I should mention that constructive mathematics, at least when done in dependent type theory, really starts to shine when things that normal mathematicians think of as propositions, are viewed as data. For example, normal mathematicians think of list membership as a predicate, but in dependent type theory, list membership is a value of an inductive family that is effectively an index of where that value occurs in that list (and in particular there could be multiple different values if that item occurs in the list at multiple different locations).
And dependent type theory is gleaming when things that normal mathematicians think of as propositions are not only values, but functions! For example a list l1 being a subset of a list l2 is a function from memberships of l1 (which, remember is data) to memberships of l2. Now you get partial evaluation when working on your proofs and things just magically unify.
There is a practical difference between Principia Mathematica taking 370 pages to prove 1 + 1 = 2 versus dependent type theory's proof `refl_equal 2`. (I'm exaggerating, but only a little).
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
And dependent type theory is gleaming when things that normal mathematicians think of as propositions are not only values, but functions! For example a list l1 being a subset of a list l2 is a function from memberships of l1 (which, remember is data) to memberships of l2. Now you get partial evaluation when working on your proofs and things just magically unify.
I write about these examples in experience I had in "Taking Propositions as Types seriously" @ <http://r6.ca/blog/20171008T222703Z.html>.
There is a practical difference between Principia Mathematica taking 370 pages to prove 1 + 1 = 2 versus dependent type theory's proof `refl_equal 2`. (I'm exaggerating, but only a little).