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Using list xs :: [Int], prove the definition does not satisfy the law

Using list xs :: [Int], prove the definition does not satisfy the law

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Question

Given the definition in Haskell:

fmap :: (a -> b) -> [a] -> [b]

fmap g xs = reverse (map g xs)

 

With Functor law:

(fmap g . fmap h) xs = fmap (g . h) xs

 

Using list xs :: [Int], prove the definition does not satisfy the law:

Expert Solution
Step 1

A functor is a function that takes a function, fmap (say) that returns another function. fmap() is also known as function map

 

Let's see the functor instance for a list with id.

fmap id (x: xs)

= (id x) : fmap xs

= x : fmap xs

 

As per the question, fmap id (x: xs) = (x: xs) which satisfies the first law, we can conclude that

= (fmap g).(fmap h) (x : xs)

= fmap g (fmap h (x: xs))

= fmap g ((h x) = fmap h xs)

= (g.h x) = fmap g(fmap h xs) which doesn't satisfies the law.

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