YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) We add the following weak dependency pairs: Strict DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, 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^2)). Strict DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1} by applications of Pre({1}) = {3}. Here rules are labeled as follows: DPs: { 1: sum^#(0()) -> c_1() , 2: sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , 3: +^#(x, 0()) -> c_3(x) , 4: +^#(x, s(y)) -> c_4(+^#(x, y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sum](x1) = [0] [0] = [5] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [0] [sum^#](x1) = [1] x1 + [7] [c_1] = [4] [c_2](x1) = [1] x1 + [3] [+^#](x1, x2) = [1] x1 + [0] [c_3](x1) = [1] x1 + [7] [c_4](x1) = [1] x1 + [7] The following symbols are considered usable {sum, +, sum^#, +^#} The order satisfies the following ordering constraints: [sum(0())] = [0] ? [5] = [0()] [sum(s(x))] = [0] >= [0] = [+(sum(x), s(x))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] >= [1] x + [0] = [s(+(x, y))] [sum^#(0())] = [12] > [4] = [c_1()] [sum^#(s(x))] = [1] x + [7] > [3] = [c_2(+^#(sum(x), s(x)))] [+^#(x, 0())] = [1] x + [0] ? [1] x + [7] = [c_3(x)] [+^#(x, s(y))] = [1] x + [0] ? [1] x + [7] = [c_4(+^#(x, 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^2)). Strict DPs: { +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sum](x1) = [0] [0] = [1] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [0] [sum^#](x1) = [1] x1 + [7] [c_1] = [0] [c_2](x1) = [1] x1 + [3] [+^#](x1, x2) = [1] x1 + [1] x2 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [7] The following symbols are considered usable {sum, +, sum^#, +^#} The order satisfies the following ordering constraints: [sum(0())] = [0] ? [1] = [0()] [sum(s(x))] = [0] >= [0] = [+(sum(x), s(x))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] >= [1] x + [0] = [s(+(x, y))] [sum^#(0())] = [8] > [0] = [c_1()] [sum^#(s(x))] = [1] x + [7] > [1] x + [3] = [c_2(+^#(sum(x), s(x)))] [+^#(x, 0())] = [1] x + [1] > [1] x + [0] = [c_3(x)] [+^#(x, s(y))] = [1] x + [1] y + [0] ? [1] x + [1] y + [7] = [c_4(+^#(x, 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^2)). Strict DPs: { +^#(x, s(y)) -> c_4(+^#(x, y)) } Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sum](x1) = [3] [0] = [5] [s](x1) = [1] x1 + [7] [+](x1, x2) = [1] x1 + [7] [sum^#](x1) = [1] x1 + [7] [c_1] = [4] [c_2](x1) = [1] x1 + [1] [+^#](x1, x2) = [1] x1 + [1] x2 + [3] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [5] The following symbols are considered usable {sum, +, sum^#, +^#} The order satisfies the following ordering constraints: [sum(0())] = [3] ? [5] = [0()] [sum(s(x))] = [3] ? [10] = [+(sum(x), s(x))] [+(x, 0())] = [1] x + [7] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [7] ? [1] x + [14] = [s(+(x, y))] [sum^#(0())] = [12] > [4] = [c_1()] [sum^#(s(x))] = [1] x + [14] >= [1] x + [14] = [c_2(+^#(sum(x), s(x)))] [+^#(x, 0())] = [1] x + [8] > [1] x + [0] = [c_3(x)] [+^#(x, s(y))] = [1] x + [1] y + [10] > [1] x + [1] y + [8] = [c_4(+^#(x, 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^2)). Strict Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Weak Trs: { +(x, 0()) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sum](x1) = [1] x1 + [4] [0] = [5] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [4] [sum^#](x1) = [1] x1 + [7] [c_1] = [4] [c_2](x1) = [1] x1 + [0] [+^#](x1, x2) = [1] x1 + [3] [c_3](x1) = [1] x1 + [3] [c_4](x1) = [1] x1 + [0] The following symbols are considered usable {sum, +, sum^#, +^#} The order satisfies the following ordering constraints: [sum(0())] = [9] > [5] = [0()] [sum(s(x))] = [1] x + [4] ? [1] x + [8] = [+(sum(x), s(x))] [+(x, 0())] = [1] x + [4] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [4] >= [1] x + [4] = [s(+(x, y))] [sum^#(0())] = [12] > [4] = [c_1()] [sum^#(s(x))] = [1] x + [7] >= [1] x + [7] = [c_2(+^#(sum(x), s(x)))] [+^#(x, 0())] = [1] x + [3] >= [1] x + [3] = [c_3(x)] [+^#(x, s(y))] = [1] x + [3] >= [1] x + [3] = [c_4(+^#(x, 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^2)). Strict Trs: { sum(s(x)) -> +(sum(x), s(x)) , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Weak Trs: { sum(0()) -> 0() , +(x, 0()) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sum](x1) = [1] x1 + [4] [0] = [7] [s](x1) = [1] x1 + [4] [+](x1, x2) = [1] x1 + [0] [sum^#](x1) = [1] x1 + [7] [c_1] = [5] [c_2](x1) = [1] x1 + [3] [+^#](x1, x2) = [1] x1 + [3] [c_3](x1) = [1] x1 + [3] [c_4](x1) = [1] x1 + [0] The following symbols are considered usable {sum, +, sum^#, +^#} The order satisfies the following ordering constraints: [sum(0())] = [11] > [7] = [0()] [sum(s(x))] = [1] x + [8] > [1] x + [4] = [+(sum(x), s(x))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] ? [1] x + [4] = [s(+(x, y))] [sum^#(0())] = [14] > [5] = [c_1()] [sum^#(s(x))] = [1] x + [11] > [1] x + [10] = [c_2(+^#(sum(x), s(x)))] [+^#(x, 0())] = [1] x + [3] >= [1] x + [3] = [c_3(x)] [+^#(x, s(y))] = [1] x + [3] >= [1] x + [3] = [c_4(+^#(x, 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^2)). Strict Trs: { +(x, s(y)) -> s(+(x, y)) } Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Weak Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x } Obligation: runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { +(x, s(y)) -> s(+(x, y)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are considered usable: Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [sum](x1) = x1 + x1^2 [0]() = 0 [s](x1) = 1 + x1 [+](x1, x2) = x1 + 2*x2 [sum^#](x1) = 3*x1 + x1^2 [c_1]() = 0 [c_2](x1) = x1 [+^#](x1, x2) = x1 [c_3](x1) = x1 [c_4](x1) = x1 The following symbols are considered usable {sum, +} This order satisfies the following ordering constraints. [sum(0())] = >= = [0()] [sum(s(x))] = 2 + 3*x + x^2 >= 3*x + x^2 + 2 = [+(sum(x), s(x))] [+(x, 0())] = x >= x = [x] [+(x, s(y))] = x + 2 + 2*y > 1 + x + 2*y = [s(+(x, y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { sum^#(0()) -> c_1() , sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) , +^#(x, 0()) -> c_3(x) , +^#(x, s(y)) -> c_4(+^#(x, y)) } Weak Trs: { sum(0()) -> 0() , sum(s(x)) -> +(sum(x), s(x)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))