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 odd(x1) = false & odd(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(false, true) = false__ t1(odd(x1), odd(x1)) = true__ 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(0) -> true even(s(X0)) -> g2(odd(X0), X0) false -> odd(0) g1(odd(0), X0) -> odd(0) g1(true, X0) -> true g2(odd(0), X0) -> odd(0) g2(true, X0) -> true odd(s(X0)) -> g1(even(X0), X0) t1(Z1, Z1) -> true__ t1(g1(Z1, X0), g1(Z1, X0)) -> true__ t1(odd(0), true) -> false__ t1(true, true) -> true__ t2(false__) -> true__ t2(true__) -> false__ with the LPO induced by t1 > s > false > odd > g1 > even > g2 > true > 0 > t2 > false__ > true__