User:Dave Menendez/Arrows

The arrow laws. Should probably be merged into Arrows.

I'm using the formulation from Ross Paterson's "Arrows and Computation", modified for use with more recent libraries. As with MonadPlus, there appear to be no laws for ArrowZero and ArrowPlus.

Category

 left identity:                        id . f = f
 right identity:                       f . id = f
 associativity:                   f . (g . h) = (f . g) . h

Arrow

For arr:

 functor-identity:                     arr id = id
 functor-composition:             arr (g . f) = arr g . arr f

For first:

 extension:                     first (arr f) = arr (f *** id)
 functor:                       first (f . g) = first f . first g
 exchange:           arr (id *** g) . first f = first f . arr (id *** g)
 unit:                      arr fst . first f = f . arr fst
 association:     arr assoc . first (first f) = first f . arr assoc

ArrowApp

 composition:        app . arr ((h .) *** id) = h . app
 reduction:         app . arr (mkPair *** id) = id
 extensionality:               app . mkPair f = f

ArrowChoice

 extension:                      left (arr f) = arr (f +++ id)
 functor:                        left (f . g) = left f . left g
 exchange:            arr (id +++ g) . left f = left f . arr (id +++ g)
 unit:                      left f . arr Left = arr Left . f
 association:    arr assocsum . left (left f) = left f . arr assocsum
 distribution:     arr distr . first (left f) = left (first f) . arr distr

ArrowLoop

 extension:                      loop (arr f) = arr (trace f)
 left tightening:          loop (f . first h) = loop f . h
 right tightening:         loop (first h . f) = h . loop f
 sliding:          loop (arr (id *** k)  . f) = loop (f . arr (id *** k))
 vanishing:                     loop (loop f) = loop (arr assoc . f . arr unassoc)
 superposing:                 second (loop f) = loop (arr unassoc . second f . arr assoc)

Utility Functions

assoc :: ((a,b),c) -> (a,(b,c))
assoc ~(~(a,b),c) = (a,(b,c))

unassoc :: (a,(b,c)) -> ((a,b),c)
unassoc ~(a,~(b,c)) = ((a,b),c)

mkPair :: Arrow a => b -> a c (b,c)
mkPair b = arr (\c -> (b,c))

assocsum :: Either (Either a b) c -> Either a (Either b c)
assocsum (Left (Left a))  = Left a
assocsum (Left (Right b)) = Right (Left b)
assocsum (Right c)        = Right (Right c)

distr :: (Either a b, c) -> Either (a,c) (b,c)
distr (Left a,  c) = Left (a,c)
distr (Right b, c) = Right (b,c)

trace :: ((b,d) -> (c,d)) -> b -> c
trace f b = let (c,d) = f (b,d) in c