row subrow | rowlist -> subrow

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

Instances

• BooleanAlgebraRecord Nil row ()
• (IsSymbol key, Cons key focus subrowTail subrow, BooleanAlgebraRecord rowlistTail row subrowTail, BooleanAlgebra focus) => BooleanAlgebraRecord (Cons key focus rowlistTail) row subrow

class HeytingAlgebra a  where

The HeytingAlgebra type class represents types that are bounded lattices with an implication operator such that the following laws hold:

• Associativity:
  • a || (b || c) = (a || b) || c
  • a && (b && c) = (a && b) && c
• Commutativity:
  • a || b = b || a
  • a && b = b && a
• Absorption:
  • a || (a && b) = a
  • a && (a || b) = a
• Idempotent:
  • a || a = a
  • a && a = a
• Identity:
  • a || ff = a
  • a && tt = a
• Implication:
  • a `implies` a = tt
  • a && (a `implies` b) = a && b
  • b && (a `implies` b) = b
  • a `implies` (b && c) = (a `implies` b) && (a `implies` c)
• Complemented:
  • not a = a `implies` ff

Members

• ff :: a
• tt :: a
• implies :: a -> a -> a
• conj :: a -> a -> a
• disj :: a -> a -> a
• not :: a -> a

Instances

• HeytingAlgebra Boolean
• HeytingAlgebra Unit
• (HeytingAlgebra b) => HeytingAlgebra (a -> b)
• (RowToList row list, HeytingAlgebraRecord list row row) => HeytingAlgebra (Record row)

class HeytingAlgebraRecord rowlist row subrow | rowlist -> subrow

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

Instances

• HeytingAlgebraRecord Nil row ()
• (IsSymbol key, Cons key focus subrowTail subrow, HeytingAlgebraRecord rowlistTail row subrowTail, HeytingAlgebra focus) => HeytingAlgebraRecord (Cons key focus rowlistTail) row subrow

Operator alias for Data.HeytingAlgebra.disj (right-associative / precedence 2)

Operator alias for Data.HeytingAlgebra.conj (right-associative / precedence 3)