/prog/ challenge redux

Create a LISP program that generates random LISP programs.

No two runs of the program should generate the same code.

Generated code must be VALID LISP and should (given random input) return a value without throwing errors in such a way that the same input always returns the same output.

I almost want to sketch out a program of the halting solver. I always wondered, the answer for a FINITE problem solver which is feed infinite code is that no, it can't determine an infinite input, but by definition, since a finite problem solver will eventually find in random code an infinite problem solver, doesn't this mean that the finite problem solver does find a solution to all programs in finite time?