Endomorphisms for non-trivial non-linear loop invariant generation

Present approaches for non-linear loop invariant generation are limited to linear (affine) systems, or they relay on non scalable methods which have high complexity. Moreover, for programs with nested loops and conditional statements that describe multivariate polynomials or multivariate fractional systems, no applicable method is known to lend itself to non-trivial non-linear invariants generation. We demonstrate a powerful computational complete method to solve this problem. Our approach avoids first-order quantifier elimination, cylindrical algebraic decomposition and Grobner bases computation, hereby circumventing difficulties met by recent methods.