typelevel / algebra
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package algebra
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package lattice
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import scala.{specialized => sp}
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/**
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  * De Morgan algebras are bounded lattices that are also equipped with
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  * a De Morgan involution.
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  *
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  * De Morgan involution obeys the following laws:
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  *
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  *  - ¬¬a = a
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  *  - ¬(x∧y) = ¬x∨¬y
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  *
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  * However, in De Morgan algebras this involution does not necessarily
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  * provide the law of the excluded middle. This means that there is no
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  * guarantee that (a ∨ ¬a) = 1. De Morgan algebra do not not necessarily
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  * provide the law of non contradiction either. This means that there is
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  * no guarantee that (a ∧ ¬a) = 0.
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  *
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  * De Morgan algebras are useful to model fuzzy logic. For a model of
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  * classical logic, see the boolean algebra type class implemented as
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  * [[Bool]].
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  */
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trait DeMorgan[@sp(Int, Long) A] extends Any with Logic[A] { self =>
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  def meet(a: A, b: A): A = and(a, b)
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  def join(a: A, b: A): A = or(a, b)
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  def imp(a: A, b: A): A = or(not(a), b)
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}
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trait DeMorganFunctions[H[A] <: DeMorgan[A]] extends
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  BoundedMeetSemilatticeFunctions[H] with
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  BoundedJoinSemilatticeFunctions[H] with
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  LogicFunctions[H]
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object DeMorgan extends DeMorganFunctions[DeMorgan] {
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  /**
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    * Access an implicit `DeMorgan[A]`.
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    */
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  @inline final def apply[@sp(Int, Long) A](implicit ev: DeMorgan[A]): DeMorgan[A] = ev
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  /**
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    * Turn a [[Bool]] into a `DeMorgan`
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    * Used for binary compatibility.
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    */
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  final def fromBool[@sp(Int, Long) A](bool: Bool[A]): DeMorgan[A] =
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    new DeMorgan[A] {
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      def and(a: A, b: A): A = bool.and(a, b)
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      def or(a: A, b: A): A = bool.or(a, b)
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      def not(a: A): A = bool.complement(a)
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      def one: A = bool.one
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      def zero: A = bool.zero
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    }
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}

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