# 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
```