We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict Trs: { not(true()) -> false() , not(false()) -> true() , odd(0()) -> false() , odd(s(x)) -> not(odd(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , +(s(x), y) -> s(+(x, y)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The input was oriented with the instance of 'Small Polynomial Path Order (PS,1-bounded)' as induced by the safe mapping safe(not) = {1}, safe(true) = {}, safe(false) = {}, safe(odd) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {} and precedence odd > not . Following symbols are considered recursive: {odd, +} The recursion depth is 1. For your convenience, here are the satisfied ordering constraints: not(; true()) > false() not(; false()) > true() odd(0();) > false() odd(s(; x);) > not(; odd(x;)) +(x, 0();) > x +(x, s(; y);) > s(; +(x, y;)) +(s(; x), y;) > s(; +(x, y;)) Hurray, we answered YES(?,O(n^1))