YES proof: The input problem is infeasible because [1] the following set of Horn clauses is satisfiable: even(x) = true ==> odd(s(x)) = true even(x) = false ==> odd(s(x)) = false odd(0) = false odd(x) = true ==> even(s(x)) = true odd(x) = false ==> even(s(x)) = false even(0) = true even(x1) = false & even(x1) = true ==> true__ = false__ true__ = false__ ==> \bottom This holds because [2] the following E does not entail the following G (Claessen-Smallbone's transformation (2018)): E: even(0) = true g1(even(x), x) = odd(s(x)) g1(false, x) = false g1(true, x) = true g2(false, x) = false g2(odd(x), x) = even(s(x)) g2(true, x) = true odd(0) = false t1(even(x1), even(x1)) = true__ t1(false, true) = false__ t2(false__) = true__ t2(true__) = false__ G: true__ = false__ This holds because [3] the following ground-complete ordered TRS entails E but does not entail G: even(s(X0)) -> g2(odd(X0), X0) g1(even(0), X0) -> even(0) g1(false, X0) -> false g2(even(0), X0) -> even(0) g2(false, X0) -> false odd(0) -> false odd(s(X0)) -> g1(even(X0), X0) t1(Z1, Z1) -> true__ t1(false, even(0)) -> false__ t1(false, false) -> true__ t1(g2(Z1, X0), g2(Z1, X0)) -> true__ t2(false__) -> true__ t2(true__) -> false__ true -> even(0) with the LPO induced by t1 > s > g2 > true > even > odd > g1 > false > 0 > t2 > false__ > true__