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: { admit(x, nil()) -> nil() , admit(x, .(u, .(v, .(w(), z)))) -> cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))) , cond(true(), y) -> y } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { admit^#(x, nil()) -> c_1() , admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) , cond^#(true(), y) -> c_3() } 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: { admit^#(x, nil()) -> c_1() , admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) , cond^#(true(), y) -> c_3() } Strict Trs: { admit(x, nil()) -> nil() , admit(x, .(u, .(v, .(w(), z)))) -> cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))) , cond(true(), y) -> y } 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(.) = {2}, Uargs(cond) = {2}, Uargs(c_2) = {1}, Uargs(cond^#) = {2} TcT has computed following constructor-restricted matrix interpretation. [admit](x1, x2) = [2] x2 + [1] [nil] = [1] [.](x1, x2) = [1] x1 + [1] x2 + [0] [w] = [2] [cond](x1, x2) = [1] x2 + [1] [=](x1, x2) = [0] [sum](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [carry](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [true] = [2] [admit^#](x1, x2) = [1] x1 + [2] x2 + [2] [c_1] = [1] [c_2](x1) = [1] x1 + [2] [cond^#](x1, x2) = [1] x2 + [1] [c_3] = [0] This order satisfies following ordering constraints: [admit(x, nil())] = [3] > [1] = [nil()] [admit(x, .(u, .(v, .(w(), z))))] = [2] u + [2] v + [2] z + [5] > [1] u + [1] v + [2] z + [4] = [cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))] [cond(true(), y)] = [1] y + [1] > [1] y + [0] = [y] 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)). Strict DPs: { admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) } Weak DPs: { admit^#(x, nil()) -> c_1() , cond^#(true(), y) -> c_3() } Weak Trs: { admit(x, nil()) -> nil() , admit(x, .(u, .(v, .(w(), z)))) -> cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))) , cond(true(), y) -> y } Obligation: innermost runtime complexity Answer: YES(?,O(1)) We estimate the number of application of {1} by applications of Pre({1}) = {}. Here rules are labeled as follows: DPs: { 1: admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) , 2: admit^#(x, nil()) -> c_1() , 3: cond^#(true(), y) -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak DPs: { admit^#(x, nil()) -> c_1() , admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) , cond^#(true(), y) -> c_3() } Weak Trs: { admit(x, nil()) -> nil() , admit(x, .(u, .(v, .(w(), z)))) -> cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))) , cond(true(), y) -> y } Obligation: innermost runtime complexity Answer: YES(?,O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { admit^#(x, nil()) -> c_1() , admit^#(x, .(u, .(v, .(w(), z)))) -> c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) , cond^#(true(), y) -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak Trs: { admit(x, nil()) -> nil() , admit(x, .(u, .(v, .(w(), z)))) -> cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))) , cond(true(), y) -> y } Obligation: innermost runtime complexity Answer: YES(?,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)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(?,O(1)) The input was oriented with the instance of 'Small Polynomial Path Order (PS)' as induced by the safe mapping and precedence empty . Following symbols are considered recursive: {} The recursion depth is 0. Further, following argument filtering is employed: empty Usable defined function symbols are a subset of: {} For your convenience, here are the satisfied ordering constraints: Hurray, we answered YES(O(1),O(n^1))