/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 146 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(4(1(1(x1))))) ->^+ 1(2(1(4(0(1(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0]. The pumping substitution is [x1 / 4(1(1(x1)))]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(0(1(x1))) -> 1(2(0(2(0(x1))))) 0(0(1(x1))) -> 2(0(2(1(0(x1))))) 0(0(1(x1))) -> 2(1(0(2(0(x1))))) 0(0(1(x1))) -> 1(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(2(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(3(x1)))))) 0(0(1(x1))) -> 1(2(0(2(0(4(x1)))))) 0(0(1(x1))) -> 1(2(0(2(2(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(3(0(x1)))))) 0(0(1(x1))) -> 1(2(0(2(5(0(x1)))))) 0(0(1(x1))) -> 1(2(0(3(2(0(x1)))))) 0(0(1(x1))) -> 1(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(3(0(x1)))))) 0(0(1(x1))) -> 1(2(2(0(4(0(x1)))))) 0(0(1(x1))) -> 1(2(3(0(2(0(x1)))))) 0(0(1(x1))) -> 1(2(5(0(2(0(x1)))))) 0(0(1(x1))) -> 1(3(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(2(0(2(x1)))))) 0(0(1(x1))) -> 2(0(1(2(2(0(x1)))))) 0(0(1(x1))) -> 2(0(1(3(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(0(2(1(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(2(x1)))))) 0(0(1(x1))) -> 2(0(2(1(0(4(x1)))))) 0(0(1(x1))) -> 2(0(2(1(2(0(x1)))))) 0(0(1(x1))) -> 2(0(2(3(1(0(x1)))))) 0(0(1(x1))) -> 2(0(2(5(1(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(0(2(x1)))))) 0(0(1(x1))) -> 2(1(0(2(2(0(x1)))))) 0(0(1(x1))) -> 2(1(0(2(5(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(2(0(4(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(2(0(x1)))))) 0(0(1(x1))) -> 2(1(4(0(4(0(x1)))))) 0(0(1(x1))) -> 2(2(0(2(1(0(x1)))))) 0(0(1(x1))) -> 2(2(0(4(1(0(x1)))))) 0(0(1(x1))) -> 2(2(1(0(2(0(x1)))))) 0(0(1(x1))) -> 3(1(2(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 0(0(1(2(0(2(x1)))))) 0(0(0(1(x1)))) -> 0(2(1(0(2(0(x1)))))) 0(0(0(1(x1)))) -> 2(0(2(1(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(2(0(2(1(x1)))))) 0(0(1(1(x1)))) -> 2(0(1(2(1(0(x1)))))) 0(0(1(1(x1)))) -> 2(0(2(0(1(1(x1)))))) 0(0(1(1(x1)))) -> 2(1(0(2(0(1(x1)))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(x1))))) 0(0(3(1(x1)))) -> 2(0(1(3(0(2(x1)))))) 0(0(3(1(x1)))) -> 2(0(2(3(1(0(x1)))))) 0(0(3(1(x1)))) -> 2(1(0(2(0(3(x1)))))) 0(0(3(1(x1)))) -> 3(2(0(2(1(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(5(1(0(x1))))) 0(0(5(1(x1)))) -> 0(5(0(2(1(x1))))) 0(0(5(1(x1)))) -> 2(5(0(0(1(x1))))) 0(0(5(1(x1)))) -> 2(5(1(0(0(x1))))) 0(0(5(1(x1)))) -> 0(1(2(5(2(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(2(5(0(x1)))))) 0(0(5(1(x1)))) -> 0(2(1(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 1(2(0(0(2(5(x1)))))) 0(0(5(1(x1)))) -> 2(0(2(1(0(5(x1)))))) 0(0(5(1(x1)))) -> 2(1(0(4(5(0(x1)))))) 0(0(5(1(x1)))) -> 2(5(0(2(1(0(x1)))))) 0(1(0(1(x1)))) -> 0(0(1(2(1(x1))))) 0(1(0(1(x1)))) -> 1(2(1(0(0(x1))))) 0(1(0(1(x1)))) -> 2(1(0(0(1(x1))))) 0(1(0(1(x1)))) -> 0(0(1(1(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(1(1(3(2(0(x1)))))) 0(1(0(1(x1)))) -> 1(1(3(2(0(0(x1)))))) 0(1(0(1(x1)))) -> 1(2(0(2(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(2(0(0(1(x1)))))) 0(1(0(1(x1)))) -> 1(2(5(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 2(1(2(1(0(0(x1)))))) 0(1(0(1(x1)))) -> 3(1(2(1(0(0(x1)))))) 0(3(0(1(x1)))) -> 1(2(3(0(2(0(x1)))))) 0(3(0(1(x1)))) -> 3(1(2(0(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(x1))))) 0(5(0(1(x1)))) -> 0(0(2(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 0(0(2(5(2(1(x1)))))) 0(5(0(1(x1)))) -> 0(1(2(0(2(5(x1)))))) 0(5(0(1(x1)))) -> 1(2(0(2(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(0(1(2(5(x1)))))) 0(5(0(1(x1)))) -> 2(0(1(5(2(0(x1)))))) 0(5(0(1(x1)))) -> 2(0(2(1(5(0(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(0(1(2(x1)))))) 0(5(0(1(x1)))) -> 2(5(0(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 1(2(5(1(0(x1))))) 0(5(1(1(x1)))) -> 2(5(0(1(1(x1))))) 0(5(1(1(x1)))) -> 1(2(1(0(4(5(x1)))))) 0(5(1(1(x1)))) -> 1(2(1(4(5(0(x1)))))) 0(5(1(1(x1)))) -> 2(0(1(2(5(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(0(2(1(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(0(1(x1)))))) 0(5(1(1(x1)))) -> 2(5(1(2(1(0(x1)))))) 0(5(1(1(x1)))) -> 2(5(2(1(0(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(5(1(0(x1))))) 0(5(5(1(x1)))) -> 0(1(2(5(2(5(x1)))))) 0(5(5(1(x1)))) -> 1(2(0(2(5(5(x1)))))) 0(5(5(1(x1)))) -> 2(0(2(5(1(5(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(0(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(2(5(5(1(0(x1)))))) 0(5(5(1(x1)))) -> 2(5(0(2(5(1(x1)))))) 0(5(5(1(x1)))) -> 2(5(2(5(0(1(x1)))))) 5(0(0(1(x1)))) -> 1(2(5(2(0(0(x1)))))) 5(0(0(1(x1)))) -> 1(4(5(0(2(0(x1)))))) 5(0(0(1(x1)))) -> 2(0(2(5(1(0(x1)))))) 5(0(0(1(x1)))) -> 2(5(1(0(2(0(x1)))))) 0(0(0(3(1(x1))))) -> 2(0(3(0(0(1(x1)))))) 0(0(0(5(1(x1))))) -> 2(5(1(0(0(0(x1)))))) 0(0(1(4(0(x1))))) -> 0(2(1(0(4(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(2(5(1(0(0(x1)))))) 0(0(1(5(0(x1))))) -> 0(5(0(2(1(0(x1)))))) 0(0(5(1(1(x1))))) -> 0(0(2(5(1(1(x1)))))) 0(0(5(1(1(x1))))) -> 2(1(0(5(0(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(0(0(2(5(1(x1)))))) 0(0(5(3(1(x1))))) -> 3(2(5(1(0(0(x1)))))) 0(1(0(1(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(0(3(1(x1))))) -> 2(0(1(3(1(0(x1)))))) 0(1(1(0(1(x1))))) -> 0(0(1(1(1(3(x1)))))) 0(1(4(1(1(x1))))) -> 1(2(1(4(0(1(x1)))))) 0(1(5(0(1(x1))))) -> 1(2(5(1(0(0(x1)))))) 0(3(0(5(1(x1))))) -> 2(5(0(0(3(1(x1)))))) 0(3(1(0(1(x1))))) -> 2(3(1(0(0(1(x1)))))) 0(3(5(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(3(5(1(1(x1))))) -> 1(2(5(1(0(3(x1)))))) 0(4(4(1(1(x1))))) -> 1(2(1(4(0(4(x1)))))) 0(4(5(5(1(x1))))) -> 5(0(2(1(4(5(x1)))))) 0(5(0(0(1(x1))))) -> 2(5(0(0(0(1(x1)))))) 0(5(0(1(0(x1))))) -> 0(0(4(1(0(5(x1)))))) 0(5(1(0(1(x1))))) -> 2(1(0(1(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(0(1(3(1(x1)))))) 0(5(1(3(1(x1))))) -> 2(5(1(0(1(3(x1)))))) 0(5(2(5(1(x1))))) -> 2(0(2(5(5(1(x1)))))) 0(5(2(5(1(x1))))) -> 2(5(5(0(2(1(x1)))))) 0(5(3(0(1(x1))))) -> 0(0(3(2(5(1(x1)))))) 0(5(3(0(1(x1))))) -> 2(0(3(0(5(1(x1)))))) 0(5(3(1(1(x1))))) -> 1(1(3(0(2(5(x1)))))) 0(5(3(1(1(x1))))) -> 1(2(5(1(3(0(x1)))))) 0(5(4(1(1(x1))))) -> 2(5(0(4(1(1(x1)))))) 5(0(1(0(1(x1))))) -> 5(1(2(1(0(0(x1)))))) 5(5(4(1(1(x1))))) -> 1(4(5(2(5(1(x1)))))) 5(5(4(1(1(x1))))) -> 2(5(1(4(5(1(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL