Datatypes

Algebraic data types

Algebraic datatypes are a family of constructions arising out of two operations, sums and products. A product encodes multiple arguments to constructors and sums encode choice between constructors.

data Unit = Unit                 -- 1
data Empty                       -- 0
data (a * b) = Product a b       -- a * b
data (a + b) = Inl a | Inr b     -- a + b
data Exp a b = Exp (a -> b)      -- a^b
data Rec f   = Rec (f (Rec f))   -- \mu

Catamorphisms

Syntax

GHC.Generics

class Generic a where
  type family Rep a :: * -> *
  to   :: a -> Rep a x
  from :: Rep a x -> a

Constructor Models

V1 Void: used for datatypes without constructors U1 Unit: used for constructors without arguments K1 Constants, additional parameters. :*: Products: encode multiple arguments to constructors :+: Sums: encode choice between constructors L1 Left hand side of a sum. R1 Right hand side of a sum. M1 Meta-information (constructor names, etc.)

newtype M1 i c f p = M1 (f p)
newtype K1 i c   p = K1 c
data U           p = U

data (:*:) a b p = a p :*: b p
data (:+:) a b p = L1 (a p) | R1 (b p)

Full Source

\clearpage

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