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ex9.txt
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1. pure id <*> x = x
2. pure (g x) = pure g <*> pure x
3. x <*> pure y = pure (\g -> g y) <*> x
4. x <*> (y <*> z) = (pure (.) <*> x <*> y) <*> z
pure = Just
Nothing <*> _ = Nothing
(Just g) <*> mx = fmap g mx
fmap id = id
fmap (g . h) = fmap g . fmap h
fmap :: (a -> b) -> Maybe a -> Maybe b
fmap g Nothing = Nothing
fmap g (Just a) = Just (g a)
Proof of 1. pure id <*> x = x
case Nothing:
pure id <*> Nothing
= {applying pure}
Just id <*> Nothing
= {applying <*>}
fmap id Nothing
= {applying fmap}
Nothing
case Just:
pure id <*> Just x
= {applying pure}
Just id <*> Just x
= {applying <*>}
fmap id (Just x)
= {applying fmap}
Just (id x)
= {applying id}
Just x
Proof of 2. pure (g x) = pure g <*> pure x
case Nothing:
pure (g Nothing)
= {applying pure}
Just (g Nothing)
= {unapplying fmap}
fmap g (Just Nothing)
= {unapplying <*>}
Just g <*> Just Nothing
= {unapplying pure}
pure g <*> pure Nothing
case Just:
pure (g (Just a))
= {applying pure}
Just (g (Just a))
= {unapplying fmap}
fmap g (Just (Just a))
= {unapplying <*>}
Just g <*> Just (Just a)
= {unapplying pure}
pure g <*> pure (Just a)
Proof of 3. x <*> pure y = pure (\g -> g y) <*> x
case Nothing:
Nothing <*> pure y
= {applying <*>}
Nothing
= {unapplying fmap}
fmap (\g -> g y) Nothing
= {unapplying <*>}
Just (\g -> g y) <*> Nothing
= {unapplying pure}
pure (\g -> g y) <*> Nothing
case Just:
Just a <*> pure y
= {applying <*>}
fmap a (pure y)
= {applying pure}
fmap a (Just y)
= {applying fmap}
Just (a y)
= {unapplying (\g -> g y)}
Just ((\g -> g y) a)
= {unapplying fmap}
fmap (\g -> g y) (Just a)
= {unapplying <*>}
Just (\g -> g y) <*> Just a
= {unapplying pure}
pure (\g -> g y) <*> Just a
Proof of 4. x <*> (y <*> z) = (pure (.) <*> x <*> y) <*> z
case Nothing:
Nothing <*> (y <*> z)
= {applying <*>}
Nothing
= {unapplying <*>}
Nothing <*> z
= {unapplying <*>}
(Nothing <*> y) <*> z
= {unapplying fmap}
((fmap . Nothing) <*> y) <*> z
= {unapplying <*>}
(Just (.) <*> Nothing <*> y) <*> z
= {unapplying pure}
(pure (.) <*> Nothing <*> y) <*> z
case Just a, Nothing:
(Just a) <*> (Nothing <*> z)
= {applying <*>}
fmap a (Nothing <*> z)
= {applicative <*>}
fmap a Nothing
= {applying fmap}
Nothing
= {unapplying <*>}
Nothing <*> z
= {unapplying fmap}
fmap (. a) Nothing <*> z
= {unapplying <*>}
(Just (. a) <*> Nothing) <*> z
= {unapplying fmap}
((fmap . (Just a)) <*> Nothing) <*> z
= {unapplying <*>}
(Just (.) <*> (Just a) <*> Nothing) <*> z
= {unapplying pure}
(pure (.) <*> (Just a) <*> Nothing) <*> z
case Just a, Just b:
(Just a) <*> ((Just b) <*> z)
= {applying <*>}
fmap a ((Just b) <*> z)
= {applying <*>}
fmap a (fmap b z)
= {unapplying .}
fmap (a . b) z
= {unapplying <*>}
Just (a . b) <*> z
= {unapplying applicative property 3}
(Just (. a) <*> (Just b)) <*> z
= {unapplying fmap}
((fmap . (Just a)) <*> (Just b)) <*> z
= {unapplying <*>}
(Just (.) <*> (Just a) <*> (Just b)) <*> z
= {unapplying pure}
(pure (.) <*> (Just a) <*> (Just b)) <*> z