-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Combinator birds.
--   
--   A catalogue of the combinator birds (Data.Aviary.Birds) - this module
--   is intended more for illustration than utility.
--   
--   (Data.Aviary.Functional) operations from Applicative, Monad etc. with
--   their types specialized to the function type. This might be usefulfor
--   <i>bird-watching</i> i.e. spotting instances where an Applicative,
--   Monad or whatever operation corresponds to a combinator bird. Again
--   this module is not intended for actual use.
--   
--   Plus a smaller set (Data.Aviary) intended to be useful, collecting
--   combinators that have already <i>escaped</i> (published elsewhere) but
--   aren't in Data.Function, along with some personal favourites.
--   
--   Changelog:
--   
--   <ol>
--   <li>2.2 added Data.Aviary.Functional - Applicative, Monad etc.
--   signatures specialized to the function type.</li>
--   <li>2.1 added <tt>dup</tt> and the <i>combiner</i> variants of
--   cardinal-prime.</li>
--   <li>2.0 added Haddock docs for <tt>(##)</tt>.</li>
--   </ol>
@package data-aviary
@version 0.2.2


-- | Functor, Applicative, Monad operations <i>specialized</i> to the
--   functional type.
--   
--   This is for reference (obviously) and is not intended for use.
module Data.Aviary.Functional
fmap :: (a -> b) -> (r -> a) -> (r -> b)
(<$>) :: (a -> b) -> (r -> a) -> (r -> b)
(<$) :: a -> (r -> b) -> (r -> a)
pure :: a -> (r -> a)
(<*>) :: (r -> a -> b) -> (r -> a) -> (r -> b)
(*>) :: (r -> a) -> (r -> b) -> (r -> b)
(<*) :: (r -> a) -> (r -> b) -> (r -> a)
(<**>) :: (r -> a) -> (r -> a -> b) -> (r -> b)
liftA :: (a -> b) -> (r -> a) -> (r -> b)
liftA2 :: (a -> b -> c) -> (r -> a) -> (r -> b) -> (r -> c)
liftA3 :: (a -> b -> c -> d) -> (r -> a) -> (r -> b) -> (r -> c) -> (r -> d)
id :: a -> a
(.) :: (b -> c) -> (a -> b) -> (a -> c)
(<<<) :: (b -> c) -> (a -> b) -> (a -> c)
(>>>) :: (a -> b) -> (b -> c) -> (a -> c)
(>>=) :: (r -> a) -> (a -> r -> b) -> (r -> b)
(>>) :: (r -> a) -> (r -> b) -> (r -> b)
return :: a -> (r -> a)
fail :: String -> (r -> a)
mapM :: (a -> r -> b) -> [a] -> r -> [b]
mapM_ :: (a -> r -> b) -> [a] -> r -> ()
forM :: [a] -> (a -> r -> b) -> r -> [b]
forM_ :: [a] -> (a -> r -> b) -> r -> ()
sequence :: [r -> a] -> r -> [a]
sequence_ :: [r -> a] -> r -> ()
(=<<) :: (a -> r -> b) -> (r -> a) -> r -> b
(>=>) :: (a -> r -> b) -> (b -> r -> c) -> a -> r -> c
(<=<) :: (b -> r -> c) -> (a -> r -> b) -> a -> r -> c
forever :: (r -> a) -> (r -> b)
join :: (r -> (r -> a)) -> r -> a
filterM :: (a -> r -> Bool) -> [a] -> r -> [a]
mapAndUnzipM :: (a -> r -> (b, c)) -> [a] -> r -> ([b], [c])
zipWithM :: (a -> b -> r -> c) -> [a] -> [b] -> r -> [c]
zipWithM_ :: (a -> b -> r -> c) -> [a] -> [b] -> r -> ()
foldM :: (a -> b -> r -> a) -> a -> [b] -> r -> a
foldM_ :: (a -> b -> r -> a) -> a -> [b] -> r -> ()
replicateM :: Int -> (r -> a) -> r -> [a]
replicateM_ :: Int -> (r -> a) -> r -> ()
when :: Bool -> (r -> ()) -> r -> ()
unless :: Bool -> (r -> ()) -> r -> ()
liftM :: (a -> b) -> (r -> a) -> r -> b
liftM2 :: (a -> b -> c) -> (r -> a) -> (r -> b) -> r -> c
liftM3 :: (a -> b -> c -> d) -> (r -> a) -> (r -> b) -> (r -> c) -> r -> d
liftM4 :: (a -> b -> c -> d -> e) -> (r -> a) -> (r -> b) -> (r -> c) -> (r -> d) -> r -> e
liftM5 :: (a -> b -> c -> d -> e -> f) -> (r -> a) -> (r -> b) -> (r -> c) -> (r -> d) -> (r -> e) -> r -> f
ap :: (r -> a -> b) -> (r -> a) -> r -> b
arr :: (b -> c) -> b -> c
first :: (b -> c) -> (b, d) -> (c, d)
second :: (b -> c) -> (d, b) -> (d, c)
(***) :: (b -> c) -> (b' -> c') -> (b, b') -> (c, c')
(&&&) :: (b -> c) -> (b -> c') -> b -> (c, c')
returnA :: b -> b
(^>>) :: (b -> c) -> (c -> d) -> (b -> d)
(>>^) :: (b -> c) -> (c -> d) -> (b -> d)
(<<^) :: (c -> d) -> (b -> c) -> (b -> d)
(^<<) :: (c -> d) -> (b -> c) -> (b -> d)
left :: (b -> c) -> (Either b d) -> (Either c d)
right :: (b -> c) -> (Either d b) -> (Either d c)
(+++) :: (b -> c) -> (b' -> c') -> (Either b b') -> (Either c c')
(|||) :: (b -> d) -> (c -> d) -> (Either b c) -> d
app :: (b -> c, b) -> c
leftApp :: (b -> c) -> (Either b d) -> (Either c d)
loop :: ((b, d) -> (c, d)) -> b -> c
extract :: (Monoid m) => (m -> a) -> a
duplicate :: (Monoid m) => (m -> a) -> m -> (m -> a)
extend :: (Monoid m) => ((m -> a) -> b) -> (m -> a) -> m -> b
liftW :: (Monoid m) => (a -> b) -> (m -> a) -> m -> b
(=>>) :: (Monoid m) => (m -> a) -> ((m -> a) -> b) -> m -> b
(.>>) :: (Monoid m) => (m -> a) -> b -> m -> b
liftCtx :: (Monoid m) => (a -> b) -> (m -> a) -> b
mapW :: (Monoid m) => ((m -> a) -> b) -> (m -> [a]) -> [b]
parallelW :: (Monoid m) => (m -> [a]) -> [m -> a]
unfoldW :: (Monoid m) => ((m -> b) -> (a, b)) -> (m -> b) -> [a]
sequenceW :: (Monoid m) => [(m -> a) -> b] -> (m -> a) -> [b]


-- | Bird monickered combinators interdefined.
--   
--   This module is intended for illustration (the type signatures!) rather
--   than utility.
--   
--   The 'long reach' Turner set { S, K, I, B, C, S', B', C' }
--   
--   The Joy et al. set { S, I, B, C, J(alt), S', B', C', J(alt)' }
module Data.Aviary.BirdsInter

-- | I combinator - identity bird / idiot bird - Haskell <a>id</a>.
idiot :: a -> a

-- | K combinator - kestrel - Haskell <a>const</a>. Corresponds to the
--   encoding of <tt>true</tt> in the lambda calculus.
--   
--   Not interdefined.
kestrel :: a -> b -> a

-- | B combinator - bluebird - Haskell (<a>.</a>).
bluebird :: (b -> c) -> (a -> b) -> a -> c

-- | C combinator - cardinal - Haskell <a>flip</a>.
cardinal :: (a -> b -> c) -> b -> a -> c

-- | A combinator - apply / applicator - Haskell (<a>$</a>).
--   
--   Note: the definition here is <tt>- C (B B I) I -</tt> and not the
--   familiar <tt>- S (S K) -</tt> which as far as Haskell is concerned has
--   a different type.
--   
--   <pre>
--   (S(SK)) :: ((a -&gt; b) -&gt; a) -&gt; (a -&gt; b) -&gt; a
--   </pre>
applicator :: (a -> b) -> a -> b

-- | Psi combinator - psi bird (?) - Haskell <tt>on</tt>.
psi :: (b -> b -> c) -> (a -> b) -> a -> a -> c

-- | B3 combinator - becard.
becard :: (c -> d) -> (b -> c) -> (a -> b) -> a -> d

-- | B1 combinator - blackbird - specs <tt>oo</tt>.
blackbird :: (c -> d) -> (a -> b -> c) -> a -> b -> d

-- | B' combinator - bluebird prime.
bluebird' :: (a -> c -> d) -> a -> (b -> c) -> b -> d

-- | B2 combinator - bunting - specs <tt>ooo</tt>.
bunting :: (d -> e) -> (a -> b -> c -> d) -> a -> b -> c -> e

-- | C' combinator - no name.
cardinal' :: (c -> a -> d) -> (b -> c) -> a -> b -> d

-- | C* combinator - cardinal once removed.
cardinalstar :: (a -> c -> b -> d) -> a -> b -> c -> d

-- | C** combinator - cardinal twice removed.
cardinalstarstar :: (a -> b -> d -> c -> e) -> a -> b -> c -> d -> e

-- | D combinator - dove.
dove :: (a -> c -> d) -> a -> (b -> c) -> b -> d

-- | D1 combinator - dickcissel.
dickcissel :: (a -> b -> d -> e) -> a -> b -> (c -> d) -> c -> e

-- | D2 combinator - dovekie.
dovekie :: (c -> d -> e) -> (a -> c) -> a -> (b -> d) -> b -> e

-- | E combinator - eagle.
eagle :: (a -> d -> e) -> a -> (b -> c -> d) -> b -> c -> e

-- | E ^ - bald eagle. For alphabetical regularity it is somewhat misnamed
--   here as eaglebald.
eaglebald :: (e -> f -> g) -> (a -> b -> e) -> a -> b -> (c -> d -> f) -> c -> d -> g

-- | F combinator - finch.
finch :: a -> b -> (b -> a -> c) -> c

-- | F* combinator - finch once removed.
finchstar :: (c -> b -> a -> d) -> a -> b -> c -> d

-- | F** combinator - finch once removed.
finchstarstar :: (a -> d -> c -> b -> e) -> a -> b -> c -> d -> e

-- | G combinator - goldfinch.
goldfinch :: (b -> c -> d) -> (a -> c) -> a -> b -> d

-- | H combinator - hummingbird.
hummingbird :: (a -> b -> a -> c) -> a -> b -> c

-- | I* combinator - identity bird once removed. Alias of
--   <a>applicator</a>, Haskell's (<a>$</a>). Type signature
idstar :: (a -> b) -> a -> b

-- | I** combinator - identity bird twice removed.
idstarstar :: (a -> b -> c) -> a -> b -> c

-- | Alternative J combinator - this is the J combintor of Joy,
--   Rayward-Smith and Burton (see. Antoni Diller 'Compiling Functional
--   Languages' page 104). It is not the J - jay combinator of the
--   literature.
jalt :: (a -> c) -> a -> b -> c

-- | J' combinator - from Joy, Rayward-Smith and Burton. See the comment to
--   <a>jalt</a>.
jalt' :: (a -> b -> d) -> a -> b -> c -> d

-- | J combinator - jay.
--   
--   This is the usual J combinator.
jay :: (a -> b -> b) -> a -> b -> a -> b

-- | Ki - kite. Corresponds to the encoding of <tt>false</tt> in the lambda
--   calculus.
kite :: a -> b -> b

-- | O combinator - owl.
owl :: ((a -> b) -> a) -> (a -> b) -> b

-- | (Big) Phi combinator - phoenix - Haskell <tt>liftM2</tt>.
phoenix :: (b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d

-- | Q4 combinator - quacky bird.
quacky :: a -> (a -> b) -> (b -> c) -> c

-- | Q combinator - queer bird.
--   
--   Haskell <tt>(##)</tt> in Peter Thiemann's Wash, reverse composition.
queer :: (a -> b) -> (b -> c) -> a -> c

-- | Q3 combinator - quirky bird.
quirky :: (a -> b) -> a -> (b -> c) -> c

-- | Q1 combinator - quixotic bird.
quixotic :: (b -> c) -> a -> (a -> b) -> c

-- | Q2 combinator - quizzical bird.
quizzical :: a -> (b -> c) -> (a -> b) -> c

-- | R combinator - robin.
robin :: a -> (b -> a -> c) -> b -> c

-- | R* combinator - robin once removed.
robinstar :: (b -> c -> a -> d) -> a -> b -> c -> d

-- | R** combinator - robin twice removed.
robinstarstar :: (a -> c -> d -> b -> e) -> a -> b -> c -> d -> e

-- | S combinator - starling.
--   
--   Haskell: Applicative's <tt>(&lt;*&gt;)</tt> on functions.
--   
--   Not interdefined.
starling :: (a -> b -> c) -> (a -> b) -> a -> c

-- | S' combinator - starling prime - Turner's big phi. Haskell:
--   Applicative's liftA2 on functions.
starling' :: (b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d

-- | T combinator - thrush. Haskell <tt>(#)</tt> in Peter Thiemann's Wash,
--   reverse application.
thrush :: a -> (a -> b) -> b

-- | V combinator - vireo.
vireo :: a -> b -> (a -> b -> b) -> b

-- | V* combinator - vireo once removed.
vireostar :: (b -> a -> b -> d) -> a -> b -> b -> d

-- | V** combinator - vireo twice removed.
vireostarstar :: (a -> c -> b -> c -> e) -> a -> b -> c -> c -> e

-- | W combinator - warbler - elementary duplicator.
warbler :: (a -> a -> b) -> a -> b

-- | W1 combinator - converse warbler. <a>warbler</a> with the arguments
--   reversed.
warbler1 :: a -> (a -> a -> b) -> b

-- | W* combinator - warbler once removed.
warblerstar :: (a -> b -> b -> c) -> a -> b -> c

-- | W** combinator - warbler twice removed.
warblerstarstar :: (a -> b -> c -> c -> d) -> a -> b -> c -> d


-- | Bird monickered combinators
--   
--   This module is intended for illustration (the type signatures!) rather
--   than utility.
--   
--   The 'long reach' Turner set { S, K, I, B, C, S', B', C' }
--   
--   The Joy et al. set { S, I, B, C, J(alt), S', B', C', J(alt)' }
module Data.Aviary.Birds

-- | I combinator - identity bird / idiot bird - Haskell <a>id</a>.
idiot :: a -> a

-- | K combinator - kestrel - Haskell <a>const</a>. Corresponds to the
--   encoding of <tt>true</tt> in the lambda calculus.
kestrel :: a -> b -> a

-- | B combinator - bluebird - Haskell (<a>.</a>).
bluebird :: (b -> c) -> (a -> b) -> a -> c

-- | C combinator - cardinal - Haskell <a>flip</a>.
cardinal :: (a -> b -> c) -> b -> a -> c

-- | A combinator - apply / applicator - Haskell (<a>$</a>).
applicator :: (a -> b) -> a -> b

-- | Psi combinator - psi bird (?) - Haskell <a>on</a>.
psi :: (b -> b -> c) -> (a -> b) -> a -> a -> c

-- | B3 combinator - becard.
becard :: (c -> d) -> (b -> c) -> (a -> b) -> a -> d

-- | B1 combinator - blackbird - specs <tt>oo</tt>.
blackbird :: (c -> d) -> (a -> b -> c) -> a -> b -> d

-- | B' combinator - bluebird prime.
bluebird' :: (a -> c -> d) -> a -> (b -> c) -> b -> d

-- | B2 combinator - bunting - specs <tt>ooo</tt>.
bunting :: (d -> e) -> (a -> b -> c -> d) -> a -> b -> c -> e

-- | C' combinator - no name.
cardinal' :: (c -> a -> d) -> (b -> c) -> a -> b -> d

-- | C* combinator - cardinal once removed.
cardinalstar :: (a -> c -> b -> d) -> a -> b -> c -> d

-- | C** combinator - cardinal twice removed.
cardinalstarstar :: (a -> b -> d -> c -> e) -> a -> b -> c -> d -> e

-- | D combinator - dove.
dove :: (a -> c -> d) -> a -> (b -> c) -> b -> d

-- | D1 combinator - dickcissel.
dickcissel :: (a -> b -> d -> e) -> a -> b -> (c -> d) -> c -> e

-- | D2 combinator - dovekie.
dovekie :: (c -> d -> e) -> (a -> c) -> a -> (b -> d) -> b -> e

-- | E combinator - eagle.
eagle :: (a -> d -> e) -> a -> (b -> c -> d) -> b -> c -> e

-- | E ^ - bald eagle. For alphabetical regularity it is somewhat misnamed
--   here as eaglebald.
eaglebald :: (e -> f -> g) -> (a -> b -> e) -> a -> b -> (c -> d -> f) -> c -> d -> g

-- | F combinator - finch.
finch :: a -> b -> (b -> a -> c) -> c

-- | F* combinator - finch once removed.
finchstar :: (c -> b -> a -> d) -> a -> b -> c -> d

-- | F** combinator - finch once removed.
finchstarstar :: (a -> d -> c -> b -> e) -> a -> b -> c -> d -> e

-- | G combinator - goldfinch.
goldfinch :: (b -> c -> d) -> (a -> c) -> a -> b -> d

-- | H combinator - hummingbird.
hummingbird :: (a -> b -> a -> c) -> a -> b -> c

-- | I* combinator - identity bird once removed Alias of <a>applicator</a>,
--   Haskell's (<a>$</a>).
idstar :: (a -> b) -> a -> b

-- | I** combinator - identity bird twice removed
idstarstar :: (a -> b -> c) -> a -> b -> c

-- | Alternative J combinator - this is the J combintor of Joy,
--   Rayward-Smith and Burton (see. Antoni Diller 'Compiling Functional
--   Languages' page 104). It is not the J - jay combinator of the
--   literature.
jalt :: (a -> c) -> a -> b -> c

-- | J' combinator - from Joy, Rayward-Smith and Burton. See the comment to
--   <a>jalt</a>.
jalt' :: (a -> b -> d) -> a -> b -> c -> d

-- | J combinator - jay.
--   
--   This is the usual J combinator.
jay :: (a -> b -> b) -> a -> b -> a -> b

-- | Ki - kite. Corresponds to the encoding of <tt>false</tt> in the lambda
--   calculus.
kite :: a -> b -> b

-- | O combinator - owl.
owl :: ((a -> b) -> a) -> (a -> b) -> b

-- | (Big) Phi combinator - phoenix - Haskell <tt>liftM2</tt>.
phoenix :: (b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d

-- | Q4 combinator - quacky bird.
quacky :: a -> (a -> b) -> (b -> c) -> c

-- | Q combinator - queer bird.
--   
--   Haskell <tt>(##)</tt> in Peter Thiemann's Wash, reverse composition.
queer :: (a -> b) -> (b -> c) -> a -> c

-- | Q3 combinator - quirky bird.
quirky :: (a -> b) -> a -> (b -> c) -> c

-- | Q1 combinator - quixotic bird.
quixotic :: (b -> c) -> a -> (a -> b) -> c

-- | Q2 combinator - quizzical bird.
quizzical :: a -> (b -> c) -> (a -> b) -> c

-- | R combinator - robin.
robin :: a -> (b -> a -> c) -> b -> c

-- | R* combinator - robin once removed.
robinstar :: (b -> c -> a -> d) -> a -> b -> c -> d

-- | R* combinator - robin twice removed.
robinstarstar :: (a -> c -> d -> b -> e) -> a -> b -> c -> d -> e

-- | S combinator - starling.
--   
--   Haskell: Applicative's <tt>(&lt;*&gt;)</tt> on functions.
--   
--   Substitution.
starling :: (a -> b -> c) -> (a -> b) -> a -> c

-- | S' combinator - starling prime - Turner's big phi. Haskell:
--   Applicative's liftA2 on functions.
starling' :: (b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d

-- | T combinator - thrush. Haskell <tt>(#)</tt> in Peter Thiemann's Wash,
--   reverse application.
thrush :: a -> (a -> b) -> b

-- | V combinator - vireo.
vireo :: a -> b -> (a -> b -> c) -> c

-- | V* combinator - vireo once removed.
vireostar :: (b -> a -> b -> d) -> a -> b -> b -> d

-- | V** combinator - vireo twice removed.
vireostarstar :: (a -> c -> b -> c -> e) -> a -> b -> c -> c -> e

-- | W combinator - warbler - elementary duplicator.
warbler :: (a -> a -> b) -> a -> b

-- | W1 combinator - converse warbler. <a>warbler</a> with the arguments
--   reversed.
warbler1 :: a -> (a -> a -> b) -> b

-- | W* combinator - warbler once removed.
warblerstar :: (a -> b -> b -> c) -> a -> b -> c

-- | W** combinator - warbler twice removed.
warblerstarstar :: (a -> b -> c -> c -> d) -> a -> b -> c -> d


-- | Plainly named combinators
--   
--   Sometimes permuted to be generally useful...
--   
--   Note the fixity of <tt>(#)</tt> and <tt>(##)</tt> is not yet
--   <i>fixed</i>. Some experience needs to be gathered as to whether the
--   precendence levels are appropriate.
module Data.Aviary

-- | T combinator - thrush
--   
--   Reverse application - the T combinator. Found in Peter Thiemann's WASH
--   and the paper 'Client-Side Web Scripting in Haskell' - Erik Meijer,
--   Daan Leijen &amp; James Hook.
(#) :: a -> (a -> b) -> b

-- | Q Combinator - the queer bitd.
--   
--   Reverse composition - found in Peter Thiemann's WASH. You might perfer
--   to use (&lt;&lt;&lt;) from Control.Categoty.
(##) :: (a -> b) -> (b -> c) -> a -> c

-- | S combinator - subst. Familiar as Applicative's (<tt>&lt;*&gt;</tt>)
--   operator, which itself is fmap:
--   
--   f (b -&gt; c) -&gt; f b -&gt; f c where f = ((-&gt;) a)
subst :: (a -> b -> c) -> (a -> b) -> a -> c

-- | The big Phi, or Turner's <tt>S'</tt> combinator. Known to Haskell
--   programmers as liftA2 and liftM2 for the Applicative and Monad
--   instances of (-&gt;).
--   
--   <pre>
--   (a1 -&gt; a2 -&gt; r) -&gt; m a1 -&gt; m a2 -&gt; m r where m = ((-&gt;) a)
--   </pre>
--   
--   Taste suggests you may prefer liftA2 especially as <tt>bigphi</tt> is
--   not a great name (calling it s' would take a very useful variable
--   name).
bigphi :: (b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d

-- | A variant of the <tt>D2</tt> or dovekie combinator - the argument
--   order has been changed to be more satisfying for Haskellers:
--   
--   <pre>
--   (appro comb f g) x y
--   </pre>
--   
--   <pre>
--   (f x) `comb` (g y)
--   </pre>
--   
--   <tt>on</tt> from Data.Function is similar but less general, where the
--   two intermediate results are formed by applying the same function to
--   the supplied arguments:
--   
--   <pre>
--   on = (appro comb f f)
--   </pre>
appro :: (c -> d -> e) -> (a -> c) -> (b -> d) -> a -> b -> e

-- | dup - duplicator aka the W combinator aka Warbler.
--   
--   <pre>
--   dup f x = f x x
--   </pre>
dup :: (a -> a -> b) -> a -> b

-- | Compose an arity 1 function with an arity 2 function. B1 - blackbird
oo :: (c -> d) -> (a -> b -> c) -> a -> b -> d

-- | Compose an arity 1 function with an arity 3 function. B2 - bunting
ooo :: (d -> e) -> (a -> b -> c -> d) -> a -> b -> c -> e

-- | Compose an arity 1 function with an arity 4 function.
oooo :: (e -> f) -> (a -> b -> c -> d -> e) -> a -> b -> c -> d -> f

-- | Combiners - similar to the cardinal' combinator.
--   
--   Mnemonically - <tt>comb</tt>(ine) after applying <tt>f</tt> to
--   <tt>x</tt> and a single identity: <tt>y</tt>.
--   
--   <pre>
--   combfi comb f x y = comb (f x) y
--   </pre>
--   
--   Equivalently:
--   
--   <pre>
--   combfi comb f = appro comb f id
--   </pre>
--   
--   But combfi is a useful introduction to the (somewhat manic, but
--   sometimes useful) higher arity versions.
combfi :: (c -> b -> d) -> (a -> c) -> a -> b -> d

-- | Extrapolation of <a>combfi</a> with another identity.
--   
--   Mnemonically - comb(ine) after applying <tt>f</tt> to <tt>x</tt> and
--   two identities: <tt>y</tt> and <tt>z</tt>.
--   
--   <pre>
--   combfii comb f x y z = comb (f x) y z
--   </pre>
combfii :: (d -> b -> c -> e) -> (a -> d) -> a -> b -> c -> e

-- | Extrapolation of <a>combfii</a> with a further identity.
--   
--   Mnemonically - comb(ine) after applying <tt>f</tt> to <tt>s</tt> and
--   three identities: <tt>t</tt> and <tt>u</tt> and <tt>v</tt>.
--   
--   <pre>
--   combfii comb f s t u v = comb (f s) t u v
--   </pre>
combfiii :: (e -> b -> c -> d -> f) -> (a -> e) -> a -> b -> c -> d -> f