/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - 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 0] x1 + [0] [0 0] [1] p(1) = [1 1] x1 + [0] [0 0] [0] p(2) = [1 1] x1 + [0] [0 0] [1] p(3) = [1 1] x1 + [0] [0 0] [0] p(4) = [1 1] x1 + [0] [0 0] [1] p(5) = [1 1] x1 + [0] [0 0] [1] Following rules are strictly oriented: 0(0(3(2(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(2(1(3(x1))))) 0(1(2(x1))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 0(4(0(2(1(x1))))) 0(1(5(2(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 0] x1 + [2] [0 0] [1] > [1 0] x1 + [1] [0 0] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(0(4(1(1(x1))))) 2(2(0(3(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 2(0(2(1(3(x1))))) 2(5(0(1(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 0] x1 + [2] [0 0] [0] > [1 0] x1 + [1] [0 0] [0] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 0] x1 + [2] [0 0] [0] > [1 0] x1 + [1] [0 0] [0] = 3(1(3(0(0(x1))))) 3(2(0(1(x1)))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 3(1(0(2(1(x1))))) 4(5(0(1(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 5(0(4(1(1(x1))))) 5(2(0(1(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 2(0(5(1(1(x1))))) 5(4(0(1(x1)))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 5(0(4(1(1(x1))))) Following rules are (at-least) weakly oriented: 0(1(0(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 0(0(1(1(5(x1))))) 0(1(4(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(1(5(4(x1))))) 0(5(2(0(x1)))) = [1 0] x1 + [2] [0 0] [1] >= [1 0] x1 + [2] [0 0] [1] = 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(2(1(1(1(x1))))) 2(2(4(1(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 2(1(1(3(4(x1))))) 2(4(1(x1))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(2(3(1(1(x1))))) 3(1(0(3(x1)))) = [1 1] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) = [1 1] x1 + [1] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 1] x1 + [1] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 3(2(1(1(1(x1))))) 3(4(1(2(x1)))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 4(2(1(1(4(x1))))) 5(1(0(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [1] = 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 5(4(3(1(1(x1))))) 5(4(1(0(x1)))) = [1 0] x1 + [2] [0 0] [1] >= [1 0] x1 + [2] [0 0] [1] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 0] x1 + [2] [0 0] [1] >= [1 0] x1 + [2] [0 0] [1] = 5(1(1(4(0(x1))))) * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - 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 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 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(3) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [1] p(4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(5) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: 0(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(1(1(3(0(x1))))) 3(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(3(0(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 5(1(1(3(0(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(2(1(3(x1))))) 0(1(0(5(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] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(1(3(1(x1))))) 0(1(2(5(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] = 2(1(1(0(5(x1))))) 0(1(4(1(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] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(0(1(1(3(x1))))) 0(1(4(5(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] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 1 0] [0] [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(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(2(3(1(x1))))) 0(3(0(1(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(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 0(2(1(3(0(x1))))) 0(3(5(1(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(5(3(1(1(x1))))) 0(4(0(1(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] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(1(3(2(x1))))) 2(1(4(1(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] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(1(3(x1))))) 2(2(4(1(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] = 2(4(2(1(1(x1))))) 2(3(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] = 2(1(1(3(4(x1))))) 2(4(1(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] = 4(2(1(1(x1)))) 2(4(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] = 4(4(2(1(1(x1))))) 2(4(3(1(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] = 4(2(3(1(1(x1))))) 2(5(0(1(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] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(3(0(0(x1))))) 3(1(2(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(5(4(x1))))) 4(1(0(1(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] = 4(0(1(1(1(x1))))) 4(1(2(1(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] = 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(1(1(3(4(x1))))) 4(2(4(1(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] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(2(1(1(4(x1))))) 4(5(0(1(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] = 5(0(4(1(1(x1))))) 5(1(3(1(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] = 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(1(1(3(5(x1))))) 5(1(4(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] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 5(5(1(1(3(x1))))) 5(2(0(1(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] = 2(0(5(1(1(x1))))) 5(2(0(5(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] = 5(0(2(1(5(x1))))) 5(3(4(1(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] = 5(4(3(1(1(x1))))) 5(4(0(1(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] = 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 5(1(1(4(0(x1))))) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - 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 0] x1 + [0] [0 0 0] [1] p(1) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(2) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(3) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(4) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(5) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: 0(1(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(1(5(x1))))) 0(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(5(4(x1))))) 2(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(1(x1))))) 2(2(4(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(2(1(1(x1))))) 2(4(1(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(3(1(1(x1))))) 3(4(1(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] > [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(2(1(1(4(x1))))) 3(5(1(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] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(1(x1))))) 4(2(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(4(x1))))) 5(1(4(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(5(1(1(3(x1))))) 5(3(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(4(3(1(1(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(4(0(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(3(0(x1))))) 0(1(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(0(2(1(x1))))) 0(1(5(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(2(x1))))) 2(2(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(2(1(3(x1))))) 2(3(1(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(4(x1))))) 2(5(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(0(0(0(x1))))) 3(0(1(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] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(0(2(1(x1))))) 4(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(4(x1))))) 4(5(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(5(x1))))) 5(2(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(5(x1))))) 5(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - 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 0] x1 + [0] [0 0 0] [1] p(1) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(2) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(3) = [1 1 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(4) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(5) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: 2(1(2(3(x1)))) = [1 1 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(2(x1))))) 2(3(1(4(x1)))) = [1 1 0] [3] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(4(x1))))) 4(1(4(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(4(x1))))) 5(1(4(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] > [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(5(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 1 0] [3] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(5(0(0(x1))))) 2(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 1 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(2(1(1(x1))))) 2(4(1(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 3(1(0(0(0(x1))))) 3(0(1(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] [1] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 1 0] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) = [1 0 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(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 3(2(1(1(4(x1))))) 3(5(1(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] [1] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(1(1(1(x1))))) 4(2(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(3(1(1(1(x1))))) 5(1(4(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(4(0(x1))))) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 2 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 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(1) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(2) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] p(3) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(4) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(5) = [1 1 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] Following rules are strictly oriented: 0(5(2(0(x1)))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 1] [1] > [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] = 2(1(5(0(0(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 1 1] [3] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 0 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(0(4(1(1(x1))))) 2(1(2(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] = 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] = 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [2] = 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 2(1(1(3(4(x1))))) 2(4(1(x1))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] = 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 1 0] [2] [0 0 0] x1 + [0] [0 0 1] [1] = 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 1 1] [2] [0 0 0] x1 + [0] [0 0 1] [0] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 0] [2] [0 0 0] x1 + [1] [0 0 1] [1] = 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) = [1 1 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) = [1 1 1] [3] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) = [1 1 2] [3] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 1 2] [3] [0 0 0] x1 + [1] [0 0 1] [1] = 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) = [1 1 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 1 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] = 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [0] = 5(1(1(4(0(x1))))) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 2 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 0] x1 + [0] [0 0 0] [1] p(1) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(2) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(3) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] p(4) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(5) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: 5(1(3(1(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(3(1(1(1(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 1 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(1(1(3(4(x1))))) 2(4(1(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1 0] [2] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(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] [1] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(0(4(x1))))) 3(0(3(0(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] [1] = 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) = [1 1 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] = 3(2(1(1(1(x1))))) 3(2(0(1(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] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(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] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] = 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 5(1(1(4(0(x1))))) * Step 7: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 0 0] [1] p(1) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 0 0] [0] p(2) = [1 0 1] [0] [0 1 0] x_1 + [0] [0 1 0] [0] p(3) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 0 1] [0] p(4) = [1 0 0] [0] [0 1 0] x_1 + [0] [0 0 0] [0] p(5) = [1 0 0] [0] [0 0 0] x_1 + [1] [0 0 0] [0] Following rules are strictly oriented: 5(2(0(5(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] = 5(0(2(1(5(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(5(0(0(x1))))) 2(1(2(3(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(3(2(x1))))) 2(1(4(1(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] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(1(3(x1))))) 2(2(4(1(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] = 2(4(2(1(1(x1))))) 2(3(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] = 2(1(1(3(4(x1))))) 2(4(1(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] = 4(2(1(1(x1)))) 2(4(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] = 4(4(2(1(1(x1))))) 2(4(3(1(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] = 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(1(3(0(0(x1))))) 3(1(0(3(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(3(0(x1))))) 3(1(2(1(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] = 1(3(2(1(1(x1))))) 3(1(2(1(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(2(1(1(1(x1))))) 3(2(0(1(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] = 1(3(0(2(1(x1))))) 3(2(0(1(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] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 3(2(1(1(4(x1))))) 3(5(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(5(4(x1))))) 4(1(0(1(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] = 4(0(1(1(1(x1))))) 4(1(2(1(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] = 2(4(1(1(1(x1))))) 4(1(4(3(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] = 4(1(1(3(4(x1))))) 4(2(4(1(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] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 4(2(1(1(4(x1))))) 4(5(0(1(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] = 5(0(4(1(1(x1))))) 5(1(0(3(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] = 5(0(1(1(3(x1))))) 5(1(0(3(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] = 5(1(1(3(0(x1))))) 5(1(3(1(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] = 5(3(1(1(1(x1))))) 5(1(4(3(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] = 4(1(1(3(5(x1))))) 5(1(4(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] = 4(1(1(5(4(x1))))) 5(1(5(3(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] = 5(5(1(1(3(x1))))) 5(2(0(1(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] = 2(0(5(1(1(x1))))) 5(3(4(1(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] = 5(4(3(1(1(x1))))) 5(4(0(1(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] = 5(0(4(1(1(x1))))) 5(4(1(0(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] = 4(1(1(5(0(x1))))) 5(4(1(0(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] = 5(1(1(4(0(x1))))) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, 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 2 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 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] p(1) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(2) = [1 1 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(3) = [1 1 0 0] [0] [0 0 1 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] p(4) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(5) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] Following rules are strictly oriented: 3(1(2(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] > [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] > [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(2(1(1(1(x1))))) 4(1(2(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] > [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(4(1(1(1(x1))))) Following rules are (at-least) weakly oriented: 0(0(3(2(x1)))) = [1 1 1 1] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(1(1(5(x1))))) 0(1(2(x1))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(1(1(x1)))) 0(1(2(2(x1)))) = [1 1 1 1] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) = [1 0 0 0] [3] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) = [1 1 1 0] [3] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(3(1(3(x1))))) 0(3(2(x1))) = [1 1 1 1] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(1(3(x1)))) 0(3(2(0(x1)))) = [1 1 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) = [1 1 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(1(1(3(4(x1))))) 2(4(1(x1))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(2(1(1(x1)))) 2(4(1(4(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] = 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) = [1 1 0 0] [2] [0 0 0 0] x1 + [2] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(1(3(0(x1))))) 3(2(0(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [0] = 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(0(1(1(1(x1))))) 4(1(4(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) = [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) = [1 1 1 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) = [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [2] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] >= [1 1 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] = 5(1(1(4(0(x1))))) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) - Signature: {0/1,2/1,3/1,4/1,5/1} / {1/1} - 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^2))