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: { *(x, +(y, z)) -> +(*(x, y), *(x, z)) , *(x, 1()) -> x , *(+(x, y), z) -> +(*(x, z), *(y, z)) , *(1(), y) -> y } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following innermost weak dependency pairs: Strict DPs: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(x, 1()) -> c_2() , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) , *^#(1(), y) -> c_4() } 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: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(x, 1()) -> c_2() , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) , *^#(1(), y) -> c_4() } Strict Trs: { *(x, +(y, z)) -> +(*(x, y), *(x, z)) , *(x, 1()) -> x , *(+(x, y), z) -> +(*(x, z), *(y, z)) , *(1(), y) -> y } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) 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^2)). Strict DPs: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(x, 1()) -> c_2() , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) , *^#(1(), y) -> c_4() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [+](x1, x2) = [1 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [1] [1] = [2] [1] [*^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 2] [2 2] [1] [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [1] [c_2] = [0] [1] [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [1] [c_4] = [0] [1] The following symbols are considered usable {*^#} The order satisfies the following ordering constraints: [*^#(x, +(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] [1 2] [2 6] [2 6] [5] ? [0 0] x + [0 0] y + [0 0] z + [3] [2 4] [2 2] [2 2] [3] = [c_1(*^#(x, y), *^#(x, z))] [*^#(x, 1())] = [0 0] x + [1] [1 2] [7] > [0] [1] = [c_2()] [*^#(+(x, y), z)] = [0 0] x + [0 0] y + [0 0] z + [1] [1 4] [1 4] [2 2] [4] ? [0 0] x + [0 0] y + [0 0] z + [3] [1 2] [1 2] [4 4] [3] = [c_3(*^#(x, z), *^#(y, z))] [*^#(1(), y)] = [0 0] y + [1] [2 2] [5] > [0] [1] = [c_4()] 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, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } Weak DPs: { *^#(x, 1()) -> c_2() , *^#(1(), y) -> c_4() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { *^#(x, 1()) -> c_2() , *^#(1(), y) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'polynomial interpretation' to orient following rules strictly. DPs: { 2: *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } Sub-proof: ---------- The following argument positions are considered usable: Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [*](x1, x2) = 3*x1 + 3*x1*x2 + 3*x1^2 + 3*x2 + 3*x2^2 [+](x1, x2) = 1 + x1 + x2 [1]() = 0 [*^#](x1, x2) = x1 + x1*x2 [c_1](x1, x2) = x1 + x2 [c_2]() = 0 [c_3](x1, x2) = x1 + x2 [c_4]() = 0 The following symbols are considered usable {*^#} This order satisfies the following ordering constraints. [*^#(x, +(y, z))] = 2*x + x*y + x*z >= 2*x + x*y + x*z = [c_1(*^#(x, y), *^#(x, z))] [*^#(+(x, y), z)] = 1 + x + y + z + x*z + y*z > x + x*z + y + y*z = [c_3(*^#(x, z), *^#(y, z))] 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(n^2)). Strict DPs: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) } Weak DPs: { *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'polynomial interpretation' to orient following rules strictly. DPs: { 1: *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) } Sub-proof: ---------- The following argument positions are considered usable: Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [*](x1, x2) = 3*x1 + 3*x1*x2 + 3*x1^2 + 3*x2 + 3*x2^2 [+](x1, x2) = 1 + x1 + x2 [1]() = 0 [*^#](x1, x2) = x1*x2 + x2 [c_1](x1, x2) = x1 + x2 [c_2]() = 0 [c_3](x1, x2) = x1 + x2 [c_4]() = 0 The following symbols are considered usable {*^#} This order satisfies the following ordering constraints. [*^#(x, +(y, z))] = x + x*y + x*z + 1 + y + z > x*y + y + x*z + z = [c_1(*^#(x, y), *^#(x, z))] [*^#(+(x, y), z)] = 2*z + x*z + y*z >= x*z + 2*z + y*z = [c_3(*^#(x, z), *^#(y, z))] 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: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } 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. { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) } 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^2))