With respect to very precise type systems, and coupling between program structure and types, one of the reason mathematics is so powerful is its flexibility
I actually have trouble with mathematics, because it is so flexible. For example, equality of sets A and B is typically shown by saying that every element of A is in B, and every element in B is in A. Why on earth do they do that? Why not just say they're the same??!
One advantage is it gives you the flexibility to demonstrate the first limb using one technique, and the second limb with a completely unrelated approach. I saw one example of showing equivalence of language defined by a class of grammars, and a language defined by constraints over sequences. You could even use a constructive and a non-constructive proof for each half.
Comments
With respect to very precise type systems, and coupling between program structure and types, one of the reason mathematics is so powerful is its flexibility
I actually have trouble with mathematics, because it is so flexible. For example, equality of sets A and B is typically shown by saying that every element of A is in B, and every element in B is in A. Why on earth do they do that? Why not just say they're the same??!
One advantage is it gives you the flexibility to demonstrate the first limb using one technique, and the second limb with a completely unrelated approach. I saw one example of showing equivalence of language defined by a class of grammars, and a language defined by constraints over sequences. You could even use a constructive and a non-constructive proof for each half.
It's a way of subdividing that is decoupled.