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


-- | Combinator birds.
--   
--   A catalogue for the combinator birds (Data.Aviary.Birds) - this module
--   is intended more for illustration than utility.
--   
--   Plus a smaller set (Data.Avery) intended to be useful, collecting
--   combinators that have already <i>escaped</i> (published elsewhere) but
--   aren't in Data.Function and other favourites.
@package data-aviary
@version 0.1.0


-- | 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
(##) :: (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.
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.
--   
--   <tt>appro</tt> is similar to the function <tt>prod</tt> from the Pair
--   calculus, but <tt>appro</tt> applies the first argument <tt> f :: (c
--   -&gt; d -&gt; e) </tt> to the two intermediate results. <tt>prod</tt>
--   always forms a pair from the intermediate results.
--   
--   <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.
appro :: (c -> d -> e) -> (a -> c) -> (b -> d) -> a -> b -> e

-- | 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