We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) , times(X, s(Y)) -> plus(X, times(Y, X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z))) , times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z))) , times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) } Strict Trs: { plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) , times(X, s(Y)) -> plus(X, times(Y, X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(plus) = {2}, Uargs(plus^#) = {2}, Uargs(c_1) = {1}, Uargs(c_2) = {1} TcT has computed the following constructor-restricted matrix interpretation. [plus](x1, x2) = [0 1] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] [times](x1, x2) = [1 2] x1 + [1 0] x2 + [0] [2 1] [2 0] [0] [s](x1) = [1 2] x1 + [2] [0 0] [0] [plus^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] [c_1](x1) = [1 0] x1 + [0] [0 1] [0] [times^#](x1, x2) = [2 1] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] [c_2](x1) = [1 0] x1 + [0] [0 1] [0] The order satisfies the following ordering constraints: [plus(plus(X, Y), Z)] = [0 1] X + [0 1] Y + [1 0] Z + [1] [0 1] [0 1] [0 1] [2] > [0 1] X + [0 1] Y + [1 0] Z + [0] [0 1] [0 1] [0 1] [2] = [plus(X, plus(Y, Z))] [times(X, s(Y))] = [1 2] X + [1 2] Y + [2] [2 1] [2 4] [4] > [1 1] X + [1 2] Y + [0] [2 1] [2 1] [1] = [plus(X, times(Y, X))] [plus^#(plus(X, Y), Z)] = [0 1] X + [0 1] Y + [1 0] Z + [1] [0 0] [0 0] [0 0] [0] > [0 1] X + [0 1] Y + [1 0] Z + [0] [0 0] [0 0] [0 0] [0] = [c_1(plus^#(X, plus(Y, Z)))] [times^#(X, s(Y))] = [2 1] X + [1 2] Y + [2] [0 0] [0 0] [0] > [1 1] X + [1 2] Y + [0] [0 0] [0 0] [0] = [c_2(plus^#(X, times(Y, X)))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z))) , times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) } Weak Trs: { plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) , times(X, s(Y)) -> plus(X, times(Y, X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z))) , times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { plus(plus(X, Y), Z) -> plus(X, plus(Y, Z)) , times(X, s(Y)) -> plus(X, times(Y, X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))