/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^1)) * Step 1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] x1 + [0] p(1) = [1] x1 + [0] p(2) = [1] x1 + [4] p(3) = [1] x1 + [1] p(4) = [1] x1 + [0] p(5) = [1] x1 + [1] Following rules are strictly oriented: 1(4(5(5(x1)))) = [1] x1 + [2] > [1] x1 + [1] = 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) = [1] x1 + [5] > [1] x1 + [4] = 0(4(1(2(4(0(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1] x1 + [1] >= [1] x1 + [8] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1] x1 + [2] >= [1] x1 + [5] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1] x1 + [2] >= [1] x1 + [5] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1] x1 + [0] >= [1] x1 + [9] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1] x1 + [1] >= [1] x1 + [5] = 0(4(5(0(2(1(x1)))))) 1(5(4(x1))) = [1] x1 + [1] >= [1] x1 + [9] = 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) = [1] x1 + [2] >= [1] x1 + [6] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1] x1 + [0] >= [1] x1 + [11] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1] x1 + [2] >= [1] x1 + [9] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1] x1 + [1] >= [1] x1 + [5] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1] x1 + [1] >= [1] x1 + [4] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1] x1 + [1] >= [1] x1 + [6] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1] x1 + [1] >= [1] x1 + [8] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1] x1 + [5] >= [1] x1 + [6] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1] x1 + [1] >= [1] x1 + [5] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1] x1 + [2] >= [1] x1 + [9] = 3(4(4(1(2(2(x1)))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Weak TRS: 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 1] x1 + [0] [0 0] [0] p(1) = [1 0] x1 + [0] [0 0] [0] p(2) = [1 0] x1 + [0] [0 0] [0] p(3) = [1 0] x1 + [0] [0 0] [0] p(4) = [1 0] x1 + [0] [0 0] [0] p(5) = [1 0] x1 + [0] [0 0] [1] Following rules are strictly oriented: 0(5(5(x1))) = [1 0] x1 + [1] [0 0] [0] > [1 0] x1 + [0] [0 0] [0] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1 1] x1 + [1] [0 0] [0] > [1 1] x1 + [0] [0 0] [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1 0] x1 + [1] [0 0] [0] > [1 0] x1 + [0] [0 0] [0] = 0(1(3(4(3(4(x1)))))) 4(3(0(5(x1)))) = [1 0] x1 + [1] [0 0] [0] > [1 0] x1 + [0] [0 0] [0] = 3(3(2(3(5(5(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1 1] x1 + [0] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = 2(1(1(0(2(0(x1)))))) 1(4(x1)) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1 1] x1 + [0] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 3(3(2(2(3(1(x1)))))) 5(4(x1)) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [0] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [1] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1 1] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 3(4(4(1(2(2(x1)))))) * Step 3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 1] x1 + [0] [0 0] [0] p(1) = [1 0] x1 + [0] [0 0] [0] p(2) = [1 0] x1 + [0] [0 0] [0] p(3) = [1 0] x1 + [0] [0 0] [0] p(4) = [1 0] x1 + [0] [0 0] [0] p(5) = [1 1] x1 + [0] [0 0] [1] Following rules are strictly oriented: 5(5(4(x1))) = [1 0] x1 + [1] [0 0] [1] > [1 0] x1 + [0] [0 0] [0] = 3(4(4(1(2(2(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1 1] x1 + [0] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1 1] x1 + [2] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1 0] x1 + [2] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1 1] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1 1] x1 + [0] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1 1] x1 + [1] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [0] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [1] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1 1] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [0] = 4(1(1(3(2(4(x1)))))) * Step 4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] x1 + [1] p(1) = [1] x1 + [0] p(2) = [1] x1 + [0] p(3) = [1] x1 + [0] p(4) = [1] x1 + [0] p(5) = [1] x1 + [1] Following rules are strictly oriented: 3(5(4(x1))) = [1] x1 + [1] > [1] x1 + [0] = 4(1(3(4(2(3(x1)))))) 5(4(x1)) = [1] x1 + [1] > [1] x1 + [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(0(x1)))) = [1] x1 + [3] > [1] x1 + [2] = 1(0(4(0(2(2(x1)))))) 5(4(4(x1))) = [1] x1 + [1] > [1] x1 + [0] = 4(1(1(3(2(4(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1] x1 + [2] >= [1] x1 + [2] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1] x1 + [3] >= [1] x1 + [1] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1] x1 + [4] >= [1] x1 + [4] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1] x1 + [3] >= [1] x1 + [1] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1] x1 + [0] >= [1] x1 + [0] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1] x1 + [1] >= [1] x1 + [3] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1] x1 + [1] >= [1] x1 + [3] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 0(4(1(2(4(0(x1)))))) 4(1(4(x1))) = [1] x1 + [0] >= [1] x1 + [0] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 3(3(2(3(5(5(x1)))))) 5(4(0(x1))) = [1] x1 + [2] >= [1] x1 + [2] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1] x1 + [2] >= [1] x1 + [3] = 5(1(5(2(1(0(x1)))))) 5(4(0(2(x1)))) = [1] x1 + [2] >= [1] x1 + [3] = 3(0(4(5(0(2(x1)))))) 5(5(4(x1))) = [1] x1 + [2] >= [1] x1 + [0] = 3(4(4(1(2(2(x1)))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] x1 + [0] p(1) = [1] x1 + [0] p(2) = [1] x1 + [0] p(3) = [1] x1 + [0] p(4) = [1] x1 + [1] p(5) = [1] x1 + [1] Following rules are strictly oriented: 1(4(5(4(x1)))) = [1] x1 + [3] > [1] x1 + [2] = 0(4(5(0(2(1(x1)))))) 4(1(4(x1))) = [1] x1 + [2] > [1] x1 + [0] = 3(3(2(2(3(1(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1] x1 + [0] >= [1] x1 + [0] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1] x1 + [2] >= [1] x1 + [1] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1] x1 + [2] >= [1] x1 + [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1] x1 + [3] >= [1] x1 + [2] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1] x1 + [1] >= [1] x1 + [1] = 3(1(1(2(2(4(x1)))))) 1(4(5(5(x1)))) = [1] x1 + [3] >= [1] x1 + [1] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1] x1 + [2] >= [1] x1 + [2] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1] x1 + [2] >= [1] x1 + [2] = 4(1(3(4(2(3(x1)))))) 4(3(0(5(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1] x1 + [2] >= [1] x1 + [1] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1] x1 + [2] >= [1] x1 + [3] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1] x1 + [2] >= [1] x1 + [2] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1] x1 + [2] >= [1] x1 + [1] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1] x1 + [2] >= [1] x1 + [2] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1] x1 + [3] >= [1] x1 + [2] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1] x1 + [3] >= [1] x1 + [2] = 3(4(4(1(2(2(x1)))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(1) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(3) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(4) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(5) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: 1(5(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(5(2(0(4(x1)))))) 5(4(0(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] = 5(1(5(2(1(0(x1)))))) 5(4(0(2(x1)))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(0(4(5(0(2(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(1(3(4(1(x1)))))) 2(5(4(0(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(0(x1)))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(4(0(2(2(x1)))))) 5(4(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(4(4(1(2(2(x1)))))) * Step 7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(1) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(3) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(4) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(5) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: 1(4(x1)) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(1(2(2(4(x1)))))) Following rules are (at-least) weakly oriented: 0(3(0(x1))) = [1 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1 0 2] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1 1 0] [3] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 0(1(3(4(3(4(x1)))))) 1(4(5(4(x1)))) = [1 1 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1 1 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 2] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 2] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 3(4(4(1(2(2(x1)))))) * Step 8: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) - Weak TRS: 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(1) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(3) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(4) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(5) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] Following rules are strictly oriented: 0(3(0(x1))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(1(0(2(0(x1)))))) Following rules are (at-least) weakly oriented: 0(5(5(x1))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) = [1 0 0] [3] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(1(3(4(3(4(x1)))))) 1(4(x1)) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 3(3(2(3(5(5(x1)))))) 5(4(x1)) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) = [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 3(4(4(1(2(2(x1)))))) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(3(0(x1))) -> 2(1(1(0(2(0(x1)))))) 0(5(5(x1))) -> 1(0(1(3(4(2(x1)))))) 0(5(5(0(x1)))) -> 0(2(0(0(3(0(x1)))))) 0(5(5(4(x1)))) -> 0(1(3(4(3(4(x1)))))) 1(4(x1)) -> 3(1(1(2(2(4(x1)))))) 1(4(5(4(x1)))) -> 0(4(5(0(2(1(x1)))))) 1(4(5(5(x1)))) -> 0(0(1(3(4(1(x1)))))) 1(5(4(x1))) -> 0(2(5(2(0(4(x1)))))) 2(5(4(0(x1)))) -> 0(4(1(2(4(0(x1)))))) 3(5(4(x1))) -> 4(1(3(4(2(3(x1)))))) 4(1(4(x1))) -> 3(3(2(2(3(1(x1)))))) 4(3(0(5(x1)))) -> 3(3(2(3(5(5(x1)))))) 5(4(x1)) -> 4(2(3(1(1(1(x1)))))) 5(4(0(x1))) -> 2(4(0(4(4(0(x1)))))) 5(4(0(x1))) -> 5(1(5(2(1(0(x1)))))) 5(4(0(0(x1)))) -> 1(0(4(0(2(2(x1)))))) 5(4(0(2(x1)))) -> 3(0(4(5(0(2(x1)))))) 5(4(4(x1))) -> 4(1(1(3(2(4(x1)))))) 5(5(4(x1))) -> 3(4(4(1(2(2(x1)))))) - Signature: {0/1,1/1,2/1,3/1,4/1,5/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))