Common monads as free monads

Free Empty ~~ Identity

Given

data Empty a

we have

data Free Empty a
     = Pure a
-- the Free constructor is impossible!

which is isomorphic to

data Identity a
     = Identity a

Given

data Identity a = Identity a

we have

data Free Identity a
     = Pure a
     | Free (Identity (Free Identity a))

which is isomorphic to

data Deferred a
     = Now a
     | Later (Deferred a)

or equivalently (if you promise to evaluate the fst element first) (Nat, a), aka Writer Nat a, with

data Nat = Z | S Nat
data Writer Nat a = Writer Nat a

Given

data Maybe a = Just a
             | Nothing

we have

data Free Maybe a
     = Pure a
     | Free (Just (Free Maybe a))
     | Free Nothing

which is equivalent to

data Hopes a
     = Confirmed a
     | Possible (Hopes a)
     | Failed

or equivalently (if you promise to evaluate the fst element first) (Nat, Maybe a), aka MaybeT (Writer Nat) a with

data Nat = Z | S Nat
data Writer Nat a = Writer Nat a
data MaybeT (Writer Nat) a = MaybeT (Nat, Maybe a)

Given

data Writer w a = Writer w a

we have

data Free (Writer w) a
     = Pure a
     | Free (Writer w (Free (Writer w) a))

which is isomorphic to

data ProgLog w a
     = Done a
     | After w (ProgLog w a)

or, equivalently, (if you promise to evaluate the log first), Writer [w] a.

Given

data Const c a = Const c

we have

data Free (Const c) a
     = Pure a
     | Free (Const c)

which is isomorphic to

data Either c a
     = Right a
     | Left c

Given

data Reader x a = Reader (x -> a)

we have

data Free (Reader x) a
     = Pure a
     | Free (x -> Free (Reader x) a)

which is isomorphic to

data Demand x a
     = Satisfied a
     | Hungry (x -> Demand x a)

or equivalently Stream x -> a with

data Stream x = Stream x (Stream x)