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benjamin-hodgson/FmapProof.hs

Last active March 1, 2019 19:53
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fmap f x <*> fmap g y = (\x y -> f x (g y)) <$> x <*> y
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FmapProof.hs
fmap f x <*> fmap g y
(pure f <*> x) <*> (pure g <*> y) -- definition of fmap
pure (.) <*> (pure f <*> x) <*> pure g <*> y -- composition
pure (.) <*> pure (.) <*> pure f <*> x <*> pure g <*> y -- composition
pure ((.) .) <*> pure f <*> x <*> pure g <*> y -- homomorphism
pure ((.) . f) <*> x <*> pure g <*> y -- homomorphism
pure ($ g) <*> (pure ((.) . f) <*> x) <*> y -- interchange
pure (.) <*> pure ($ g) <*> pure ((.) . f) <*> x <*> y -- composition
pure (($ g) .) <*> pure ((.) . f) <*> x <*> y -- homomorphism
pure (($ g) . ((.) . f)) <*> x <*> y -- homomorphism
pure ((. g) . f) <*> x <*> y -- i cheated and used pointfree.io
(\x y -> f x (g y)) <$> x <*> y -- definition of (.) and <$>
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