Elem is like a promoted elem: it allows us to check whether a type constructor t is present in a list of type constructors ts.

t ∉ ts is shorthand for asserting that a type constructor t is not present in a list of types constructors ts.

An applicative functor f indexed by a monoid (M,<>,Identity)

If you import Prelude hiding (<*>) and pure, you can redefine (<*>) and pure to use their indexed equivalents. For example,

import Prelude hiding ((<*>), pure)

pure :: IxApplicative f -> a -> f Identity a
pure = ipure

(<*>) :: IxApplicative f => f i (a -> b) -> f j a -> f (i <> j) b
(<*>) = iap

A monad m indexed by a monoid (M,<>,Identity)

You can use do-notation with IxMonad by enabling the RebindableSyntax extension and redefining (>>=), (>>), and return. For example,

{-#LANGUAGE RebindableSyntax #-}

import Prelude hiding ((>>=), (>>), return)

(>>=) :: IxMonad m => m i a -> (a -> m j b) -> m (i <> j) b
(>>=) = ibind

(>>) :: IxMonad m => m i a -> m j b -> m (i <> j) b
a >> b = a >>= const b

return :: IxApplicative m => a -> m Identity a
return = ipure

This is the technique employed by the Text.XML.Twiml.Syntax module.

A free monad indexed by a monoid (M,++,[])

Lift an indexed functor into IxFree

Promoted list concatenation

toAttrValue transforms a data type into a String that can be set as the value of an attribute.