# Type witness

### From HaskellWiki

A **type witness** is a value that represents one of a range of possible types. When implemented as generalised algebraic datatypes (GADTs), type witness can be used to perform dynamic casts.

Types `a` and `b` may or may not be the same type. We wish to perform the cast `a -> Maybe b`, or more generally, `p a -> Maybe (p b)` for any `p`, depending on whether `a` and `b` are the same type. This can be done with GADTs if we have "witnesses" for the two types:

dynamicCast :: Witness a -> Witness b -> p a -> Maybe (p b)

A simple witness type might be defined over the Int, Bool and Char types like so:

data Witness a where IntWitness :: Witness Int BoolWitness :: Witness Bool CharWitness :: Witness Char

dynamicCast IntWitness IntWitness pa = Just pa dynamicCast BoolWitness BoolWitness pa = Just pa dynamicCast CharWitness CharWitness pa = Just pa dynamicCast _ _ _ = Nothing

If we wish to add type constructors such as the list constructor, that can be done by composing type constructors:

data Witness a where IntWitness :: Witness Int BoolWitness :: Witness Bool CharWitness :: Witness Char ListWitness :: Witness a -> Witness [a]

data Compose p q a = MkCompose (p (q a))

dynamicCast IntWitness IntWitness pa = Just pa dynamicCast BoolWitness BoolWitness pa = Just pa dynamicCast CharWitness CharWitness pa = Just pa dynamicCast (ListWitness wa) (ListWitness wb) pla = do MkCompose plb <- dynamicCast wa wb (MkCompose pla) return plb dynamicCast _ _ _ = Nothing

By using more complex composition types in addition to `Compose`, it is possible for witnesses to allow type constructors of more complex kinds.