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https://github.com/mikeizbicki/ifcxt

constraint level if statements
https://github.com/mikeizbicki/ifcxt

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constraint level if statements

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# IfCxt



This package introduces the function:

```

ifCxt :: IfCxt cxt => proxy cxt -> (cxt => a) -> a -> a

```

This function acts like an `if` statement where the `proxy cxt` parameter is the condition.

If the type checker can satisfy the `cxt` constraint, then the second argument `cxt => a` is returned;

otherwise, the third argument `a` is returned.



Before seeing more details about how `ifCxt` is implemented,

let's look at three examples of how to use it.



### Example 1: show every type



The `cxtShow` function below is polymorphic over the type `a`.

If `a` is an instance of `Show`, then `cxtShow a` evaluates to `show a`;

but if `a` is not an instance of `Show`, `cxtShow a` evaluates to `<>`.



```

cxtShow :: forall a. IfCxt (Show a) => a -> String

cxtShow a = ifCxt (Proxy::Proxy (Show a))

    (show a)

    "<>"

```

In ghci:



```

ghci> cxtShow (1 :: Int)

"1"

```

```

ghci> cxtShow (id :: a -> a)

"<>"

```



### Example 2: make your code asymptotically efficient



The `nub` function removes duplicate elements from lists.

It can be defined as:



```

nub :: Eq a => [a] -> [a]

nub []     =  []

nub (x:xs) =  x : nub (filter (x/=) xs)

```

This function takes time O(n^2).

But if we also have an `Ord` constraint, we can define a much more efficient version that takes time O(n log n):



```

nubOrd :: Ord a => [a] -> [a]

nubOrd = go . sort

    where

        go (x1:x2:xs)

            | x1==x2    =      go (x2:xs)

            | otherwise = x1 : go (x2:xs)

        go [x] = [x]

        go []  = []

```

Now, we can use the `ifCxt` function to define a version of `nub` that will automatically select the most efficient implementation for whatever type we happen to run it on:



```

cxtNub :: forall a. (Eq a, IfCxt (Ord a)) => [a] -> [a]

cxtNub = ifCxt (Proxy::Proxy (Ord a)) nubOrd nub

```



### Example 3: make your code numerically stable



The simplest way to sum a list of numbers is:

```

sumSimple :: Num a => [a] -> a

sumSimple = foldl' (+) 0

```

This method has numerical stability issues on floating point representations.

[Kahan summation](https://en.wikipedia.org/wiki/Kahan_summation_algorithm) is a more accurate technique shown below:

```

sumKahan :: Num a => [a] -> a

sumKahan = snd . foldl' go (0,0)

    where

        go (c,t) i = ((t'-t)-y,t')

            where

                y = i-c

                t' = t+y

```

Because Kahan summation does a lot more work than simple summation, we would prefer not to run it on non-floating point types.

The `sumCxt` function below accomplishes this:

```

cxtSum :: forall a. (Num a, IfCxt (Floating a)) => [a] -> a

cxtSum = ifCxt (Proxy::Proxy (Floating a)) sumKahan sumSimple

```

Notice that the `ifCxt` function is conditioning on the `Floating a` constraint,

which isn't actually *used* by the `sumKahan` function.



## How it works



The magic of the technique is in the `IfCxt` class:

```

class IfCxt (cxt :: Constraint) where

    ifCxt :: proxy cxt -> (cxt => a) -> a -> a

```

(Notice that making a constraint an instance of a class requires the`ConstraintKinds` extension,

and the higher order `(cxt => a)` parameter requires the `RankNTypes` extension.)



There is a "global" instance defined as:

```

instance {-# OVERLAPPABLE #-} IfCxt cxt where ifCxt _ t f = f

```

What this says is that if no more specific instance is available, then the "global" `ifCxt` function will be used, which always returns the `f` (false) parameter.



Then for every instance of every other class, we need to define an overlapping `IfCxt` instance that always returns the `t` (true) parameter.

For example, for `Show Int`, we define:

```

instance {-# OVERLAPS #-} IfCxt (Show Int) where ifCxt _ t f = t

```



This is a lot of boilerplate, so the template haskell function `mkIfCxtInstances` can be used to define these instances automatically.

Unfortunately, due to a [bug in template haskell](https://ghc.haskell.org/trac/ghc/ticket/9699) we cannot enumerate all the classes currently in scope.

So you must manually call `mkIfCxtInstances` on each class you want `ifCxt` to work with.