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One way to look at this is that you have types which classify a structure/theory. In some sense, it works like this—you have generators which allow you to inductively define a large set of "things", then you have laws which cut those things down.

For instance, the theory of monoids gives rise to the generators

    zero : AMonoid
    +    : AMonoid -> AMonoid -> AMonoid
satisfying the laws

    zero + a = a
    a + zero = a
    a + (b + c) = (a + b) + c
Types can be refined and combined in principled ways by glomming the generators and laws together. Ultimately, what you're looking for is a notion of the set generated by all of the generators satisfying all of the laws plus any "coherence" laws which help to define what happens when two laws interact.

Unfortunately, nobody is very good at this today. It's a goal to be able to automatically achieve the above system in such a way that isn't O(n^2) effort to encode.




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