“Going bananas with recursion schemes for fixed point data types”
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The document discusses representing and evaluating recursive expressions using functional programming techniques. It defines an Exp algebraic data type to represent expressions like integers, doubles, sums, products, divisions and squares. It then implements functions to evaluate and optimize expressions represented as Exp values. An example expression is given involving multiplication, summation and division.
![sealed trait Exp
final case class IntValue(v: Int) extends Exp
final case class DecValue(v: Double) extends Exp
final case class Sum(exp1: Exp, exp2: Exp) extends Exp
final case class Multiply(exp1: Exp, exp2: Exp) extends Exp
final case class Divide(exp1: Exp, exp2: Exp) extends Exp
final case class Square(exp: Exp) extends Exp
sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-54-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[?](IntValue[?](10),
IntValue[?](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-55-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[?](IntValue[Unit](10),
IntValue[Unit](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-56-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[?](IntValue[Unit](10),
IntValue[Unit](5))
)
Exp[Unit]](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-57-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[?](IntValue[Unit](10),
IntValue[Unit](5))
)
Exp[Unit]](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-58-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[?](IntValue[Unit](10),
IntValue[Unit](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-59-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[?] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-60-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-61-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[?] =
Divide[?](
DecValue[?](5.2),
Sum[?](IntValue[?](10),
IntValue[?](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-62-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[?] =
Divide[?](
DecValue[?](5.2),
Sum[?](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-63-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[?] =
Divide[?](
DecValue[?](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-64-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[?] =
Divide[?](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-65-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[?] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-66-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-67-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-68-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-69-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp3: Exp = from(input)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)
val exp3: Exp[?] = from(input)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-70-320.jpg)
![sealed trait Exp[A]
final case class IntValue[A](v: Int) extends Exp[A]
final case class DecValue[A](v: Double) extends Exp[A]
final case class Sum[A](exp1: A, exp2: A) extends Exp[A]
final case class Multiply[A](exp1: A, exp2: A) extends Exp[A]
final case class Divide[A](exp1: A, exp2: A) extends Exp[A]
final case class Square[A](exp: A) extends Exp[A]
val exp1: Exp =
Sum(IntValue(10),
IntValue(5))
)
val exp2: Exp =
Divide(
DecValue(5.2),
Sum(IntValue(10),
IntValue(5))
)
)
val exp3: Exp = from(input)
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](IntValue[Unit](10),
IntValue[Unit](5))
)
)
val exp3: Exp[Exp[Exp[Exp[Exp[Exp[Exp[Exp[Exp[Ex](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-71-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-75-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](
IntValue[Unit](10),
IntValue[Unit](5)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-76-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](
Fix(IntValue[Unit](10)),
Fix(IntValue[Unit](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-77-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Exp[Exp[Unit]] =
Sum[Exp[Unit]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-78-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Exp[Exp[Unit]] =
Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-79-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Exp[Exp[Unit]] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-80-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-81-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](
IntValue[Unit](10),
IntValue[Unit](5))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-82-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Exp[Unit]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-83-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
DecValue[Exp[Unit]](5.2),
Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-84-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
Fix(DecValue[Fix[Exp]](5.2)),
Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
)
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-85-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Divide[Exp[Exp[Unit]]](
Fix(DecValue[Fix[Exp]](5.2)),
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
))
)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-86-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Exp[Exp[Exp[Unit]]] =
Fix(Divide[Fix[Exp]](
Fix(DecValue[Fix[Exp]](5.2)),
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-87-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Fix[Exp] =
Fix(Divide[Fix[Exp]](
Fix(DecValue[Fix[Exp]](5.2)),
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-88-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5))
))
val exp2: Fix[Exp] =
Fix(Divide[Fix[Exp]](
Fix(DecValue[Fix[Exp]](5.2)),
Fix(Sum[Fix[Exp]](
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](5)))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-89-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
val exp2: Fix[Exp] =
Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-90-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
val exp2: Fix[Exp] =
Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-91-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
val exp2: Fix[Exp] =
Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-92-320.jpg)
![case class Fix[F[_]](unFix: F[Fix[F]])
val exp1: Fix[Exp] =
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
val exp2: Fix[Exp] =
Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))
val exp3: Fix[Exp] = from(input)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-93-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-113-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_], T](a: A[T])](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-114-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_], T](a: A[T])(trans: T => Int)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-115-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_], T](a: A[T])(trans: T => Int): A[Int] = ???](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-116-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = ???](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-117-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-118-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans)
case class Container[T](t: T)
val r: Container[Int] =
foo[Container, String](Container("bar"))(str => str.length)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-119-320.jpg)
![Functor 101
trait Functor[F[_]] {
def map[A, B](fa: F[A])(f: A => B): F[B]
}
def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans)
implicit val functor: Functor[Container] = new Functor[Container] {
def map[A, B](c: Container[A])(f: A => B): Container[B] = Container(f(c.t))
}
case class Container[T](t: T)
val r: Container[Int] =
foo[Container, String](Container("bar"))(str => str.length)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-120-320.jpg)
![sealed trait Exp
final case class IntValue(v: Int) extends Exp
final case class DecValue(v: Double) extends Exp
final case class Sum(exp1: Exp, exp2: Exp) extends Exp
final case class Multiply(exp1: Exp, exp2: Exp) extends Exp
final case class Divide(exp1: Exp, exp2: Exp) extends Exp
final case class Square(exp: Exp) extends Exp
implicit val functor: Functor[Exp] = new Functor[Exp] {
def map[A, B](exp: Exp[A])(f: A => B): Exp[B] =](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-121-320.jpg)
![sealed trait Exp
final case class IntValue(v: Int) extends Exp
final case class DecValue(v: Double) extends Exp
final case class Sum(exp1: Exp, exp2: Exp) extends Exp
final case class Multiply(exp1: Exp, exp2: Exp) extends Exp
final case class Divide(exp1: Exp, exp2: Exp) extends Exp
final case class Square(exp: Exp) extends Exp
implicit val functor: Functor[Exp] = new Functor[Exp] {
def map[A, B](exp: Exp[A])(f: A => B): Exp[B] = exp match {
case Sum(a1, a2) => Sum(f(a1), f(a2))
case Multiply(a1, a2) => Multiply(f(a1), f(a2))
case Divide(a1, a2) => Divide(f(a1), f(a2))
case Square(a) => Square(f(a))
case IntValue(v) => IntValue(v)
case DecValue(v) => DecValue(v)
}
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-122-320.jpg)
![F-algebra
type Algebra[F[_], A] = F[A] => A](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-126-320.jpg)
![type Algebra[F[_], A] = F[A] => A
val evaluate: Algebra[Exp, Double] = {
case IntValue(v) => v.toDouble
case DecValue(v) => v
case Sum(a1, a2) => a1 + a2
case Multiply(a1, a2) => a1 * a2
case Square(a) => a * a
case Divide(a1, a2) => a1 / a2
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-127-320.jpg)
![type Algebra[F[_], A] = F[A] => A
val evaluate: Algebra[Exp, Double] = {
case IntValue(v) => v.toDouble
case DecValue(v) => v
case Sum(a1, a2) => a1 + a2
case Multiply(a1, a2) => a1 * a2
case Square(a) => a * a
case Divide(a1, a2) => a1 / a2
}
val mkStr: Algebra[Exp, String] = {
case IntValue(v) => v.toString
case DecValue(v) => v.toString
case Sum(a1, a2) => s"($a1 + $a2)"
case Multiply(a1, a2) => s"($a1 + $a2)"
case Square(a) => s"($a)^2"
case Divide(a1, a2) => s"($a1 + $a2)"
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-128-320.jpg)
![val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double
case IntValue(v) => v.toDouble
case DecValue(v) => v
case Sum(a1, a2) => a1 + a2
case Multiply(a1, a2) => a1 * a2
case Square(a) => a * a
case Divide(a1, a2) => a1 / a2
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-129-320.jpg)
![val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double
case IntValue(v) => v.toDouble
case DecValue(v) => v
case Sum(a1, a2) => a1 + a2
case Multiply(a1, a2) => a1 * a2
case Square(a) => a * a
case Divide(a1, a2) => a1 / a2
}
val exp2: Fix[Exp] = Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-130-320.jpg)
![val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double
case IntValue(v) => v.toDouble
case DecValue(v) => v
case Sum(a1, a2) => a1 + a2
case Multiply(a1, a2) => a1 * a2
case Square(a) => a * a
case Divide(a1, a2) => a1 / a2
}
val exp2: Fix[Exp] = Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))
> exp2.cata(evaluate)
0.3466666666666667](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-131-320.jpg)
![val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double
case IntValue(v) => v.toString
case DecValue(v) => v.toString
case Sum(a1, a2) => s"($a1 + $a2)"
case Multiply(a1, a2) => s"($a1 + $a2)"
case Square(a) => s"($a)^2"
case Divide(a1, a2) => s"($a1 + $a2)"
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-132-320.jpg)
![val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double
case IntValue(v) => v.toString
case DecValue(v) => v.toString
case Sum(a1, a2) => s"($a1 + $a2)"
case Multiply(a1, a2) => s"($a1 + $a2)"
case Square(a) => s"($a)^2"
case Divide(a1, a2) => s"($a1 + $a2)"
}
val exp2: Fix[Exp] = Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-133-320.jpg)
![val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double
case IntValue(v) => v.toString
case DecValue(v) => v.toString
case Sum(a1, a2) => s"($a1 + $a2)"
case Multiply(a1, a2) => s"($a1 + $a2)"
case Square(a) => s"($a)^2"
case Divide(a1, a2) => s"($a1 + $a2)"
}
val exp2: Fix[Exp] = Fix(Divide(
Fix(DecValue(5.2)),
Fix(Sum(
Fix(IntValue(10)),
Fix(IntValue(5))
))
))
> exp2.cata(mkStr)
(5.2 + (10 + 5))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-134-320.jpg)
![val optimize: Exp => Exp = {
case Multiply(exp1, exp2)
if(exp1 == exp2) => Square(optimize(exp1))
case IntValue(v) => IntValue(v)
case DecValue(v) => DecValue(v)
case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2))
case Multiply(exp1, exp2) => Multiply(optimize(exp1), optimize(exp2))
case Square(exp) => Square(optimize(exp))
case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2))
}
val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp]
case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1)))
case other => Fix(other)
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-136-320.jpg)
![val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp]
case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1)))
case other => Fix(other)
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-137-320.jpg)
![val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp]
case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1)))
case other => Fix(other)
}
val aTimesAExp: Fix[Exp] =
Fix(Multiply(
Fix(Sum(
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](20))
)),
Fix(Sum(
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](20))
))
))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-138-320.jpg)
![val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp]
case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1)))
case other => Fix(other)
}
val aTimesAExp: Fix[Exp] =
Fix(Multiply(
Fix(Sum(
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](20))
)),
Fix(Sum(
Fix(IntValue[Fix[Exp]](10)),
Fix(IntValue[Fix[Exp]](20))
))
))
> aTimesAExp.cata(optimize)
Fix(Square(Fix(Sum(Fix(IntValue(10)),Fix(IntValue(20))))))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-139-320.jpg)
![type Coalgebra[F[_], A] = A => F[A]](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-145-320.jpg)
![type Coalgebra[F[_], A] = A => F[A]
val divisors: Coalgebra[Exp, Int] = {
case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2)
case n => IntValue(n)
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-146-320.jpg)
![type Coalgebra[F[_], A] = A => F[A]
val divisors: Coalgebra[Exp, Int] = {
case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2)
case n => IntValue(n)
}
> 12.ana[Fix, Exp](divisors)](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-147-320.jpg)
![type Coalgebra[F[_], A] = A => F[A]
val divisors: Coalgebra[Exp, Int] = {
case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2)
case n => IntValue(n)
}
> 12.ana[Fix, Exp](divisors)
Fix(Multiply(Fix(IntValue(2)),Fix(Multiply(Fix(IntValue(2)),Fix(Multiply(Fix(IntVal
ue(2)),Fix(IntValue(7))))))))](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-148-320.jpg)
![val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double]
val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-155-320.jpg)
![val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double]
val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double
describe("divisors") {
it("once evaluated will give initial value") {
forAll(positiveInt) { n =>
n.ana(divisiors).cata(evaluate) shouldEqual(n)
}
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-156-320.jpg)
![val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double]
val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double
describe("divisors") {
it("once evaluated will give initial value") {
forAll(positiveInt) { n =>
n.hylo(evaluate, divisors) shouldEqual(n)
}
}](https://image.slidesharecdn.com/goingbananaswithrecursionschemesforfixedpointdatatypes-161117154745/85/Going-bananas-with-recursion-schemes-for-fixed-point-data-types-157-320.jpg)
“Going bananas with recursion schemes for fixed point data types”
- 1. “Going bananas with recursion schemes for fixed point data types”
- 4. Elegance of Functional Programming
- 5. Recursion
- 9. Can we actually can Build some Stuff?
- 13. Recursive Structures are common
- 14. Recursive Structures are common ● ○ ■ ■ ○ ■ ■
- 15. Recursive Structures are common ● ○ ■ ■ ○ ■ ■ ● ○ ○ ○
- 16. Recursive Structures are common ● ○ ■ ■ ○ ■ ■ ● ○ ○ ○ ●
- 17. Expressions!
- 18. Expressions!
- 19. Expressions!
- 20. Expressions!
- 21. Expressions!
- 22. Expressions!
- 23. Expressions!
- 24. Expressions!
- 25. Expressions!
- 27. data Exp = IntVal Int | DecVal Double | Sum Exp Exp | Multiply Exp Exp | Divide Exp Exp | Square Exp
- 28. data Exp = IntVal Int | DecVal Double | Sum Exp Exp | Multiply Exp Exp | Divide Exp Exp | Square Exp sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp
- 29. data Exp = IntVal Int | DecVal Double | Sum Exp Exp | Multiply Exp Exp | Divide Exp Exp | Square Exp sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp
- 30. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val evaluate: Exp => Double =
- 31. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) }
- 32. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val mkString: Exp => String = { case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(exp1, exp2) => s"( ${mkStr(exp1)} + ${mkStr(exp2)})" case Multiply(exp1, exp2) => s"( ${mkStr(exp1)} * ${mkStr(exp2)})" case Square(exp) => s"(${mkStr(exp)})^2" case Divide(exp1, exp2) => s"( ${mkStr(exp1)} / ${mkStr(exp2)})" }
- 33. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1))
- 34. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1)) case IntValue(v) => IntValue(v) case DecValue(v) => DecValue(v) case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2)) case Multiply(exp1, exp2) => Multiply(optimize(exp1), optimize(exp2)) case Square(exp) => Square(optimize(exp)) case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2)) }
- 35. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1)) case IntValue(v) => IntValue(v) case DecValue(v) => DecValue(v) case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2)) case Multiply(exp1, exp2) => Multiply(exp1, optimize(exp2)) case Square(exp) => Square(optimize(exp)) case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2)) }
- 36. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1)) case IntValue(v) => optimize(IntValue(v)) case DecValue(v) => optimize(DecValue(v)) case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2)) case Multiply(exp1, exp2) => Multiply(optimize(exp1), optimize(exp2)) case Square(exp) => Square(optimize(exp)) case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2)) }
- 37. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 38. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 39. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 40. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 41. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 42. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 43. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 44. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 45. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 46. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 47. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 48. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 49. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 50. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 51. val mkString: Exp => String = { case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(exp1, exp2) => s"(${mkStr(exp1)} + ${mkStr(exp2)})" case Multiply(exp1, exp2) => s"(${mkStr(exp1)} * ${mkStr(exp2)})" case Square(exp) => s"(${mkStr(exp)})^2" case Divide(exp1, exp2) => s"( ${mkStr(exp1)} / ${mkStr(exp2)})" } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 52. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 53. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp
- 54. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A]
- 55. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[?](IntValue[?](10), IntValue[?](5)) )
- 56. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[?](IntValue[Unit](10), IntValue[Unit](5)) )
- 57. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[?](IntValue[Unit](10), IntValue[Unit](5)) ) Exp[Unit]
- 58. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[?](IntValue[Unit](10), IntValue[Unit](5)) ) Exp[Unit]
- 59. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[?](IntValue[Unit](10), IntValue[Unit](5)) )
- 60. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[?] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) )
- 61. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) )
- 62. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[?] = Divide[?]( DecValue[?](5.2), Sum[?](IntValue[?](10), IntValue[?](5)) ) )
- 63. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[?] = Divide[?]( DecValue[?](5.2), Sum[?](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 64. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[?] = Divide[?]( DecValue[?](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 65. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[?] = Divide[?]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 66. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[?] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 67. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 68. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 69. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) )
- 70. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp3: Exp = from(input) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) ) val exp3: Exp[?] = from(input)
- 71. sealed trait Exp[A] final case class IntValue[A](v: Int) extends Exp[A] final case class DecValue[A](v: Double) extends Exp[A] final case class Sum[A](exp1: A, exp2: A) extends Exp[A] final case class Multiply[A](exp1: A, exp2: A) extends Exp[A] final case class Divide[A](exp1: A, exp2: A) extends Exp[A] final case class Square[A](exp: A) extends Exp[A] val exp1: Exp = Sum(IntValue(10), IntValue(5)) ) val exp2: Exp = Divide( DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) val exp3: Exp = from(input) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]](IntValue[Unit](10), IntValue[Unit](5)) ) ) val exp3: Exp[Exp[Exp[Exp[Exp[Exp[Exp[Exp[Exp[Ex
- 73. Fix
- 75. case class Fix[F[_]](unFix: F[Fix[F]])
- 76. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]]( IntValue[Unit](10), IntValue[Unit](5) )
- 77. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]]( Fix(IntValue[Unit](10)), Fix(IntValue[Unit](5)) )
- 78. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Exp[Exp[Unit]] = Sum[Exp[Unit]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )
- 79. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Exp[Exp[Unit]] = Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )
- 80. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Exp[Exp[Unit]] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) ))
- 81. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) ))
- 82. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]]( IntValue[Unit](10), IntValue[Unit](5)) ) )
- 83. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Exp[Unit]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) ) )
- 84. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( DecValue[Exp[Unit]](5.2), Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) ) )
- 85. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( Fix(DecValue[Fix[Exp]](5.2)), Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) ) )
- 86. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Divide[Exp[Exp[Unit]]]( Fix(DecValue[Fix[Exp]](5.2)), Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) )) )
- 87. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Exp[Exp[Exp[Unit]]] = Fix(Divide[Fix[Exp]]( Fix(DecValue[Fix[Exp]](5.2)), Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) )) ))
- 88. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Fix[Exp] = Fix(Divide[Fix[Exp]]( Fix(DecValue[Fix[Exp]](5.2)), Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) )) ))
- 89. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5)) )) val exp2: Fix[Exp] = Fix(Divide[Fix[Exp]]( Fix(DecValue[Fix[Exp]](5.2)), Fix(Sum[Fix[Exp]]( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](5))) )) ))
- 90. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) ))
- 91. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) ))
- 92. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) ))
- 93. case class Fix[F[_]](unFix: F[Fix[F]]) val exp1: Fix[Exp] = Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) )) val exp3: Fix[Exp] = from(input)
- 94. val evaluate: Exp => Double = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(exp1, exp2) => evaluate(exp1) + evaluate(exp2) case Multiply(exp1, exp2) => evaluate(exp1) * evaluate(exp2) case Square(exp) => val v = evaluate(exp) v * v case Divide(exp1, exp2) => evaluate(exp1) / evaluate(exp2) } val exp = Multiply( Sum(IntValue(10), DecValue(2.5)), Divide(DecValue(5.2), Sum(IntValue(10), IntValue(5)) ) ) Multiply Sum Divide Int(10) Dec(2.5) Dec(5.2) Sum Int(10) Int(5)
- 95. Need something that will traverse the structure…
- 96. Catamorphism
- 111. Missing Ingredients
- 112. Functor 101
- 113. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] }
- 114. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_], T](a: A[T])
- 115. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_], T](a: A[T])(trans: T => Int)
- 116. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_], T](a: A[T])(trans: T => Int): A[Int] = ???
- 117. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = ???
- 118. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans)
- 119. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans) case class Container[T](t: T) val r: Container[Int] = foo[Container, String](Container("bar"))(str => str.length)
- 120. Functor 101 trait Functor[F[_]] { def map[A, B](fa: F[A])(f: A => B): F[B] } def foo[A[_] : Functor, T](a: A[T])(trans: T => Int): A[Int] = a.map(trans) implicit val functor: Functor[Container] = new Functor[Container] { def map[A, B](c: Container[A])(f: A => B): Container[B] = Container(f(c.t)) } case class Container[T](t: T) val r: Container[Int] = foo[Container, String](Container("bar"))(str => str.length)
- 121. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp implicit val functor: Functor[Exp] = new Functor[Exp] { def map[A, B](exp: Exp[A])(f: A => B): Exp[B] =
- 122. sealed trait Exp final case class IntValue(v: Int) extends Exp final case class DecValue(v: Double) extends Exp final case class Sum(exp1: Exp, exp2: Exp) extends Exp final case class Multiply(exp1: Exp, exp2: Exp) extends Exp final case class Divide(exp1: Exp, exp2: Exp) extends Exp final case class Square(exp: Exp) extends Exp implicit val functor: Functor[Exp] = new Functor[Exp] { def map[A, B](exp: Exp[A])(f: A => B): Exp[B] = exp match { case Sum(a1, a2) => Sum(f(a1), f(a2)) case Multiply(a1, a2) => Multiply(f(a1), f(a2)) case Divide(a1, a2) => Divide(f(a1), f(a2)) case Square(a) => Square(f(a)) case IntValue(v) => IntValue(v) case DecValue(v) => DecValue(v) } }
- 123. Missing Ingredients
- 124. Missing Ingredients
- 125. Missing Ingredients
- 126. F-algebra type Algebra[F[_], A] = F[A] => A
- 127. type Algebra[F[_], A] = F[A] => A val evaluate: Algebra[Exp, Double] = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(a1, a2) => a1 + a2 case Multiply(a1, a2) => a1 * a2 case Square(a) => a * a case Divide(a1, a2) => a1 / a2 }
- 128. type Algebra[F[_], A] = F[A] => A val evaluate: Algebra[Exp, Double] = { case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(a1, a2) => a1 + a2 case Multiply(a1, a2) => a1 * a2 case Square(a) => a * a case Divide(a1, a2) => a1 / a2 } val mkStr: Algebra[Exp, String] = { case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(a1, a2) => s"($a1 + $a2)" case Multiply(a1, a2) => s"($a1 + $a2)" case Square(a) => s"($a)^2" case Divide(a1, a2) => s"($a1 + $a2)" }
- 129. val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(a1, a2) => a1 + a2 case Multiply(a1, a2) => a1 * a2 case Square(a) => a * a case Divide(a1, a2) => a1 / a2 }
- 130. val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(a1, a2) => a1 + a2 case Multiply(a1, a2) => a1 * a2 case Square(a) => a * a case Divide(a1, a2) => a1 / a2 } val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) ))
- 131. val evaluate: Algebra[Exp, Double] = { // Exp[Double] => Double case IntValue(v) => v.toDouble case DecValue(v) => v case Sum(a1, a2) => a1 + a2 case Multiply(a1, a2) => a1 * a2 case Square(a) => a * a case Divide(a1, a2) => a1 / a2 } val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) )) > exp2.cata(evaluate) 0.3466666666666667
- 132. val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(a1, a2) => s"($a1 + $a2)" case Multiply(a1, a2) => s"($a1 + $a2)" case Square(a) => s"($a)^2" case Divide(a1, a2) => s"($a1 + $a2)" }
- 133. val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(a1, a2) => s"($a1 + $a2)" case Multiply(a1, a2) => s"($a1 + $a2)" case Square(a) => s"($a)^2" case Divide(a1, a2) => s"($a1 + $a2)" } val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) ))
- 134. val mkStr: Algebra[Exp, String] = { // Exp[Double] => Double case IntValue(v) => v.toString case DecValue(v) => v.toString case Sum(a1, a2) => s"($a1 + $a2)" case Multiply(a1, a2) => s"($a1 + $a2)" case Square(a) => s"($a)^2" case Divide(a1, a2) => s"($a1 + $a2)" } val exp2: Fix[Exp] = Fix(Divide( Fix(DecValue(5.2)), Fix(Sum( Fix(IntValue(10)), Fix(IntValue(5)) )) )) > exp2.cata(mkStr) (5.2 + (10 + 5))
- 135. val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1)) case IntValue(v) => IntValue(v) case DecValue(v) => DecValue(v) case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2)) case Multiply(exp1, exp2) => Multiply(optimize(exp1), optimize(exp2)) case Square(exp) => Square(optimize(exp)) case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2)) }
- 136. val optimize: Exp => Exp = { case Multiply(exp1, exp2) if(exp1 == exp2) => Square(optimize(exp1)) case IntValue(v) => IntValue(v) case DecValue(v) => DecValue(v) case Sum(exp1, exp2) => Sum(optimize(exp1), optimize(exp2)) case Multiply(exp1, exp2) => Multiply(optimize(exp1), optimize(exp2)) case Square(exp) => Square(optimize(exp)) case Divide(exp1, exp2) => Divide(optimize(exp1), optimize(exp2)) } val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp] case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1))) case other => Fix(other) }
- 137. val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp] case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1))) case other => Fix(other) }
- 138. val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp] case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1))) case other => Fix(other) } val aTimesAExp: Fix[Exp] = Fix(Multiply( Fix(Sum( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](20)) )), Fix(Sum( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](20)) )) ))
- 139. val optimize: Algebra[Exp, Fix[Exp]] = { // Exp[Fix[Exp]] => Fix[Exp] case Multiply(Fix(a1), Fix(a2)) if(a1 == a2) => Fix(Square(Fix(a1))) case other => Fix(other) } val aTimesAExp: Fix[Exp] = Fix(Multiply( Fix(Sum( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](20)) )), Fix(Sum( Fix(IntValue[Fix[Exp]](10)), Fix(IntValue[Fix[Exp]](20)) )) )) > aTimesAExp.cata(optimize) Fix(Square(Fix(Sum(Fix(IntValue(10)),Fix(IntValue(20))))))
- 140. ZOO of morphisims
- 141. Anamorphism ●
- 142. Anamorphism ● ●
- 143. Anamorphism ● ● ●
- 145. type Coalgebra[F[_], A] = A => F[A]
- 146. type Coalgebra[F[_], A] = A => F[A] val divisors: Coalgebra[Exp, Int] = { case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2) case n => IntValue(n) }
- 147. type Coalgebra[F[_], A] = A => F[A] val divisors: Coalgebra[Exp, Int] = { case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2) case n => IntValue(n) } > 12.ana[Fix, Exp](divisors)
- 148. type Coalgebra[F[_], A] = A => F[A] val divisors: Coalgebra[Exp, Int] = { case n if(n % 2 == 0 && n != 2) => Multiply(2, n / 2) case n => IntValue(n) } > 12.ana[Fix, Exp](divisors) Fix(Multiply(Fix(IntValue(2)),Fix(Multiply(Fix(IntValue(2)),Fix(Multiply(Fix(IntVal ue(2)),Fix(IntValue(7))))))))
- 149. Hylomorphism
- 150. Hylomorphism ●
- 151. Hylomorphism ● ●
- 152. Hylomorphism ● ● ●
- 153. Hylomorphism ● ● ●
- 154. Hylomorphism ● ● ●
- 155. val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double] val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double
- 156. val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double] val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double describe("divisors") { it("once evaluated will give initial value") { forAll(positiveInt) { n => n.ana(divisiors).cata(evaluate) shouldEqual(n) } }
- 157. val divisors: Coalgebra[Exp, Int] = { … } // Double => Exp[Double] val evaluate: Algebra[Exp, Double] = { … } // Exp[Double] => Double describe("divisors") { it("once evaluated will give initial value") { forAll(positiveInt) { n => n.hylo(evaluate, divisors) shouldEqual(n) } }
- 158. What else?
- 159. To sum it up
- 160. To sum it up ●
- 161. To sum it up ● ●
- 162. To sum it up ● ● ● ○ ○
- 163. To sum it up ● ● ● ○ ○ ●
- 164. To sum it up ● ● ● ○ ○ ● ●
- 165. To sum it up ● ● ● ○ ○ ● ● ●
- 166. References
- 167. Pawel Szulc