/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), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 63 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 722 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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(2(x_1)) -> 2(encArg(x_1)) encArg(3(x_1)) -> 3(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763] {(50,51,[0_1|0, 1_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0]), (50,52,[3_1|1]), (50,59,[3_1|1]), (50,66,[3_1|1]), (50,72,[3_1|1]), (50,76,[5_1|1]), (50,81,[3_1|1]), (50,86,[5_1|1]), (50,91,[3_1|1]), (50,98,[3_1|1]), (50,104,[4_1|1]), (50,111,[2_1|1, 3_1|1, 4_1|1, 5_1|1, 0_1|1, 1_1|1]), (50,112,[0_1|2]), (50,117,[3_1|2]), (50,124,[1_1|2]), (50,131,[0_1|2]), (50,137,[4_1|2]), (50,143,[5_1|2]), (50,150,[0_1|2]), 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(612,196,[0_1|2]), (612,202,[0_1|2]), (612,209,[0_1|2]), (612,226,[0_1|2]), (612,252,[0_1|2]), (612,290,[0_1|2]), (612,297,[0_1|2]), (612,304,[0_1|2]), (612,310,[0_1|2]), (612,316,[0_1|2]), (612,330,[0_1|2]), (612,344,[0_1|2]), (612,350,[0_1|2]), (612,357,[0_1|2]), (612,364,[0_1|2]), (612,382,[0_1|2]), (612,401,[0_1|2]), (612,426,[0_1|2]), (612,466,[0_1|2]), (612,473,[0_1|2]), (612,510,[0_1|2]), (612,529,[0_1|2]), (612,587,[0_1|2]), (612,117,[3_1|2]), (612,124,[1_1|2]), (612,137,[4_1|2]), (612,143,[5_1|2]), (612,156,[3_1|2]), (612,163,[5_1|2]), (612,168,[5_1|2]), (612,187,[3_1|2]), (612,214,[3_1|2]), (612,219,[3_1|2]), (612,233,[3_1|2]), (612,238,[3_1|2]), (612,245,[3_1|2]), (612,259,[3_1|2]), (612,266,[3_1|2]), (612,272,[3_1|2]), (612,276,[3_1|2]), (612,283,[3_1|2]), (612,323,[2_1|2]), (612,337,[5_1|2]), (612,371,[5_1|2]), (612,376,[5_1|2]), (612,388,[2_1|2]), (612,395,[3_1|2]), (612,408,[3_1|2]), (612,415,[3_1|2]), (612,420,[3_1|2]), (612,433,[5_1|2]), (612,439,[3_1|2]), (612,446,[4_1|2]), (612,453,[5_1|2]), (612,460,[5_1|2]), (612,625,[0_1|3]), (612,620,[2_1|3]), (613,614,[4_1|2]), (614,615,[2_1|2]), (615,616,[3_1|2]), (616,617,[1_1|2]), (617,618,[4_1|2]), (618,619,[1_1|2]), (618,480,[2_1|2]), (618,485,[2_1|2]), (618,490,[1_1|2]), (618,497,[2_1|2]), (618,503,[2_1|2]), (618,510,[0_1|2]), (618,517,[5_1|2]), (618,706,[2_1|3]), (619,111,[0_1|2]), (619,143,[0_1|2, 5_1|2]), (619,163,[0_1|2, 5_1|2]), (619,168,[0_1|2, 5_1|2]), (619,337,[0_1|2, 5_1|2]), (619,371,[0_1|2, 5_1|2]), (619,376,[0_1|2, 5_1|2]), (619,433,[0_1|2, 5_1|2]), (619,453,[0_1|2, 5_1|2]), (619,460,[0_1|2, 5_1|2]), (619,517,[0_1|2]), (619,543,[0_1|2]), (619,548,[0_1|2]), (619,580,[0_1|2]), (619,606,[0_1|2]), (619,613,[0_1|2]), (619,157,[0_1|2]), (619,112,[0_1|2]), (619,117,[3_1|2]), (619,124,[1_1|2]), (619,131,[0_1|2]), (619,137,[4_1|2]), (619,150,[0_1|2]), (619,156,[3_1|2]), (619,174,[0_1|2]), (619,181,[0_1|2]), (619,187,[3_1|2]), (619,191,[0_1|2]), (619,196,[0_1|2]), (619,202,[0_1|2]), (619,209,[0_1|2]), (619,214,[3_1|2]), (619,219,[3_1|2]), (619,226,[0_1|2]), (619,233,[3_1|2]), (619,238,[3_1|2]), (619,245,[3_1|2]), (619,252,[0_1|2]), (619,259,[3_1|2]), (619,266,[3_1|2]), (619,272,[3_1|2]), (619,276,[3_1|2]), (619,283,[3_1|2]), (619,290,[0_1|2]), (619,297,[0_1|2]), (619,304,[0_1|2]), (619,310,[0_1|2]), (619,316,[0_1|2]), (619,323,[2_1|2]), (619,330,[0_1|2]), (619,344,[0_1|2]), (619,350,[0_1|2]), (619,357,[0_1|2]), (619,364,[0_1|2]), (619,382,[0_1|2]), (619,388,[2_1|2]), (619,395,[3_1|2]), (619,401,[0_1|2]), (619,408,[3_1|2]), (619,415,[3_1|2]), (619,420,[3_1|2]), (619,426,[0_1|2]), (619,439,[3_1|2]), (619,446,[4_1|2]), (619,466,[0_1|2]), (619,473,[0_1|2]), (619,625,[0_1|3]), (619,620,[2_1|3]), (620,621,[3_1|3]), (621,622,[1_1|3]), (622,623,[3_1|3]), (623,624,[0_1|3]), (624,126,[0_1|3]), (625,626,[2_1|3]), (626,627,[3_1|3]), (627,628,[1_1|3]), (628,629,[3_1|3]), (629,125,[0_1|3]), (629,448,[0_1|3]), (629,145,[0_1|3]), (630,631,[2_1|3]), (631,632,[3_1|3]), (632,633,[1_1|3]), (633,634,[3_1|3]), (634,112,[0_1|3]), (634,131,[0_1|3]), (634,150,[0_1|3]), (634,174,[0_1|3]), (634,181,[0_1|3]), (634,191,[0_1|3]), (634,196,[0_1|3]), (634,202,[0_1|3]), (634,209,[0_1|3]), (634,226,[0_1|3]), (634,252,[0_1|3]), (634,290,[0_1|3]), (634,297,[0_1|3]), (634,304,[0_1|3]), (634,310,[0_1|3]), (634,316,[0_1|3]), (634,330,[0_1|3]), (634,344,[0_1|3]), (634,350,[0_1|3]), (634,357,[0_1|3]), (634,364,[0_1|3]), (634,382,[0_1|3]), (634,401,[0_1|3]), (634,426,[0_1|3]), (634,466,[0_1|3]), (634,473,[0_1|3]), (634,510,[0_1|3]), (634,529,[0_1|3]), (634,587,[0_1|3]), (634,125,[0_1|3]), (634,145,[0_1|3]), (635,636,[3_1|3]), (636,637,[1_1|3]), (637,638,[3_1|3]), (638,639,[0_1|3]), (638,625,[0_1|3]), (639,144,[1_1|3]), (639,581,[1_1|3]), (640,641,[5_1|3]), (641,642,[4_1|3]), (642,643,[4_1|3]), (643,644,[3_1|3]), (644,645,[1_1|3]), (645,157,[4_1|3]), (646,647,[5_1|3]), (647,648,[4_1|3]), (648,649,[3_1|3]), (649,650,[1_1|3]), (650,651,[3_1|3]), (651,652,[2_1|3]), (652,157,[0_1|3]), (653,654,[3_1|3]), (654,655,[1_1|3]), (655,656,[3_1|3]), (656,657,[0_1|3]), (657,227,[0_1|3]), (657,305,[0_1|3]), (657,331,[0_1|3]), (657,358,[0_1|3]), (657,365,[0_1|3]), (657,383,[0_1|3]), (657,126,[0_1|3]), (658,659,[1_1|3]), (659,660,[3_1|3]), (660,661,[0_1|3]), (660,658,[3_1|3]), (660,662,[0_1|3]), (660,667,[0_1|3]), (660,673,[0_1|3]), (661,124,[2_1|3]), (661,490,[2_1|3]), (661,524,[2_1|3]), (661,600,[2_1|3]), (661,601,[2_1|3]), (662,663,[2_1|3]), (663,664,[3_1|3]), (664,665,[3_1|3]), (665,666,[1_1|3]), (666,124,[3_1|3]), (666,490,[3_1|3]), (666,524,[3_1|3]), (666,600,[3_1|3]), (666,601,[3_1|3]), (667,668,[4_1|3]), (668,669,[3_1|3]), (669,670,[1_1|3]), (670,671,[3_1|3]), (671,672,[2_1|3]), (672,124,[2_1|3]), (672,490,[2_1|3]), (672,524,[2_1|3]), (672,600,[2_1|3]), (672,601,[2_1|3]), (673,674,[2_1|3]), (674,675,[3_1|3]), (675,676,[2_1|3]), (676,677,[2_1|3]), (677,678,[1_1|3]), (678,679,[3_1|3]), (679,124,[2_1|3]), (679,490,[2_1|3]), (679,524,[2_1|3]), (679,600,[2_1|3]), (679,601,[2_1|3]), (680,681,[3_1|3]), (681,682,[2_1|3]), (682,683,[2_1|3]), (683,684,[1_1|3]), (684,491,[4_1|3]), (685,686,[2_1|3]), (686,687,[1_1|3]), (687,688,[0_1|3]), (688,689,[4_1|3]), (689,491,[3_1|3]), (690,691,[3_1|3]), (691,692,[2_1|3]), (692,693,[2_1|3]), (693,694,[1_1|3]), (694,143,[5_1|3]), (694,163,[5_1|3]), (694,168,[5_1|3]), (694,337,[5_1|3]), (694,371,[5_1|3]), (694,376,[5_1|3]), (694,433,[5_1|3]), (694,453,[5_1|3]), (694,460,[5_1|3]), (694,517,[5_1|3]), (694,543,[5_1|3]), (694,548,[5_1|3]), (694,580,[5_1|3]), (694,606,[5_1|3]), (694,613,[5_1|3]), (695,696,[3_1|3]), (696,697,[1_1|3]), (697,698,[3_1|3]), (698,699,[0_1|3]), (698,625,[0_1|3]), (699,124,[1_1|3]), (699,490,[1_1|3]), (699,524,[1_1|3]), (699,600,[1_1|3]), (699,144,[1_1|3]), (699,581,[1_1|3]), (699,620,[2_1|3]), (700,701,[3_1|3]), (701,702,[1_1|3]), (702,703,[0_1|3]), (703,704,[4_1|3]), (704,705,[3_1|3]), (705,491,[1_1|3]), (706,707,[3_1|3]), (707,708,[1_1|3]), (708,709,[3_1|3]), (709,710,[0_1|3]), (710,112,[0_1|3]), (710,131,[0_1|3]), (710,150,[0_1|3]), (710,174,[0_1|3]), (710,181,[0_1|3]), (710,191,[0_1|3]), (710,196,[0_1|3]), (710,202,[0_1|3]), (710,209,[0_1|3]), (710,226,[0_1|3]), (710,252,[0_1|3]), (710,290,[0_1|3]), (710,297,[0_1|3]), (710,304,[0_1|3]), (710,310,[0_1|3]), (710,316,[0_1|3]), (710,330,[0_1|3]), (710,344,[0_1|3]), (710,350,[0_1|3]), (710,357,[0_1|3]), (710,364,[0_1|3]), (710,382,[0_1|3]), (710,401,[0_1|3]), (710,426,[0_1|3]), (710,466,[0_1|3]), (710,473,[0_1|3]), (710,510,[0_1|3]), (710,529,[0_1|3]), (710,587,[0_1|3]), (710,338,[0_1|3]), (710,227,[0_1|3]), (710,305,[0_1|3]), (710,331,[0_1|3]), (710,358,[0_1|3]), (710,365,[0_1|3]), (710,383,[0_1|3]), (710,126,[0_1|3]), (711,712,[0_1|3]), (712,713,[4_1|3]), (713,714,[4_1|3]), (714,715,[3_1|3]), (715,716,[2_1|3]), (716,324,[0_1|3]), (717,718,[2_1|3]), (718,719,[0_1|3]), (719,720,[4_1|3]), (720,721,[4_1|3]), (721,722,[5_1|3]), (722,137,[4_1|3]), (722,446,[4_1|3]), (722,536,[4_1|3]), (722,553,[4_1|3]), (722,593,[4_1|3]), (722,372,[4_1|3]), (722,377,[4_1|3]), (722,434,[4_1|3]), (722,454,[4_1|3]), (722,461,[4_1|3]), (722,518,[4_1|3]), (722,549,[4_1|3]), (722,614,[4_1|3]), (722,158,[4_1|3]), (723,724,[2_1|3]), (724,725,[0_1|3]), (725,726,[4_1|3]), (726,143,[5_1|3]), (726,163,[5_1|3]), (726,168,[5_1|3]), (726,337,[5_1|3]), (726,371,[5_1|3]), (726,376,[5_1|3]), (726,433,[5_1|3]), (726,453,[5_1|3]), (726,460,[5_1|3]), (726,517,[5_1|3]), (726,543,[5_1|3]), (726,548,[5_1|3]), (726,580,[5_1|3]), (726,606,[5_1|3]), (726,613,[5_1|3]), (727,728,[1_1|3]), (728,729,[3_1|3]), (729,730,[0_1|3]), (729,187,[3_1|2]), (729,191,[0_1|2]), (729,196,[0_1|2]), (729,202,[0_1|2]), (729,209,[0_1|2]), (729,214,[3_1|2]), (729,219,[3_1|2]), (729,226,[0_1|2]), (729,233,[3_1|2]), (729,238,[3_1|2]), (729,245,[3_1|2]), (729,252,[0_1|2]), (729,259,[3_1|2]), (729,266,[3_1|2]), (729,272,[3_1|2]), (729,276,[3_1|2]), (729,283,[3_1|2]), (729,290,[0_1|2]), (729,297,[0_1|2]), (729,658,[3_1|3]), (729,662,[0_1|3]), (729,667,[0_1|3]), (729,673,[0_1|3]), (729,680,[0_1|3]), (729,685,[3_1|3]), (730,111,[2_1|3]), (730,124,[2_1|3]), (730,490,[2_1|3]), (730,524,[2_1|3]), (730,600,[2_1|3]), (731,732,[2_1|3]), (732,733,[3_1|3]), (733,734,[3_1|3]), (734,735,[1_1|3]), (735,111,[3_1|3]), (735,124,[3_1|3]), (735,490,[3_1|3]), (735,524,[3_1|3]), (735,600,[3_1|3]), (736,737,[4_1|3]), (737,738,[3_1|3]), (738,739,[1_1|3]), (739,740,[3_1|3]), (740,741,[2_1|3]), (741,111,[2_1|3]), (741,124,[2_1|3]), (741,490,[2_1|3]), (741,524,[2_1|3]), (741,600,[2_1|3]), (742,743,[2_1|3]), (743,744,[3_1|3]), (744,745,[2_1|3]), (745,746,[2_1|3]), (746,747,[1_1|3]), (747,748,[3_1|3]), (748,111,[2_1|3]), (748,124,[2_1|3]), (748,490,[2_1|3]), (748,524,[2_1|3]), (748,600,[2_1|3]), (749,750,[3_1|3]), (750,751,[2_1|3]), (751,752,[2_1|3]), (752,753,[1_1|3]), (753,137,[4_1|3]), (753,446,[4_1|3]), (753,536,[4_1|3]), (753,553,[4_1|3]), (753,593,[4_1|3]), (753,491,[4_1|3]), (754,755,[2_1|3]), (755,756,[1_1|3]), (756,757,[0_1|3]), (757,758,[4_1|3]), (758,137,[3_1|3]), (758,446,[3_1|3]), (758,536,[3_1|3]), (758,553,[3_1|3]), (758,593,[3_1|3]), (758,491,[3_1|3]), (759,760,[2_1|3]), (760,761,[3_1|3]), (761,762,[1_1|3]), (762,763,[3_1|3]), (763,338,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(3(3(4(x1)))) ->^+ 5(4(3(2(3(0(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [x1 / 3(3(4(x1)))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(1(0(x1))) -> 0(2(3(1(3(0(x1)))))) 0(2(1(x1))) -> 3(1(3(0(2(x1))))) 0(2(1(x1))) -> 0(2(3(3(1(3(x1)))))) 0(2(1(x1))) -> 0(4(3(1(3(2(2(x1))))))) 0(2(1(x1))) -> 0(2(3(2(2(1(3(2(x1)))))))) 1(0(0(x1))) -> 2(3(1(3(0(0(x1)))))) 0(0(2(0(x1)))) -> 0(0(4(4(3(2(0(x1))))))) 0(1(2(1(x1)))) -> 0(3(1(3(1(3(2(x1))))))) 0(1(3(5(x1)))) -> 0(5(4(4(3(1(4(x1))))))) 0(1(3(5(x1)))) -> 3(5(4(3(1(3(2(0(x1)))))))) 0(1(5(1(x1)))) -> 5(3(1(3(0(1(x1)))))) 0(2(1(4(x1)))) -> 0(3(2(2(1(4(x1)))))) 0(2(1(4(x1)))) -> 3(2(1(0(4(3(x1)))))) 0(2(2(1(x1)))) -> 3(2(0(2(3(1(x1)))))) 0(2(2(5(x1)))) -> 3(1(3(2(2(0(4(5(x1)))))))) 0(2(4(4(x1)))) -> 3(2(0(4(4(5(4(x1))))))) 0(2(4(5(x1)))) -> 3(2(0(4(5(x1))))) 0(3(3(4(x1)))) -> 5(4(3(2(3(0(x1)))))) 0(3(5(1(x1)))) -> 5(4(3(2(2(0(1(x1))))))) 1(0(2(4(x1)))) -> 2(3(0(3(1(4(x1)))))) 1(1(2(5(x1)))) -> 1(3(2(2(1(5(x1)))))) 0(0(0(3(4(x1))))) -> 0(0(4(4(5(2(3(0(x1)))))))) 0(0(1(1(5(x1))))) -> 5(0(3(1(3(1(0(5(x1)))))))) 0(1(2(2(4(x1))))) -> 4(3(2(1(2(3(0(x1))))))) 0(1(5(1(4(x1))))) -> 5(3(1(0(4(3(1(x1))))))) 0(2(2(0(1(x1))))) -> 0(4(3(0(2(3(2(1(x1)))))))) 0(2(2(2(5(x1))))) -> 3(2(2(5(4(3(2(0(x1)))))))) 0(2(2(5(0(x1))))) -> 3(2(2(0(5(2(3(0(x1)))))))) 0(3(4(1(4(x1))))) -> 2(3(3(1(0(5(4(4(x1)))))))) 0(4(1(3(4(x1))))) -> 3(1(3(0(4(4(5(x1))))))) 0(4(2(2(4(x1))))) -> 3(2(2(0(4(4(x1)))))) 0(4(2(4(1(x1))))) -> 3(1(0(4(4(3(2(x1))))))) 0(5(2(1(5(x1))))) -> 5(4(5(2(3(1(0(x1))))))) 1(0(2(2(4(x1))))) -> 2(2(0(4(3(1(4(x1))))))) 1(0(2(2(4(x1))))) -> 2(3(3(2(1(0(4(3(x1)))))))) 1(2(2(0(1(x1))))) -> 4(3(2(1(0(2(3(1(x1)))))))) 1(2(3(5(0(x1))))) -> 5(2(3(1(3(0(x1)))))) 1(2(4(3(5(x1))))) -> 5(4(3(1(3(2(x1)))))) 0(0(1(3(3(5(x1)))))) -> 0(3(2(3(0(1(5(x1))))))) 0(0(2(2(3(0(x1)))))) -> 0(2(3(0(3(2(0(x1))))))) 0(0(2(2(5(1(x1)))))) -> 0(3(0(2(3(2(1(5(x1)))))))) 0(0(3(5(3(0(x1)))))) -> 0(5(0(3(1(3(3(0(x1)))))))) 0(0(4(0(5(1(x1)))))) -> 0(0(4(0(4(3(1(5(x1)))))))) 0(0(4(1(3(0(x1)))))) -> 0(0(3(2(3(0(1(4(x1)))))))) 0(1(0(3(4(5(x1)))))) -> 3(1(0(4(3(5(0(4(x1)))))))) 0(1(4(0(2(5(x1)))))) -> 0(5(2(0(4(4(1(x1))))))) 0(1(5(3(4(0(x1)))))) -> 0(4(4(2(0(3(1(5(x1)))))))) 0(2(1(5(3(5(x1)))))) -> 3(2(1(3(5(1(5(0(x1)))))))) 0(3(5(2(4(1(x1)))))) -> 0(0(5(4(3(2(1(x1))))))) 0(5(0(1(5(4(x1)))))) -> 3(1(3(0(0(5(4(5(x1)))))))) 0(5(1(1(3(4(x1)))))) -> 4(1(0(4(5(4(3(1(x1)))))))) 0(5(3(4(1(4(x1)))))) -> 5(4(0(4(3(1(3(x1))))))) 0(5(3(4(1(5(x1)))))) -> 0(3(2(1(0(5(4(5(x1)))))))) 0(5(5(0(1(5(x1)))))) -> 0(5(5(3(2(1(5(0(x1)))))))) 1(0(2(4(0(5(x1)))))) -> 1(4(0(3(2(3(0(5(x1)))))))) 1(1(4(0(1(0(x1)))))) -> 0(3(1(3(1(4(1(0(x1)))))))) 1(2(1(0(2(1(x1)))))) -> 1(1(3(2(0(2(1(x1))))))) 1(2(4(0(3(4(x1)))))) -> 3(2(0(4(3(3(1(4(x1)))))))) 1(2(4(3(3(0(x1)))))) -> 4(0(2(3(3(1(3(0(x1)))))))) 1(2(4(3(3(5(x1)))))) -> 3(2(3(2(5(4(3(1(x1)))))))) 1(2(4(3(4(5(x1)))))) -> 3(1(3(2(4(5(4(x1))))))) 1(2(4(5(1(0(x1)))))) -> 0(4(2(1(3(1(5(x1))))))) 1(2(4(5(3(4(x1)))))) -> 4(3(2(2(1(3(5(4(x1)))))))) 1(2(5(5(3(0(x1)))))) -> 5(2(3(3(1(3(5(0(x1)))))))) 0(0(2(4(5(3(5(x1))))))) -> 2(0(5(4(3(5(3(0(x1)))))))) 0(1(0(5(3(5(1(x1))))))) -> 1(0(0(5(5(3(1(3(x1)))))))) 0(1(2(2(5(2(0(x1))))))) -> 5(1(0(3(2(2(2(0(x1)))))))) 0(2(1(5(4(0(1(x1))))))) -> 0(0(5(1(2(3(1(4(x1)))))))) 0(2(3(4(1(1(4(x1))))))) -> 3(1(0(4(4(1(2(0(x1)))))))) 0(2(4(1(1(5(1(x1))))))) -> 3(1(0(5(4(1(1(2(x1)))))))) 0(2(5(0(4(0(1(x1))))))) -> 0(5(1(0(4(3(2(0(x1)))))))) 0(2(5(5(3(5(1(x1))))))) -> 0(5(5(1(5(5(3(2(x1)))))))) 0(4(0(3(5(2(0(x1))))))) -> 0(4(0(4(5(3(2(0(x1)))))))) 0(4(1(1(4(1(5(x1))))))) -> 0(4(3(1(4(1(1(5(x1)))))))) 0(4(1(4(0(3(5(x1))))))) -> 3(1(5(0(0(4(3(4(x1)))))))) 0(5(1(3(4(2(4(x1))))))) -> 5(4(4(0(3(1(5(2(x1)))))))) 1(0(3(3(3(4(0(x1))))))) -> 0(3(0(3(3(1(4(5(x1)))))))) 1(0(3(4(4(3(5(x1))))))) -> 5(4(3(1(3(0(2(4(x1)))))))) 1(2(0(4(1(3(5(x1))))))) -> 5(4(2(3(1(4(1(0(x1)))))))) 1(2(4(0(5(3(5(x1))))))) -> 5(1(3(2(3(5(0(4(x1)))))))) The (relative) TRS S consists of the following rules: 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(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: INNERMOST