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Strilanc comments on Progress on automated mathematical theorem proving? - Less Wrong
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Constraint programming would be extremely powerful. They aren't a silver bullet, but they would make a ton of hard problems a lot easier. Particularly when it comes to proving whole-program correctness properties like "does not use more than X memory".
For example, when a human programmer is designing a public key cryptosystem, they're trying to satisfy properties like "there is no tractable Turing Machine that takes a public key as input and outputs the corresponding private key with non-negligible probability". Doing that is really, really hard compared to just writing down what we want to satisfy.
But that's going to take an insanely powerful theorem-prover. Also, this is more the theorem-proving end than automated program end.
It's true that it would need to be insanely powerful. Even clever humans haven't managed to prove that one-way hash functions exist (though we know a function that is one-way if and only if they exist...).
Note theorem proving is isomorphic to programming. Even if it wasn't quite so equivalent, you could just ask for a constructive proof of a program satisfying whatever and the proof would contain the program.