let R be the TRS under consideration f(g(_1),s(0),_2) -> f(g(s(0)),_2,g(_1)) is in elim_R(R) let l0 be the left-hand side of this rule p0 = 1.0 is a position in l0 we have l0|p0 = 0 g(0) -> 0 is in R let r'0 be the right-hand side of this rule theta0 = {} is a mgu of l0|p0 and r'0 ==> f(g(_1),s(g(0)),_2) -> f(g(s(0)),_2,g(_1)) is in EU_R^1 let l1 be the left-hand side of this rule p1 = 1 is a position in l1 we have l1|p1 = s(g(0)) g(s(_3)) -> s(g(_3)) is in R let r'1 be the right-hand side of this rule theta1 = {_3/0} is a mgu of l1|p1 and r'1 ==> f(g(_1),g(s(0)),_2) -> f(g(s(0)),_2,g(_1)) is in EU_R^2 let l be the left-hand side and r be the right-hand side of this rule let p = epsilon let theta = {_1/s(0), _2/g(s(0))} let theta' = {} we have r|p = f(g(s(0)),_2,g(_1)) and theta'(theta(l)) = theta(r|p) so, theta(l) = f(g(s(0)),g(s(0)),g(s(0))) is non-terminating w.r.t. R Termination disproved by the backward process proof stopped at iteration i=2, depth k=3 15 rule(s) generated