We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere(2018) so that it works for any gadget with high enough rank . We apply our generalizedtheorem to solve two open problems: We present the first result that demonstrates a separation in proof powerfor cutting planes with unbounded versus polynomially bounded coefficients . * We give the first explicit separation between monotonerary formulas andmonotone real formulas . Previously only anon-explicit separation was known . An important technical ingredient, which may be of independent interest, is that we show that the . standard decision treecomplexity and the parity decision tree complexity of the . corresponding decision . tree complexity are equal . In particular, this implies that the standard . decision tree complexity and . the . parity decision Tree complexity of . the correspondingfalsified clause search problem are equal are equal. In addition, we give an explicit family of functionsthat can be computed with monotronary formulas of nearly linear