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Proof of functor laws for Maybe

In this section I want to give a short example of how equational reasoning can be used to proof certain properties of a given piece of code in Haskell.

So without further ado let's begin:

Known facts

The Functor instance declaration of the type Maybe is defined as:

instance  Functor Maybe  where
    fmap _ Nothing       = Nothing       -- (1)
    fmap f (Just a)      = Just (f a)    -- (2)

The composition operator (.) is defined as:

(.) :: (b -> c) -> (a -> b) -> a -> c
f . g x = f (g x)                        -- (3)

The Identity function id is defined as:

id :: a -> a
id x =  x                                -- (4)

Claim

The claim is that Maybe fulfils the two functor laws:

1.: fmap id = id
2.: fmap (f . g) = (fmap f . fmap g)

Proof of the first law

Claim: fmap id m = id m, for any m of type Maybe a.

Proof. On cases of m.

Case 1: m = Nothing.

fmap id m = fmap id Nothing -- by expansion of m
          = Nothing         -- by applying equation (1)
          = id m            -- by definition m, by applying equation (4)

Case 2: m = Just a.

fmap id m = fmap id (Just a) -- by expansion of m
          = Just (id a)      -- by applying equation (2)
          = Just a           -- by expansion of id (equation (4))
          = m                -- by definition of m
          = id m             -- by applying equation (4)

Therefore, fmap id m = id m in all cases.∎

Proof of the second law

Claim: fmap (f . g) m = (fmap f . fmap g) m, for any m of type Maybe a.

Proof. On cases of m.

Case 1: m = Nothing.

fmap (f . g) m      = fmap (f . g) Nothing    -- by expansion of m
                    = Nothing                 -- by applying equation (1)
(fmap f . fmap g) m = fmap f (fmap g Nothing) -- by applying equation (4) and expanding m
                    = fmap f Nothing          -- by applying equation (1)
                    = Nothing                 -- by applying equation (1)

Case 2: m = Just a.

fmap (f . g) m      = fmap (f . g) (Just a)    -- by expansion of m
                    = Just ((f . g) a)         -- by applying equation (2)
(fmap f . fmap g) m = fmap f (fmap g (Just a)) -- by applying equation (4) and expanding m
                    = fmap f (Just (g a))      -- by applying equation (2)
                    = Just (f (g a)            -- by applying equation (2)
                    = Just ((f . g) a)         -- by applying equation (3)

Therefore, fmap (f . g) m = (fmap f . fmap g) m in all cases. ∎

Conclusion

You'll see this kind of reasoning quite a lot in Haskell documentation and online discussions. The simple reason is: if you can prove something you don't have to test it.