/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: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: 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 + [1] p(2) = [1] x1 + [0] Following rules are strictly oriented: 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [3] = 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [4] > [1] x1 + [2] = 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(2(1(0(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [3] > [1] x1 + [2] = 2(2(1(0(2(x1))))) Following rules are (at-least) weakly oriented: 0(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(2(1(0(2(x1))))) 1(2(2(0(2(x1))))) = [1] x1 + [2] >= [1] x1 + [2] = 1(0(2(2(2(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 2(0(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [2] >= [1] x1 + [2] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [2] >= [1] x1 + [2] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [2] >= [1] x1 + [2] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [2] >= [1] x1 + [2] = 2(2(2(1(0(x1))))) * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) - Weak TRS: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2} + 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] [1] p(1) = [1 0] x1 + [1] [0 0] [1] p(2) = [1 1] x1 + [0] [0 0] [0] Following rules are strictly oriented: 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] > [1 1] x1 + [2] [0 0] [1] = 1(1(0(2(2(x1))))) 1(2(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] > [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] > [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] > [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] > [1 1] x1 + [2] [0 0] [0] = 2(2(2(1(0(x1))))) Following rules are (at-least) weakly oriented: 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [1] = 1(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [3] [0 0] [0] = 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [1] >= [1 1] x1 + [2] [0 0] [0] = 2(2(2(1(0(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [3] [0 0] [0] = 2(0(1(2(2(x1))))) * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) - Weak TRS: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2} + 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 + [2] p(2) = [1] x1 + [0] Following rules are strictly oriented: 2(1(1(0(2(x1))))) = [1] x1 + [5] > [1] x1 + [4] = 2(0(1(0(2(x1))))) Following rules are (at-least) weakly oriented: 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [4] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [5] = 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [4] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [4] = 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [4] = 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [5] = 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [4] = 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [5] = 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [4] = 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [5] = 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [5] = 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) = [1] x1 + [6] >= [1] x1 + [3] = 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [4] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [5] = 1(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) = [1] x1 + [4] >= [1] x1 + [3] = 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [5] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [3] >= [1] x1 + [3] = 2(2(2(1(0(x1))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) - Weak TRS: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2} + 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 0] [1] [0 0 1] x1 + [0] [0 0 0] [1] p(1) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(2) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] > [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] > [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(2(2(x1))))) Following rules are (at-least) weakly oriented: 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [1] = 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [3] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] x1 + [1] [0 0 0] [1] = 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(2(1(0(x1))))) 1(2(2(0(2(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] = 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [2] [0 0 0] x1 + [0] [0 0 0] [0] = 2(2(2(1(0(x1))))) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: derivational complexity wrt. signature {0,1,2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))