> If the CAS "shows" that MIN(r(*)...) >0 has no solutions, > we are done, but there are here 2 possibilities > - a solution does not exist > - the CAS was not able to find one > Which one should we consider? >
You are right. It could be a bug if CAS said there is no solution, but there was one and it just could not find it.
Any way, Mathematica said there is no solution for MIN(...)>0 so, may be it has a bug or may be it is true. We do not know, you are right.
So one needs to do real proof, like in school, on paper and pencil to be real sure. (ofcourse, a proof by human can also have a bug, so we need to have review by others of the proof, just to be more sure it is correct)