Personally, dependent types never worked for me the way I thought they would when I first learnt about them (I was quite enthusiastic about them and about const

Personally, dependent types never worked for me the way I thought they would when I first learnt about them (I was quite enthusiastic about them and about constructive mathematics initially). I found them too verbose, and too nitpicky. Your list membership is a great example: Usually, I am not concerned with where in the list does the element occur, list membership as a predicate is perfectly natural and usually adequate (but admittedly, if that's the only use case, better use a set than a list). I think an elegant mathematical argument wins not only by what is said, but also by what can be left unsaid. But yes, being able to work with functions as data is also very important! That's why I am pursuing a combination of simple type theory and set theory as the foundation of a theorem prover, as I have outlined here: http://www.practal.com http://www.practal.com