cognitive psychology. PhD 
Combinatorics problem: you're arranging 10 people (A .. J) in a row. A and B must sit next to each other. How many arrangements? How could writing this problem as a program help scaffold the problem-solving process?
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But subtractive is hard to reason about analytically -- how to compute len(..)? Then we rewrite additively: permutations(['AB'] + 'C' .. 'J') + permutations(['BA'] + 'C' .. 'J') Now we can easily derive the formula permutations(9) * 2.
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Maybe this is actually a compiler problem: how can you rewrite the subtractive program (which naturally falls out of the problem statement) into an additive one (which is easily countable)? i.e. how do you remove all conditional expressions?
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hmm... makes me think of the examples ive seen about squiggol (https://en.wikipedia.org/wiki/Bird%E2%80%93Meertens_formalism …), which i know less about than i ought to
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this kind of dichotomy is also definitely very present when working in a proof assistant—i wonder whether it's been pinned down in the proof automation/engineering literature?
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I love how you framed the additive vs. subtractive dichotomy which I think reaches into a whole class of problems. Would love to hear if anyone knows of any documented literature on this. As you have rightly pointed out, this seems like it plays out in multiple domains.
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I could imagine a nice set of rewrite rules that captures most of the reasoning, and one could teach these rules to students. This sketch connects the subtractive and additive versions. The two equalities allow you to simplify perm . keep in different ways.pic.twitter.com/I16CP0w1eJ
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In many domains, rewrites reify insight. In my experience, they make some tricks more tangible, generalizable, and easier to prove and reason about.
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