Sure,

A program that looks at a pattern, emulates it and then enumerates the various edge-cases, looking for further patterns from the programmer's answer.

It would still require effort but it could be a lot easier than present approaches and it might even find edge-cases a human would miss.

    > A program that looks at a pattern [...]

This is what programming already is, the "pattern" being code. A compiler is a program that looks at a code pattern and, if it's a good compiler, it can tell you if you've missed edge-cases through compile time errors.

That's the traditional focus of programming, but I wouldn't say that makes it the whole point. I very much expect programming to change over time