What Does “Without Loss of Generality” Mean, and How Do We Detect It

When one goes from a geometrical statement to an algebraic statement, the immediate translation is to replace every point by a pair of coordinates, if in the plane (or more as required). A statement with N points is then a statement with 2N (or 3N or more) variables, and the complexity of tools like cylindrical algebraic decomposition is doubly exponential in the number of variables. Hence one says “without loss of generality, A is at (0,0)” and so on. How might one automate this, in particular the detection that a “without loss of generality” argument is possible, or turn it into a procedure (and possibly even a formal proof)?