Comment by wizzwizz4

These questions are hard to understand.

1. Yes, the Sledgehammer suite contains three testing systems that I believe are concolic (quickcheck, nitpick, nunchaku), but they only work due to hand-coded support for the mathematical constructs in question. They'd be really inefficient for a formalised software environment, because they'd be operating at a much lower-level than the abstractions of the software environment, unless dedicated support for that software environment were provided.

2. Quickcheck is a fuzz-testing framework, but it doesn't help to construct anything (except as far as it constructs examples / counterexamples). Are you thinking of something that automatically finds and defines intermediary lemmas for arbitrary areas of mathematics? Because I'm not aware of any particular work in that direction: if computers could do that, there'd be little need for mathematicians.

3. By thinking really hard, then writing down the mathematics. Same way you write computer programs, really, except there's a lot more architectural work. (Most of the time, the computer can brute-force a proof for you, so you need only choose appropriate intermediary lemmas.)

4. I can't parse the question, but I suspect you're thinking of the meta-logic / object-logic distinction. The actual steps in the term rewriting are not represented in the object logic: the meta-logic simply asserts that it's valid to perform these steps. (Not even that: it just does them, accountable to none other than itself.) Isabelle's meta-logic is software, written in a programming language called ML.