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[#46] Document typeclasses laws #47

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Oct 21, 2020
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[#46] Document typeclasses laws #47

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[#46] Document typeclasses laws
chshersh Oct 21, 2020
ee6e66a
Make docs shorter
chshersh Oct 21, 2020
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57 changes: 55 additions & 2 deletions src/Prolens.hs
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Original file line number Diff line number Diff line change
Expand Up @@ -194,7 +194,7 @@ Moreover, @p in@ must have 'Functor' instance first to implement the
Instances of 'Profunctor' should satisfy the following laws:

* __Identity:__ @'dimap' 'id' 'id' ≡ 'id'@
* __Composition:__ @'dimap' (inAB . inBC) (outYZ . outXY) ≡ 'dimap' outBC outYZ . 'dimap' outAB outXY@
* __Composition:__ @'dimap' (inAB . inBC) (outYZ . outXY) ≡ 'dimap' inBC outYZ . 'dimap' inAB outXY@

@since 0.0.0.0
-}
Expand Down Expand Up @@ -239,6 +239,20 @@ The second element of a pair (variable of type @c@) can be of any
type, and you can decide what type it should be. This is convenient
for implementing various functions. E.g. 'lens' uses this fact.

Instances of 'Strong' should satisfy the following laws:

* __'first' via 'second' swap:__ @'first' ≡ 'dimap' 'Data.Tuple.swap' 'Data.Tuple.swap' . 'second'@
* __'second' via 'first' swap:__ @'second' ≡ 'dimap' 'Data.Tuple.swap' 'Data.Tuple.swap' . 'first'@

* __Fst functor:__ @'dimap' 'fst' 'id' ≡ 'fmap' 'fst' . 'first'@
* __Snd functor:__ @'dimap' 'snd' 'id' ≡ 'fmap' 'snd' . 'second'@

* __Distribution over 'first':__ @'dimap' ('second' f) 'id' . 'first' ≡ 'fmap' ('second' f) . 'first'@
* __Distribution over 'second':__ @'dimap' ('first' f) 'id' . 'second' ≡ 'fmap' ('first' f) . 'second'@

* __Associativity of 'first':__ @'first' . 'first' ≡ 'dimap' (\\((a, b), c) -> (a, (b, c))) (\\(a, (b, c)) -> ((a, b), c)) . 'first'@
* __Associativity of 'second':__ @'second' . 'second' ≡ 'dimap' (\\(a, (b, c)) -> ((a, b), c)) (\\((a, b), c) -> (a, (b, c))) . 'second'@

@since 0.0.0.0
-}
class Profunctor p => Strong p where
Expand Down Expand Up @@ -272,6 +286,29 @@ The other element of 'Either' (variable of type @c@) can be of any
type, and you can decide what type it should be. This is convenient
for implementing various functions. E.g. 'prism' uses this fact.


Assuming, we have the following functions in scope:

@
swapEither :: Either a b -> Either b a
unnestLeft :: Either (Either a b) c -> Either a (Either b c)
unnestRight :: Either a (Either b c) -> Either (Either a b) c
@

Instances of 'Choice' should satisfy the following laws:

* __'left' via 'right' swap:__ @'left' ≡ 'dimap' swapEither swapEither . 'right'@
* __'right' via 'left' swap:__ @'right' ≡ 'dimap' swapEither swapEither . 'left'@

* __'Left' functor:__ @'fmap' 'Left' ≡ 'dimap' 'Left' 'id' . 'left'@
* __'Right' functor:__ @'fmap' 'Right' ≡ 'dimap' 'Right' 'id' . 'right'@

* __Distribution over 'left':__ @'dimap' ('right' f) 'id' . 'left' ≡ 'fmap' ('right' f) . 'left'@
* __Distribution over 'right':__ @'dimap' ('left' f) 'id' . 'right' ≡ 'fmap' ('left' f) . 'right'@

* __Associativity of 'left':__ @'left' . 'left' ≡ 'dimap' unnestLeft unnestRight . 'left'@
* __Associativity of 'right':__ @'right' . 'right' ≡ 'dimap' unnestRight unnestLeft . 'right'@

@since 0.0.0.0
-}
class Profunctor p => Choice p where
Expand Down Expand Up @@ -309,11 +346,27 @@ instance (Applicative m) => Choice (Fun m) where
{- | 'Monoidal' is 'Strong' 'Profunctor' that can be appended. It is
similar to 'Monoid's but for higher-kinded types.

Instances of 'Monoidal' should satisfy the following laws:

* __Right identity:__ @'pappend' f 'pempty' ≡ 'first' f@
* __Left identity:__ @'pappend' 'pempty' f ≡ 'second' f@
* __Associativity:__ @'pappend' f ('pappend' g h) ⋍ 'pappend' ('pappend' f g) h@

⚠️ __Note:__ The @⋍@ operator in the __associativity__ law is equality
ignoring the structure. The law is written in that way because
'pappend' returns a tuple and the order of nested tuples depends on
the order of 'pappend' applications. In practice, this means, that if
you want to check the law, you reorder tuples in the following way:

@
'pappend' f ('pappend' g h) ≡ 'dimap' (\\(a, (b, c)) -> ((a, b), c)) (\\((a, b), c) -> (a, (b, c))) ('pappend' ('pappend' f g) h)
@

@since 0.0.0.0
-}
class Strong p => Monoidal p where
pappend :: p a b -> p c d -> p (a, c) (b, d)
pempty :: p a a
pempty :: p a a

-- | @since 0.0.0.0
instance Monoidal (->) where
Expand Down