> To build a theorem prover you need to take away some capability (namely, the ability to do general recursion - the base language must be total and can't be Turing complete), not add new capabilities. In Haskell everything can be "undefined" which means that you can prove everything (even things that are supposed to be false).

Despite what the fanatical constructivists (as opposed to the ones who simply think it's pragmatically nice) seem to want us to think, it turns out that you can prove interesting things with LEM (AKA call/cc) and classical logic.