证明成分（（->）r）类型的适用法律

Question

The Composition Applicative Law is as follows:

pure (.) <*> u <*> v <*> w = u <*> (v <*> w)

Here's my attempt at proving the Composition law for the ((->) r) type:

RHS:

u <*> (v <*> w)
u <*> ( \y -> v y (w y) )
\x -> u x ( (\y -> v y (w y)) x )
\x -> u x ( v x (w x)) -- (A)

LHS:

pure (.) <*> u <*> v <*> w
const (.) <*> u <*> v <*> w
(\f -> const (.) f (u f)) <*> v <*> w
(\f -> (.) (u f)) <*> v <*> w
(\g -> (\f -> (.) (u f)) g (v g)) <*> w
\x -> (\g -> (\f -> (.) (u f)) g (v g)) x (w x)
-- Expanding labmda by applying to x
\x -> ((\f -> (.) (u f)) x (v x)) (w x)
\x -> (( (.) (u x)) (v x)) (w x)

\x -> ((u x) . (v x)) (w x) -- (B)

I don't think (A) & (B) are equivalent, so where did I make a mistake? I would appreciate any help or suggestions.

Answer 1

You're almost there. You just need to use the definition of (.) , which is

(f . g) x = f (g x)

After substituting that definition in the last line of your LHS calculation, you should have two obviously equal lambdas.