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)) -> *(otimes(x, y), z) , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) , *(1(), y) -> y , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(1(), y) -> c_3() , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(1(), y) -> c_3() , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } Strict Trs: { *(x, *(y, z)) -> *(otimes(x, y), z) , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) , *(1(), y) -> y , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(1(), y) -> c_3() , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 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}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [*](x1, x2) = [1 1] x1 + [1 1] x2 + [1] [2 2] [2 2] [1] [otimes](x1, x2) = [1 2] x1 + [1 2] x2 + [2] [0 1] [0 1] [1] [1] = [2] [1] [+](x1, x2) = [1 2] x1 + [1 2] x2 + [2] [0 1] [0 1] [1] [oplus](x1, x2) = [1 1] x1 + [1 1] x2 + [1] [0 1] [0 1] [1] [*^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 2] [2 1] [1] [c_1](x1) = [1 0] x1 + [1] [0 1] [1] [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [1] [c_3] = [0] [1] [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [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] [4 4] [4 4] [4] ? [0 0] x + [0 0] y + [0 0] z + [2] [1 4] [1 4] [2 1] [6] = [c_1(*^#(otimes(x, y), z))] [*^#(x, oplus(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] [1 2] [2 3] [2 3] [4] ? [0 0] x + [0 0] y + [0 0] z + [3] [2 4] [2 1] [2 1] [3] = [c_2(*^#(x, y), *^#(x, z))] [*^#(1(), y)] = [0 0] y + [1] [2 1] [5] > [0] [1] = [c_3()] [*^#(+(x, y), z)] = [0 0] x + [0 0] y + [0 0] z + [1] [1 4] [1 4] [2 1] [5] ? [0 0] x + [0 0] y + [0 0] z + [3] [1 2] [1 2] [4 2] [3] = [c_4(*^#(x, z), *^#(y, z))] 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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } Weak DPs: { *^#(1(), y) -> c_3() } 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. { *^#(1(), y) -> c_3() } 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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_4(*^#(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(*^#(otimes(x, y), z)) , 2: *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , 3: *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } Sub-proof: ---------- The following argument positions are considered usable: Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [*](x1, x2) = 2 + 2*x1 + 2*x2 + 2*x2^2 [otimes](x1, x2) = 0 [1]() = 0 [+](x1, x2) = 2 + x1 + x2 [oplus](x1, x2) = 2 + x1 + x2 [*^#](x1, x2) = 2*x1 + 2*x1*x2 + 2*x2 [c_1](x1) = 2*x1 [c_2](x1, x2) = x1 + x2 [c_3]() = 0 [c_4](x1, x2) = x1 + x2 The following symbols are considered usable {*^#} This order satisfies the following ordering constraints. [*^#(x, *(y, z))] = 6*x + 4*x*y + 4*x*z + 4*x*z^2 + 4 + 4*y + 4*z + 4*z^2 > 4*z = [c_1(*^#(otimes(x, y), z))] [*^#(x, oplus(y, z))] = 6*x + 2*x*y + 2*x*z + 4 + 2*y + 2*z > 4*x + 2*x*y + 2*y + 2*x*z + 2*z = [c_2(*^#(x, y), *^#(x, z))] [*^#(+(x, y), z)] = 4 + 2*x + 2*y + 6*z + 2*x*z + 2*y*z > 2*x + 2*x*z + 4*z + 2*y + 2*y*z = [c_4(*^#(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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_4(*^#(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(*^#(otimes(x, y), z)) , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) , *^#(+(x, y), z) -> c_4(*^#(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))