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: runtime complexity Answer: YES(O(1),O(n^1)) We add the 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(y) } 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(y) } 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: 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}, Uargs(c_3) = {1} TcT has computed the following constructor-restricted matrix interpretation. [admit](x1, x2) = [2 0] x2 + [0] [0 0] [0] [nil] = [2] [0] [.](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] [w] = [2] [0] [cond](x1, x2) = [0 1] x1 + [1 0] x2 + [0] [0 0] [0 2] [0] [=](x1, x2) = [0] [0] [sum](x1, x2, x3) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [0 0] [0] [carry](x1, x2, x3) = [0] [0] [true] = [0] [1] [admit^#](x1, x2) = [2 1] x1 + [2 0] x2 + [2] [1 1] [0 0] [2] [c_1] = [1] [1] [c_2](x1) = [1 0] x1 + [2] [0 1] [2] [cond^#](x1, x2) = [1 0] x2 + [2] [0 0] [2] [c_3](x1) = [1 0] x1 + [1] [0 1] [1] The following symbols are considered usable {admit, cond, admit^#, cond^#} The order satisfies the following ordering constraints: [admit(x, nil())] = [4] [0] > [2] [0] = [nil()] [admit(x, .(u, .(v, .(w(), z))))] = [2 0] u + [2 0] v + [2 0] z + [4] [0 0] [0 0] [0 0] [0] > [1 0] u + [1 0] v + [2 0] z + [2] [0 0] [0 0] [0 0] [0] = [cond(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z)))))] [cond(true(), y)] = [1 0] y + [1] [0 2] [0] > [1 0] y + [0] [0 1] [0] = [y] [admit^#(x, nil())] = [2 1] x + [6] [1 1] [2] > [1] [1] = [c_1()] [admit^#(x, .(u, .(v, .(w(), z))))] = [2 1] x + [2 0] u + [2 0] v + [2 0] z + [6] [1 1] [0 0] [0 0] [0 0] [2] ? [1 0] u + [1 0] v + [2 0] z + [6] [0 0] [0 0] [0 0] [4] = [c_2(cond^#(=(sum(x, u, v), w()), .(u, .(v, .(w(), admit(carry(x, u, v), z))))))] [cond^#(true(), y)] = [1 0] y + [2] [0 0] [2] ? [1 0] y + [1] [0 1] [1] = [c_3(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),O(n^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)))))) , cond^#(true(), y) -> c_3(y) } Weak DPs: { admit^#(x, nil()) -> c_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: runtime complexity Answer: YES(O(1),O(n^1)) We estimate the number of application of {1} by applications of Pre({1}) = {2}. 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: cond^#(true(), y) -> c_3(y) , 3: admit^#(x, nil()) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { cond^#(true(), y) -> c_3(y) } 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)))))) } 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: runtime complexity Answer: YES(O(1),O(n^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)))))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { cond^#(true(), y) -> c_3(y) } 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: 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: { cond^#(true(), y) -> c_3(y) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: cond^#(true(), y) -> c_3(y) } Sub-proof: ---------- The following argument positions are usable: none TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [admit](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [.](x1, x2) = [1] x1 + [1] x2 + [0] [w] = [0] [cond](x1, x2) = [7] x1 + [7] x2 + [0] [=](x1, x2) = [1] x1 + [1] 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] = [3] [admit^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [cond^#](x1, x2) = [3] x1 + [7] x2 + [3] [c_3](x1) = [7] x1 + [3] The following symbols are considered usable {cond^#} The order satisfies the following ordering constraints: [cond^#(true(), y)] = [7] y + [12] > [7] y + [3] = [c_3(y)] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { cond^#(true(), y) -> c_3(y) } Obligation: 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. { cond^#(true(), y) -> c_3(y) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))