YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { f(a()) -> f(c(a())) , f(a()) -> f(d(a())) , f(c(X)) -> X , f(c(a())) -> f(d(b())) , f(c(b())) -> f(d(a())) , f(d(X)) -> X , e(g(X)) -> e(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(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: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(X)) } Strict Trs: { f(a()) -> f(c(a())) , f(a()) -> f(d(a())) , f(c(X)) -> X , f(c(a())) -> f(d(b())) , f(c(b())) -> f(d(a())) , f(d(X)) -> X , e(g(X)) -> e(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^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(n^1)). Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(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(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_7) = {1} TcT has computed following constructor-restricted matrix interpretation. [a] = [0] [c](x1) = [0] [d](x1) = [0] [b] = [0] [g](x1) = [1] x1 + [1] [f^#](x1) = [0] [c_1](x1) = [1] x1 + [2] [c_2](x1) = [1] x1 + [2] [c_3] = [1] [c_4](x1) = [1] x1 + [2] [c_5](x1) = [1] x1 + [2] [c_6] = [1] [e^#](x1) = [2] x1 + [2] [c_7](x1) = [1] x1 + [1] This order satisfies following ordering constraints: 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)). Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() } Weak DPs: { e^#(g(X)) -> c_7(e^#(X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We estimate the number of application of {3,6} by applications of Pre({3,6}) = {1,2,4,5}. Here rules are labeled as follows: DPs: { 1: f^#(a()) -> c_1(f^#(c(a()))) , 2: f^#(a()) -> c_2(f^#(d(a()))) , 3: f^#(c(X)) -> c_3() , 4: f^#(c(a())) -> c_4(f^#(d(b()))) , 5: f^#(c(b())) -> c_5(f^#(d(a()))) , 6: f^#(d(X)) -> c_6() , 7: e^#(g(X)) -> c_7(e^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) } Weak DPs: { f^#(c(X)) -> c_3() , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We estimate the number of application of {2,3,4} by applications of Pre({2,3,4}) = {1}. Here rules are labeled as follows: DPs: { 1: f^#(a()) -> c_1(f^#(c(a()))) , 2: f^#(a()) -> c_2(f^#(d(a()))) , 3: f^#(c(a())) -> c_4(f^#(d(b()))) , 4: f^#(c(b())) -> c_5(f^#(d(a()))) , 5: f^#(c(X)) -> c_3() , 6: f^#(d(X)) -> c_6() , 7: e^#(g(X)) -> c_7(e^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict DPs: { f^#(a()) -> c_1(f^#(c(a()))) } Weak DPs: { f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We estimate the number of application of {1} by applications of Pre({1}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(a()) -> c_1(f^#(c(a()))) , 2: f^#(a()) -> c_2(f^#(d(a()))) , 3: f^#(c(X)) -> c_3() , 4: f^#(c(a())) -> c_4(f^#(d(b()))) , 5: f^#(c(b())) -> c_5(f^#(d(a()))) , 6: f^#(d(X)) -> c_6() , 7: e^#(g(X)) -> c_7(e^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(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. { f^#(a()) -> c_1(f^#(c(a()))) , f^#(a()) -> c_2(f^#(d(a()))) , f^#(c(X)) -> c_3() , f^#(c(a())) -> c_4(f^#(d(b()))) , f^#(c(b())) -> c_5(f^#(d(a()))) , f^#(d(X)) -> c_6() , e^#(g(X)) -> c_7(e^#(X)) } 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))