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    <p>Being an idle goodfornothing, I thought I would make a computer do the expansion for me. So into GHCi, I typed</p> <pre><code>let pu x = "(\\_ -&gt; " ++ x ++ ")" let f &gt;*&lt; a = "(\\g -&gt; " ++ f ++ " g (" ++ a ++ " g))" </code></pre> <p>So now I have funny versions of <code>pure</code> and <code>&lt;*&gt;</code> which map strings which look like expressions to string which look like more complicated expressions. I then defined, similarly, the analogue of <code>sequenceA</code>, replacing functions by strings.</p> <pre><code>let sqa [] = pu "[]" ; sqa (f : fs) = (pu "(:)" &gt;*&lt; f) &gt;*&lt; sqa fs </code></pre> <p>I was then able to generate the expanded form of the example as follows</p> <pre><code>putStrLn $ sqa ["(+3)","(+2)"] ++ " 3" </code></pre> <p>which duly printed</p> <pre><code>(\g -&gt; (\g -&gt; (\_ -&gt; (:)) g ((+3) g)) g ((\g -&gt; (\g -&gt; (\_ -&gt; (:)) g ((+2) g)) g ((\_ -&gt; []) g)) g)) 3 </code></pre> <p>This last, copied to the prompt, yielded</p> <pre><code>[6,5] </code></pre> <p>Comparing the output from my "metaprogram" with the attempt in the question shows a shorter initial prefix of lambdas, arising from a shallower nesting of <code>&lt;*&gt;</code> operations. Remember, it's</p> <pre><code>(pure (:) &lt;*&gt; (+3)) &lt;*&gt; ((pure (:) &lt;*&gt; (+2)) &lt;*&gt; pure []) </code></pre> <p>so the outer <code>(:)</code> should be only three lambdas deep. I suspect the proposed expansion may correspond to a differently bracketed version of the above, perhaps</p> <pre><code>pure (:) &lt;*&gt; (+3) &lt;*&gt; pure (:) &lt;*&gt; (+2) &lt;*&gt; pure [] </code></pre> <p>Indeed, when I evaluate</p> <pre><code>putStrLn $ pu "(:)" &gt;*&lt; "(+3)" &gt;*&lt; pu "(:)" &gt;*&lt; "(+2)" &gt;*&lt; pu "[]" ++ " 3 " </code></pre> <p>I get</p> <pre><code>(\g -&gt; (\g -&gt; (\g -&gt; (\g -&gt; (\_ -&gt; (:)) g ((+3) g)) g ((\_ -&gt; (:)) g)) g ((+2) g)) g ((\_ -&gt; []) g)) 3 </code></pre> <p>which looks like it matches the (updated)</p> <pre><code>(\d -&gt; (\c -&gt; (\b -&gt; (\a -&gt; (\_ -&gt; (:)) a ((+3) a)) b ((\_ -&gt; (:)) b)) c ((+2) c)) d ((\_ -&gt; []) d)) 3 </code></pre> <p>I hope this machine-assisted investigation helps to clarify what's going on.</p>
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    1. COYour methodology is over my head, but that's definitely the answer to my first question. Any thoughts about the second? (Translating to (.) notation)
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    2. COMy methodology is just to mechanize the process of "writing out the expansion" by generating the relevant expressions as a string. Where `pure` computes the constant function, `pu` computes the *text* of the constant function. Meanwhile my main thought about translation in terms of constant functions is that it won't be particularly enlightening to do so. Let's start just by expressing `<*>` via composition, and...it's a nuisance because the 3rd (*environment*) argument is copied. We need a duplicator, e.g. `join f a = f a a`, and then `(<*>) = (join .) . ((.) . flip)` which ain't much light.
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    3. COIn case you think I figured that out by myself, I'd better point you to this handy page about combinatory logic: http://en.wikipedia.org/wiki/B,C,K,W_System The `Applicative` instance for `((->) a)` takes `pure` as the K combinator, `<*>` as the S combinator. Translating applicative-style expressions to composition amounts to translating SK-combinator terms to BCKW-combinator terms, which is doable (both systems are complete) but grungy.
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