/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: DecomposeCP. WORST_CASE(?,O(n^2)) + 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: innermost derivational complexity wrt. signature {0,1,2} + Applied Processor: DecomposeCP {onSelectionCP_ = any strict-rules, withBoundCP_ = RelativeMul, withCP_ = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Nothing} to orient following rules strictly: 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))))) The Processor induces the complexity certificate TIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?) Observe that Problem (R) is 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 multiplication. Problem (S) - 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))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost derivational complexity wrt. signature {0,1,2} ** Step 1.a: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: innermost derivational complexity wrt. signature {0,1,2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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] 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 1.a:2: Assumption. WORST_CASE(?,O(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: innermost derivational complexity wrt. signature {0,1,2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ** Step 1.b:1: WeightGap. 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))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost 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 + [7] p(2) = [1] x1 + [0] Following rules are strictly oriented: 1(1(2(0(2(x1))))) = [1] x1 + [15] > [1] x1 + [9] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] > [1] x1 + [9] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] > [1] x1 + [9] = 1(0(2(2(0(x1))))) 2(1(1(0(2(x1))))) = [1] x1 + [15] > [1] x1 + [9] = 2(0(1(0(2(x1))))) Following rules are (at-least) weakly oriented: 0(1(2(0(2(x1))))) = [1] x1 + [9] >= [1] x1 + [9] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [9] >= [1] x1 + [15] = 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [9] >= [1] x1 + [9] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1] x1 + [9] >= [1] x1 + [9] = 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] >= [1] x1 + [15] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] >= [1] x1 + [15] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] >= [1] x1 + [15] = 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1] x1 + [15] >= [1] x1 + [15] = 1(2(1(0(2(x1))))) 1(2(2(0(2(x1))))) = [1] x1 + [8] >= [1] x1 + [8] = 1(0(2(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [8] >= [1] x1 + [8] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [8] >= [1] x1 + [8] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [8] >= [1] x1 + [8] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1] x1 + [8] >= [1] x1 + [8] = 2(2(2(1(0(x1))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b: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(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(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(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: 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))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost 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 0] x1 + [0] [0 0] [1] p(1) = [1 0] x1 + [0] [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 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 0(1(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 1(1(0(2(2(x1))))) 1(2(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] > [1 1] x1 + [0] [0 0] [1] = 1(0(2(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] > [1 1] x1 + [1] [0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] > [1 0] x1 + [1] [0 0] [0] = 2(2(2(1(0(x1))))) Following rules are (at-least) weakly oriented: 0(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [0] [0 0] [1] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [1] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 1(2(1(0(2(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [1] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 2(0(1(0(2(x1))))) ** Step 1.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 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(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) - Weak 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))))) 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(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(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))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost 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 0] x1 + [0] [0 0] [0] p(1) = [1 1] x1 + [0] [0 0] [0] p(2) = [1 0] x1 + [0] [0 0] [1] Following rules are strictly oriented: 0(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] > [1 0] x1 + [0] [0 0] [0] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] > [1 0] x1 + [0] [0 0] [0] = 0(2(2(1(0(x1))))) Following rules are (at-least) weakly oriented: 0(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 0(1(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 1(2(1(0(2(x1))))) 1(2(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1(0(2(2(2(x1))))) 2(1(1(0(2(x1))))) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [1] = 2(0(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [1] >= [1 0] x1 + [0] [0 0] [1] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [1] >= [1 0] x1 + [0] [0 0] [1] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 0] x1 + [1] [0 0] [1] >= [1 0] x1 + [0] [0 0] [1] = 2(2(2(1(0(x1))))) ** Step 1.b:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) - Weak 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(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(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))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost 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 0] x1 + [0] [0 0] [1] p(1) = [1 0] x1 + [1] [0 0] [0] p(2) = [1 1] x1 + [0] [0 0] [0] Following rules are strictly oriented: 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] > [1 1] x1 + [2] [0 0] [0] = 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] > [1 1] x1 + [2] [0 0] [0] = 1(2(1(0(2(x1))))) Following rules are (at-least) weakly oriented: 0(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [2] [0 0] [1] = 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 1] x1 + [1] [0 0] [1] = 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [1] >= [1 0] x1 + [1] [0 0] [1] = 0(2(2(1(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 0] x1 + [2] [0 0] [0] = 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) = [1 1] x1 + [3] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 1(1(0(2(2(x1))))) 1(2(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 1(0(2(2(2(x1))))) 2(1(1(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(0(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [2] [0 0] [0] = 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 1] x1 + [1] [0 0] [0] = 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) = [1 1] x1 + [2] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = 2(2(2(1(0(x1))))) ** Step 1.b:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak 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))))) - Signature: {0/1,1/1,2/1} / {} - Obligation: innermost 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^2))