/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^4)) * Step 1: DecomposeCP. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: 0(0(1(x1))) -> 0(2(3(0(1(x1))))) 0(0(1(x1))) -> 0(4(0(5(4(1(x1)))))) 0(0(1(x1))) -> 2(1(0(0(3(4(x1)))))) 0(0(1(x1))) -> 4(0(5(4(0(1(x1)))))) 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 0(1(0(x1))) -> 0(0(2(5(4(1(x1)))))) 0(1(0(x1))) -> 1(0(0(5(4(x1))))) 0(1(1(x1))) -> 1(0(3(4(1(x1))))) 0(1(1(x1))) -> 5(0(3(4(1(1(x1)))))) 0(1(2(5(0(x1))))) -> 1(5(4(0(2(0(x1)))))) 0(1(3(0(x1)))) -> 0(2(0(2(1(3(x1)))))) 0(1(4(2(0(x1))))) -> 1(0(4(2(3(0(x1)))))) 0(1(5(0(x1)))) -> 0(0(5(4(1(5(x1)))))) 0(1(5(0(x1)))) -> 0(5(4(2(1(0(x1)))))) 0(3(0(1(x1)))) -> 0(0(4(1(3(0(x1)))))) 0(3(1(0(x1)))) -> 0(0(2(3(1(x1))))) 0(3(1(1(x1)))) -> 5(1(1(0(3(4(x1)))))) 1(4(5(1(0(x1))))) -> 5(4(2(1(1(0(x1)))))) 5(0(1(x1))) -> 0(1(4(5(4(4(x1)))))) 5(0(1(x1))) -> 0(5(4(1(x1)))) 5(0(1(x1))) -> 0(5(4(1(4(4(x1)))))) 5(0(1(x1))) -> 2(5(4(0(1(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(x1))))) 5(0(1(x1))) -> 5(0(4(3(0(1(x1)))))) 5(0(1(0(x1)))) -> 5(0(0(4(1(3(x1)))))) 5(0(1(4(0(x1))))) -> 1(4(5(4(0(0(x1)))))) 5(1(0(x1))) -> 0(5(0(2(2(1(x1)))))) 5(1(0(x1))) -> 1(4(0(5(2(3(x1)))))) 5(1(0(x1))) -> 1(5(0(4(4(2(x1)))))) 5(1(0(x1))) -> 4(4(1(0(4(5(x1)))))) 5(1(0(x1))) -> 5(0(2(2(1(x1))))) 5(1(0(x1))) -> 5(0(5(4(1(x1))))) 5(1(1(x1))) -> 1(1(4(5(4(4(x1)))))) 5(1(1(x1))) -> 1(1(5(4(x1)))) 5(1(1(x1))) -> 1(5(3(4(1(x1))))) 5(1(1(x1))) -> 3(5(2(3(1(1(x1)))))) 5(1(1(x1))) -> 4(1(2(1(5(4(x1)))))) 5(1(1(x1))) -> 5(4(1(1(x1)))) 5(1(2(0(x1)))) -> 1(4(0(5(4(2(x1)))))) 5(1(2(0(x1)))) -> 5(0(4(2(2(1(x1)))))) 5(1(4(0(x1)))) -> 1(5(4(0(2(3(x1)))))) 5(1(4(0(x1)))) -> 4(5(2(1(3(0(x1)))))) 5(1(5(1(x1)))) -> 5(4(1(5(1(x1))))) 5(3(0(1(x1)))) -> 0(1(5(2(3(x1))))) 5(3(1(0(x1)))) -> 1(3(0(4(3(5(x1)))))) 5(3(1(0(x1)))) -> 1(4(3(5(0(x1))))) 5(3(1(0(x1)))) -> 1(5(0(4(3(x1))))) 5(3(1(0(x1)))) -> 5(4(3(1(0(x1))))) 5(3(1(1(x1)))) -> 1(1(5(3(3(4(x1)))))) 5(5(1(0(0(x1))))) -> 5(5(0(4(1(0(x1)))))) - Signature: {0/1,1/1,5/1} / {2/1,3/1,4/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: DecomposeCP {onSelectionCP_ = any strict-rules, withBoundCP_ = RelativeComp, withCP_ = NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Nothing} to orient following rules strictly: 0(0(1(x1))) -> 0(2(3(0(1(x1))))) 0(0(1(x1))) -> 0(4(0(5(4(1(x1)))))) 0(0(1(x1))) -> 2(1(0(0(3(4(x1)))))) 0(0(1(x1))) -> 4(0(5(4(0(1(x1)))))) 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 0(1(0(x1))) -> 0(0(2(5(4(1(x1)))))) 0(1(0(x1))) -> 1(0(0(5(4(x1))))) 0(1(1(x1))) -> 1(0(3(4(1(x1))))) 0(1(1(x1))) -> 5(0(3(4(1(1(x1)))))) 0(1(2(5(0(x1))))) -> 1(5(4(0(2(0(x1)))))) 0(1(3(0(x1)))) -> 0(2(0(2(1(3(x1)))))) 0(1(4(2(0(x1))))) -> 1(0(4(2(3(0(x1)))))) 0(1(5(0(x1)))) -> 0(0(5(4(1(5(x1)))))) 0(1(5(0(x1)))) -> 0(5(4(2(1(0(x1)))))) 0(3(0(1(x1)))) -> 0(0(4(1(3(0(x1)))))) 0(3(1(0(x1)))) -> 0(0(2(3(1(x1))))) 0(3(1(1(x1)))) -> 5(1(1(0(3(4(x1)))))) 1(4(5(1(0(x1))))) -> 5(4(2(1(1(0(x1)))))) 5(0(1(x1))) -> 0(1(4(5(4(4(x1)))))) 5(0(1(x1))) -> 0(5(4(1(x1)))) 5(0(1(x1))) -> 0(5(4(1(4(4(x1)))))) 5(0(1(x1))) -> 2(5(4(0(1(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(x1))))) 5(0(1(x1))) -> 5(0(4(3(0(1(x1)))))) 5(0(1(0(x1)))) -> 5(0(0(4(1(3(x1)))))) 5(0(1(4(0(x1))))) -> 1(4(5(4(0(0(x1)))))) 5(1(0(x1))) -> 0(5(0(2(2(1(x1)))))) 5(1(0(x1))) -> 1(4(0(5(2(3(x1)))))) 5(1(0(x1))) -> 1(5(0(4(4(2(x1)))))) 5(1(0(x1))) -> 4(4(1(0(4(5(x1)))))) 5(1(0(x1))) -> 5(0(2(2(1(x1))))) 5(1(0(x1))) -> 5(0(5(4(1(x1))))) 5(1(1(x1))) -> 1(1(4(5(4(4(x1)))))) 5(1(1(x1))) -> 1(1(5(4(x1)))) 5(1(1(x1))) -> 1(5(3(4(1(x1))))) 5(1(1(x1))) -> 3(5(2(3(1(1(x1)))))) 5(1(1(x1))) -> 4(1(2(1(5(4(x1)))))) 5(1(1(x1))) -> 5(4(1(1(x1)))) 5(1(2(0(x1)))) -> 1(4(0(5(4(2(x1)))))) 5(1(2(0(x1)))) -> 5(0(4(2(2(1(x1)))))) 5(1(4(0(x1)))) -> 1(5(4(0(2(3(x1)))))) 5(1(4(0(x1)))) -> 4(5(2(1(3(0(x1)))))) 5(1(5(1(x1)))) -> 5(4(1(5(1(x1))))) 5(3(0(1(x1)))) -> 0(1(5(2(3(x1))))) 5(3(1(0(x1)))) -> 1(3(0(4(3(5(x1)))))) 5(3(1(0(x1)))) -> 1(4(3(5(0(x1))))) 5(3(1(0(x1)))) -> 1(5(0(4(3(x1))))) 5(3(1(0(x1)))) -> 5(4(3(1(0(x1))))) 5(3(1(1(x1)))) -> 1(1(5(3(3(4(x1)))))) 5(5(1(0(0(x1))))) -> 5(5(0(4(1(0(x1)))))) The Processor induces the complexity certificate TIME (?,O(n^4)) BEST_CASE TIME (?,?) SPACE(?,?) Observe that weak rules from Problem (R) are non-size-increasing. Once the complexity of (R) has been assessed, it suffices to consider only rules whose complexity has not been estimated in (R) resulting in the following Problem (S). Overall the certificate is obtained by composition. Problem (S) - Signature: {0/1,1/1,5/1} / {2/1,3/1,4/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} ** Step 1.a:1: NaturalMI. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: 0(0(1(x1))) -> 0(2(3(0(1(x1))))) 0(0(1(x1))) -> 0(4(0(5(4(1(x1)))))) 0(0(1(x1))) -> 2(1(0(0(3(4(x1)))))) 0(0(1(x1))) -> 4(0(5(4(0(1(x1)))))) 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 0(1(0(x1))) -> 0(0(2(5(4(1(x1)))))) 0(1(0(x1))) -> 1(0(0(5(4(x1))))) 0(1(1(x1))) -> 1(0(3(4(1(x1))))) 0(1(1(x1))) -> 5(0(3(4(1(1(x1)))))) 0(1(2(5(0(x1))))) -> 1(5(4(0(2(0(x1)))))) 0(1(3(0(x1)))) -> 0(2(0(2(1(3(x1)))))) 0(1(4(2(0(x1))))) -> 1(0(4(2(3(0(x1)))))) 0(1(5(0(x1)))) -> 0(0(5(4(1(5(x1)))))) 0(1(5(0(x1)))) -> 0(5(4(2(1(0(x1)))))) 0(3(0(1(x1)))) -> 0(0(4(1(3(0(x1)))))) 0(3(1(0(x1)))) -> 0(0(2(3(1(x1))))) 0(3(1(1(x1)))) -> 5(1(1(0(3(4(x1)))))) 1(4(5(1(0(x1))))) -> 5(4(2(1(1(0(x1)))))) 5(0(1(x1))) -> 0(1(4(5(4(4(x1)))))) 5(0(1(x1))) -> 0(5(4(1(x1)))) 5(0(1(x1))) -> 0(5(4(1(4(4(x1)))))) 5(0(1(x1))) -> 2(5(4(0(1(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(x1))))) 5(0(1(x1))) -> 5(0(4(3(0(1(x1)))))) 5(0(1(0(x1)))) -> 5(0(0(4(1(3(x1)))))) 5(0(1(4(0(x1))))) -> 1(4(5(4(0(0(x1)))))) 5(1(0(x1))) -> 0(5(0(2(2(1(x1)))))) 5(1(0(x1))) -> 1(4(0(5(2(3(x1)))))) 5(1(0(x1))) -> 1(5(0(4(4(2(x1)))))) 5(1(0(x1))) -> 4(4(1(0(4(5(x1)))))) 5(1(0(x1))) -> 5(0(2(2(1(x1))))) 5(1(0(x1))) -> 5(0(5(4(1(x1))))) 5(1(1(x1))) -> 1(1(4(5(4(4(x1)))))) 5(1(1(x1))) -> 1(1(5(4(x1)))) 5(1(1(x1))) -> 1(5(3(4(1(x1))))) 5(1(1(x1))) -> 3(5(2(3(1(1(x1)))))) 5(1(1(x1))) -> 4(1(2(1(5(4(x1)))))) 5(1(1(x1))) -> 5(4(1(1(x1)))) 5(1(2(0(x1)))) -> 1(4(0(5(4(2(x1)))))) 5(1(2(0(x1)))) -> 5(0(4(2(2(1(x1)))))) 5(1(4(0(x1)))) -> 1(5(4(0(2(3(x1)))))) 5(1(4(0(x1)))) -> 4(5(2(1(3(0(x1)))))) 5(1(5(1(x1)))) -> 5(4(1(5(1(x1))))) 5(3(0(1(x1)))) -> 0(1(5(2(3(x1))))) 5(3(1(0(x1)))) -> 1(3(0(4(3(5(x1)))))) 5(3(1(0(x1)))) -> 1(4(3(5(0(x1))))) 5(3(1(0(x1)))) -> 1(5(0(4(3(x1))))) 5(3(1(0(x1)))) -> 5(4(3(1(0(x1))))) 5(3(1(1(x1)))) -> 1(1(5(3(3(4(x1)))))) 5(5(1(0(0(x1))))) -> 5(5(0(4(1(0(x1)))))) - Signature: {0/1,1/1,5/1} / {2/1,3/1,4/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just first alternative for decompose on 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 1 0 0] [0] [0 0 0 1] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] p(1) = [1 0 0 0] [0] [0 1 1 0] x1 + [1] [0 0 0 1] [0] [0 0 0 1] [1] p(2) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(3) = [1 0 0 0] [0] [0 1 1 0] x1 + [0] [0 0 0 0] [0] [0 0 0 1] [0] p(4) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(5) = [1 0 0 1] [0] [0 0 0 1] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] Following rules are strictly oriented: 0(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(2(3(0(1(x1))))) 0(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(4(0(5(4(1(x1)))))) 0(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(1(0(0(3(4(x1)))))) 0(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(0(5(4(0(1(x1)))))) 0(1(0(x1))) = [1 1 0 2] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(0(2(1(2(x1))))) 0(1(0(x1))) = [1 1 0 2] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(0(2(5(4(1(x1)))))) 0(1(0(x1))) = [1 1 0 2] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(0(0(5(4(x1))))) 0(1(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 1] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(0(3(4(1(x1))))) 0(1(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 1] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(3(4(1(1(x1)))))) 0(1(2(5(0(x1))))) = [1 1 0 1] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(5(4(0(2(0(x1)))))) 0(1(3(0(x1)))) = [1 1 0 2] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(2(0(2(1(3(x1)))))) 0(1(4(2(0(x1))))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(0(4(2(3(0(x1)))))) 0(1(5(0(x1)))) = [1 1 0 3] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(0(5(4(1(5(x1)))))) 0(1(5(0(x1)))) = [1 1 0 3] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(5(4(2(1(0(x1)))))) 0(3(0(1(x1)))) = [1 1 1 2] [3] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(0(4(1(3(0(x1)))))) 0(3(1(0(x1)))) = [1 1 0 3] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(0(2(3(1(x1))))) 0(3(1(1(x1)))) = [1 1 1 2] [3] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 1 0] [2] [0 0 0 0] x1 + [2] [0 0 0 0] [2] [0 0 0 0] [2] = 5(1(1(0(3(4(x1)))))) 1(4(5(1(0(x1))))) = [1 1 0 1] [1] [0 0 0 1] x1 + [2] [0 0 0 0] [0] [0 0 0 0] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(4(2(1(1(0(x1)))))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] = 0(1(4(5(4(4(x1)))))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(5(4(1(x1)))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(5(4(1(4(4(x1)))))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 2(5(4(0(1(x1))))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(2(1(2(x1))))) 5(0(1(x1))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 1 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(4(3(0(1(x1)))))) 5(0(1(0(x1)))) = [1 1 0 3] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(0(4(1(3(x1)))))) 5(0(1(4(0(x1))))) = [1 1 0 1] [2] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 1 0 1] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(4(5(4(0(0(x1)))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 0(5(0(2(2(1(x1)))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(4(0(5(2(3(x1)))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(5(0(4(4(2(x1)))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 1] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(4(1(0(4(5(x1)))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(2(2(1(x1))))) 5(1(0(x1))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(5(4(1(x1))))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [2] [0 0 0 0] [1] [0 0 0 0] [2] = 1(1(4(5(4(4(x1)))))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [2] [0 0 0 0] [1] [0 0 0 0] [2] = 1(1(5(4(x1)))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(5(3(4(1(x1))))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 3(5(2(3(1(1(x1)))))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(1(2(1(5(4(x1)))))) 5(1(1(x1))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(4(1(1(x1)))) 5(1(2(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(4(0(5(4(2(x1)))))) 5(1(2(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(0(4(2(2(1(x1)))))) 5(1(4(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(5(4(0(2(3(x1)))))) 5(1(4(0(x1)))) = [1 1 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 4(5(2(1(3(0(x1)))))) 5(1(5(1(x1)))) = [1 0 0 2] [3] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 1] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(4(1(5(1(x1))))) 5(3(0(1(x1)))) = [1 1 1 1] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] = 0(1(5(2(3(x1))))) 5(3(1(0(x1)))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 1] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(3(0(4(3(5(x1)))))) 5(3(1(0(x1)))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 0 1] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(4(3(5(0(x1))))) 5(3(1(0(x1)))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [0] [0 0 0 0] [1] = 1(5(0(4(3(x1))))) 5(3(1(0(x1)))) = [1 1 0 1] [1] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(4(3(1(0(x1))))) 5(3(1(1(x1)))) = [1 0 0 1] [2] [0 0 0 1] x1 + [2] [0 0 0 1] [2] [0 0 0 1] [2] > [1 0 0 0] [0] [0 0 0 0] x1 + [2] [0 0 0 0] [1] [0 0 0 0] [2] = 1(1(5(3(3(4(x1)))))) 5(5(1(0(0(x1))))) = [1 1 0 3] [2] [0 0 0 1] x1 + [1] [0 0 0 1] [1] [0 0 0 1] [1] > [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = 5(5(0(4(1(0(x1)))))) Following rules are (at-least) weakly oriented: ** Step 1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(0(1(x1))) -> 0(2(3(0(1(x1))))) 0(0(1(x1))) -> 0(4(0(5(4(1(x1)))))) 0(0(1(x1))) -> 2(1(0(0(3(4(x1)))))) 0(0(1(x1))) -> 4(0(5(4(0(1(x1)))))) 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 0(1(0(x1))) -> 0(0(2(5(4(1(x1)))))) 0(1(0(x1))) -> 1(0(0(5(4(x1))))) 0(1(1(x1))) -> 1(0(3(4(1(x1))))) 0(1(1(x1))) -> 5(0(3(4(1(1(x1)))))) 0(1(2(5(0(x1))))) -> 1(5(4(0(2(0(x1)))))) 0(1(3(0(x1)))) -> 0(2(0(2(1(3(x1)))))) 0(1(4(2(0(x1))))) -> 1(0(4(2(3(0(x1)))))) 0(1(5(0(x1)))) -> 0(0(5(4(1(5(x1)))))) 0(1(5(0(x1)))) -> 0(5(4(2(1(0(x1)))))) 0(3(0(1(x1)))) -> 0(0(4(1(3(0(x1)))))) 0(3(1(0(x1)))) -> 0(0(2(3(1(x1))))) 0(3(1(1(x1)))) -> 5(1(1(0(3(4(x1)))))) 1(4(5(1(0(x1))))) -> 5(4(2(1(1(0(x1)))))) 5(0(1(x1))) -> 0(1(4(5(4(4(x1)))))) 5(0(1(x1))) -> 0(5(4(1(x1)))) 5(0(1(x1))) -> 0(5(4(1(4(4(x1)))))) 5(0(1(x1))) -> 2(5(4(0(1(x1))))) 5(0(1(x1))) -> 5(0(2(1(2(x1))))) 5(0(1(x1))) -> 5(0(4(3(0(1(x1)))))) 5(0(1(0(x1)))) -> 5(0(0(4(1(3(x1)))))) 5(0(1(4(0(x1))))) -> 1(4(5(4(0(0(x1)))))) 5(1(0(x1))) -> 0(5(0(2(2(1(x1)))))) 5(1(0(x1))) -> 1(4(0(5(2(3(x1)))))) 5(1(0(x1))) -> 1(5(0(4(4(2(x1)))))) 5(1(0(x1))) -> 4(4(1(0(4(5(x1)))))) 5(1(0(x1))) -> 5(0(2(2(1(x1))))) 5(1(0(x1))) -> 5(0(5(4(1(x1))))) 5(1(1(x1))) -> 1(1(4(5(4(4(x1)))))) 5(1(1(x1))) -> 1(1(5(4(x1)))) 5(1(1(x1))) -> 1(5(3(4(1(x1))))) 5(1(1(x1))) -> 3(5(2(3(1(1(x1)))))) 5(1(1(x1))) -> 4(1(2(1(5(4(x1)))))) 5(1(1(x1))) -> 5(4(1(1(x1)))) 5(1(2(0(x1)))) -> 1(4(0(5(4(2(x1)))))) 5(1(2(0(x1)))) -> 5(0(4(2(2(1(x1)))))) 5(1(4(0(x1)))) -> 1(5(4(0(2(3(x1)))))) 5(1(4(0(x1)))) -> 4(5(2(1(3(0(x1)))))) 5(1(5(1(x1)))) -> 5(4(1(5(1(x1))))) 5(3(0(1(x1)))) -> 0(1(5(2(3(x1))))) 5(3(1(0(x1)))) -> 1(3(0(4(3(5(x1)))))) 5(3(1(0(x1)))) -> 1(4(3(5(0(x1))))) 5(3(1(0(x1)))) -> 1(5(0(4(3(x1))))) 5(3(1(0(x1)))) -> 5(4(3(1(0(x1))))) 5(3(1(1(x1)))) -> 1(1(5(3(3(4(x1)))))) 5(5(1(0(0(x1))))) -> 5(5(0(4(1(0(x1)))))) - Signature: {0/1,1/1,5/1} / {2/1,3/1,4/1} - Obligation: derivational complexity wrt. signature {0,1,2,3,4,5} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ** Step 1.b:1: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {0/1,1/1,5/1} / {2/1,3/1,4/1} - 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^4))