/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 61 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 80 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 3(3(2(x1))) 1(2(3(2(x1)))) -> 3(4(4(2(x1)))) 0(5(1(4(1(x1))))) -> 4(1(4(3(x1)))) 4(3(0(2(2(x1))))) -> 4(1(1(4(5(x1))))) 5(5(1(5(2(x1))))) -> 4(3(5(2(x1)))) 0(4(3(3(4(4(1(x1))))))) -> 4(4(2(5(0(2(2(x1))))))) 2(0(4(1(2(2(1(3(x1)))))))) -> 2(3(0(4(5(5(1(1(x1)))))))) 1(2(2(1(5(2(1(2(1(x1))))))))) -> 1(2(2(2(0(2(4(4(3(x1))))))))) 4(5(1(4(3(4(3(5(4(3(x1)))))))))) -> 4(3(2(0(2(4(3(2(3(x1))))))))) 1(4(1(2(5(3(4(3(3(2(2(x1))))))))))) -> 0(0(1(4(0(4(5(2(3(0(4(x1))))))))))) 4(5(1(3(2(2(5(4(3(5(4(x1))))))))))) -> 4(0(1(1(5(3(5(4(2(2(4(x1))))))))))) 5(1(4(0(1(5(5(3(3(0(3(2(x1)))))))))))) -> 3(0(5(3(2(0(1(0(4(1(2(x1))))))))))) 0(4(1(1(3(3(2(5(4(2(2(1(3(x1))))))))))))) -> 1(0(1(4(3(4(4(2(3(4(2(2(1(x1))))))))))))) 5(1(3(3(5(3(1(3(2(1(2(0(4(x1))))))))))))) -> 2(2(3(2(3(2(3(5(2(5(1(4(x1)))))))))))) 0(5(3(5(3(3(3(3(4(5(5(5(4(4(x1)))))))))))))) -> 0(1(0(3(2(3(4(0(5(5(2(4(0(x1))))))))))))) 1(5(0(1(0(4(4(2(2(3(4(1(4(1(x1)))))))))))))) -> 3(4(5(0(0(4(0(3(5(0(4(1(5(4(x1)))))))))))))) 5(1(0(5(2(2(2(3(3(2(5(1(5(1(x1)))))))))))))) -> 5(4(0(1(4(3(2(2(3(3(3(5(5(1(x1)))))))))))))) 5(1(2(2(4(0(2(4(2(5(2(1(4(0(5(x1))))))))))))))) -> 0(1(4(3(0(5(3(4(3(3(1(4(1(5(x1)))))))))))))) 0(4(0(3(2(0(2(1(2(0(0(2(4(2(3(4(x1)))))))))))))))) -> 3(2(1(3(3(4(5(5(4(0(3(2(1(2(3(x1))))))))))))))) 1(5(1(3(3(3(0(4(0(2(3(1(5(1(4(2(x1)))))))))))))))) -> 3(4(0(5(0(4(4(0(2(1(3(1(4(0(4(2(x1)))))))))))))))) 5(1(1(0(0(3(2(5(0(3(4(2(1(2(5(1(x1)))))))))))))))) -> 5(1(3(2(1(0(1(0(5(5(3(1(1(4(1(0(x1)))))))))))))))) 3(2(1(2(4(2(1(1(3(3(3(5(2(2(0(4(4(x1))))))))))))))))) -> 3(2(3(1(0(2(0(2(4(5(5(4(1(0(0(2(5(0(x1)))))))))))))))))) 4(1(0(3(0(4(3(2(2(1(3(2(2(4(0(2(4(x1))))))))))))))))) -> 4(2(5(3(3(3(1(2(4(5(3(5(3(5(1(3(4(x1))))))))))))))))) 3(0(0(2(1(1(3(5(1(2(2(2(5(1(0(0(0(1(x1)))))))))))))))))) -> 3(5(1(0(4(0(1(2(2(5(0(3(4(3(5(5(4(3(x1)))))))))))))))))) 4(5(5(1(5(3(5(3(2(0(4(4(2(1(0(3(5(3(x1)))))))))))))))))) -> 4(4(0(5(1(3(5(5(3(4(4(0(0(4(3(0(0(0(x1)))))))))))))))))) 4(5(5(2(5(1(0(2(1(0(1(4(4(4(2(1(5(1(x1)))))))))))))))))) -> 4(3(2(3(0(2(5(3(4(1(4(4(5(1(1(4(0(x1))))))))))))))))) 0(2(0(2(2(0(1(1(2(4(1(1(0(3(3(2(1(4(1(4(x1)))))))))))))))))))) -> 2(2(3(3(0(2(1(3(5(3(4(4(1(2(4(4(4(4(4(0(x1)))))))))))))))))))) 3(4(1(1(0(3(4(0(5(5(5(5(3(5(2(3(2(3(1(3(x1)))))))))))))))))))) -> 3(4(3(4(3(1(0(1(1(4(5(5(2(3(2(3(0(2(3(x1))))))))))))))))))) 4(5(2(3(5(4(5(0(5(1(2(3(0(1(1(0(3(5(0(3(0(x1))))))))))))))))))))) -> 4(4(3(1(4(4(0(5(3(5(2(1(4(2(4(1(0(2(4(5(0(x1))))))))))))))))))))) 5(1(4(0(0(3(4(2(3(0(3(5(4(0(4(2(4(0(0(5(0(x1))))))))))))))))))))) -> 5(0(3(2(2(0(4(1(1(5(3(0(1(5(0(1(3(2(2(3(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 3(3(2(x1))) 1(2(3(2(x1)))) -> 3(4(4(2(x1)))) 0(5(1(4(1(x1))))) -> 4(1(4(3(x1)))) 4(3(0(2(2(x1))))) -> 4(1(1(4(5(x1))))) 5(5(1(5(2(x1))))) -> 4(3(5(2(x1)))) 0(4(3(3(4(4(1(x1))))))) -> 4(4(2(5(0(2(2(x1))))))) 2(0(4(1(2(2(1(3(x1)))))))) -> 2(3(0(4(5(5(1(1(x1)))))))) 1(2(2(1(5(2(1(2(1(x1))))))))) -> 1(2(2(2(0(2(4(4(3(x1))))))))) 4(5(1(4(3(4(3(5(4(3(x1)))))))))) -> 4(3(2(0(2(4(3(2(3(x1))))))))) 1(4(1(2(5(3(4(3(3(2(2(x1))))))))))) -> 0(0(1(4(0(4(5(2(3(0(4(x1))))))))))) 4(5(1(3(2(2(5(4(3(5(4(x1))))))))))) -> 4(0(1(1(5(3(5(4(2(2(4(x1))))))))))) 5(1(4(0(1(5(5(3(3(0(3(2(x1)))))))))))) -> 3(0(5(3(2(0(1(0(4(1(2(x1))))))))))) 0(4(1(1(3(3(2(5(4(2(2(1(3(x1))))))))))))) -> 1(0(1(4(3(4(4(2(3(4(2(2(1(x1))))))))))))) 5(1(3(3(5(3(1(3(2(1(2(0(4(x1))))))))))))) -> 2(2(3(2(3(2(3(5(2(5(1(4(x1)))))))))))) 0(5(3(5(3(3(3(3(4(5(5(5(4(4(x1)))))))))))))) -> 0(1(0(3(2(3(4(0(5(5(2(4(0(x1))))))))))))) 1(5(0(1(0(4(4(2(2(3(4(1(4(1(x1)))))))))))))) -> 3(4(5(0(0(4(0(3(5(0(4(1(5(4(x1)))))))))))))) 5(1(0(5(2(2(2(3(3(2(5(1(5(1(x1)))))))))))))) -> 5(4(0(1(4(3(2(2(3(3(3(5(5(1(x1)))))))))))))) 5(1(2(2(4(0(2(4(2(5(2(1(4(0(5(x1))))))))))))))) -> 0(1(4(3(0(5(3(4(3(3(1(4(1(5(x1)))))))))))))) 0(4(0(3(2(0(2(1(2(0(0(2(4(2(3(4(x1)))))))))))))))) -> 3(2(1(3(3(4(5(5(4(0(3(2(1(2(3(x1))))))))))))))) 1(5(1(3(3(3(0(4(0(2(3(1(5(1(4(2(x1)))))))))))))))) -> 3(4(0(5(0(4(4(0(2(1(3(1(4(0(4(2(x1)))))))))))))))) 5(1(1(0(0(3(2(5(0(3(4(2(1(2(5(1(x1)))))))))))))))) -> 5(1(3(2(1(0(1(0(5(5(3(1(1(4(1(0(x1)))))))))))))))) 3(2(1(2(4(2(1(1(3(3(3(5(2(2(0(4(4(x1))))))))))))))))) -> 3(2(3(1(0(2(0(2(4(5(5(4(1(0(0(2(5(0(x1)))))))))))))))))) 4(1(0(3(0(4(3(2(2(1(3(2(2(4(0(2(4(x1))))))))))))))))) -> 4(2(5(3(3(3(1(2(4(5(3(5(3(5(1(3(4(x1))))))))))))))))) 3(0(0(2(1(1(3(5(1(2(2(2(5(1(0(0(0(1(x1)))))))))))))))))) -> 3(5(1(0(4(0(1(2(2(5(0(3(4(3(5(5(4(3(x1)))))))))))))))))) 4(5(5(1(5(3(5(3(2(0(4(4(2(1(0(3(5(3(x1)))))))))))))))))) -> 4(4(0(5(1(3(5(5(3(4(4(0(0(4(3(0(0(0(x1)))))))))))))))))) 4(5(5(2(5(1(0(2(1(0(1(4(4(4(2(1(5(1(x1)))))))))))))))))) -> 4(3(2(3(0(2(5(3(4(1(4(4(5(1(1(4(0(x1))))))))))))))))) 0(2(0(2(2(0(1(1(2(4(1(1(0(3(3(2(1(4(1(4(x1)))))))))))))))))))) -> 2(2(3(3(0(2(1(3(5(3(4(4(1(2(4(4(4(4(4(0(x1)))))))))))))))))))) 3(4(1(1(0(3(4(0(5(5(5(5(3(5(2(3(2(3(1(3(x1)))))))))))))))))))) -> 3(4(3(4(3(1(0(1(1(4(5(5(2(3(2(3(0(2(3(x1))))))))))))))))))) 4(5(2(3(5(4(5(0(5(1(2(3(0(1(1(0(3(5(0(3(0(x1))))))))))))))))))))) -> 4(4(3(1(4(4(0(5(3(5(2(1(4(2(4(1(0(2(4(5(0(x1))))))))))))))))))))) 5(1(4(0(0(3(4(2(3(0(3(5(4(0(4(2(4(0(0(5(0(x1))))))))))))))))))))) -> 5(0(3(2(2(0(4(1(1(5(3(0(1(5(0(1(3(2(2(3(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 3(3(2(x1))) 1(2(3(2(x1)))) -> 3(4(4(2(x1)))) 0(5(1(4(1(x1))))) -> 4(1(4(3(x1)))) 4(3(0(2(2(x1))))) -> 4(1(1(4(5(x1))))) 5(5(1(5(2(x1))))) -> 4(3(5(2(x1)))) 0(4(3(3(4(4(1(x1))))))) -> 4(4(2(5(0(2(2(x1))))))) 2(0(4(1(2(2(1(3(x1)))))))) -> 2(3(0(4(5(5(1(1(x1)))))))) 1(2(2(1(5(2(1(2(1(x1))))))))) -> 1(2(2(2(0(2(4(4(3(x1))))))))) 4(5(1(4(3(4(3(5(4(3(x1)))))))))) -> 4(3(2(0(2(4(3(2(3(x1))))))))) 1(4(1(2(5(3(4(3(3(2(2(x1))))))))))) -> 0(0(1(4(0(4(5(2(3(0(4(x1))))))))))) 4(5(1(3(2(2(5(4(3(5(4(x1))))))))))) -> 4(0(1(1(5(3(5(4(2(2(4(x1))))))))))) 5(1(4(0(1(5(5(3(3(0(3(2(x1)))))))))))) -> 3(0(5(3(2(0(1(0(4(1(2(x1))))))))))) 0(4(1(1(3(3(2(5(4(2(2(1(3(x1))))))))))))) -> 1(0(1(4(3(4(4(2(3(4(2(2(1(x1))))))))))))) 5(1(3(3(5(3(1(3(2(1(2(0(4(x1))))))))))))) -> 2(2(3(2(3(2(3(5(2(5(1(4(x1)))))))))))) 0(5(3(5(3(3(3(3(4(5(5(5(4(4(x1)))))))))))))) -> 0(1(0(3(2(3(4(0(5(5(2(4(0(x1))))))))))))) 1(5(0(1(0(4(4(2(2(3(4(1(4(1(x1)))))))))))))) -> 3(4(5(0(0(4(0(3(5(0(4(1(5(4(x1)))))))))))))) 5(1(0(5(2(2(2(3(3(2(5(1(5(1(x1)))))))))))))) -> 5(4(0(1(4(3(2(2(3(3(3(5(5(1(x1)))))))))))))) 5(1(2(2(4(0(2(4(2(5(2(1(4(0(5(x1))))))))))))))) -> 0(1(4(3(0(5(3(4(3(3(1(4(1(5(x1)))))))))))))) 0(4(0(3(2(0(2(1(2(0(0(2(4(2(3(4(x1)))))))))))))))) -> 3(2(1(3(3(4(5(5(4(0(3(2(1(2(3(x1))))))))))))))) 1(5(1(3(3(3(0(4(0(2(3(1(5(1(4(2(x1)))))))))))))))) -> 3(4(0(5(0(4(4(0(2(1(3(1(4(0(4(2(x1)))))))))))))))) 5(1(1(0(0(3(2(5(0(3(4(2(1(2(5(1(x1)))))))))))))))) -> 5(1(3(2(1(0(1(0(5(5(3(1(1(4(1(0(x1)))))))))))))))) 3(2(1(2(4(2(1(1(3(3(3(5(2(2(0(4(4(x1))))))))))))))))) -> 3(2(3(1(0(2(0(2(4(5(5(4(1(0(0(2(5(0(x1)))))))))))))))))) 4(1(0(3(0(4(3(2(2(1(3(2(2(4(0(2(4(x1))))))))))))))))) -> 4(2(5(3(3(3(1(2(4(5(3(5(3(5(1(3(4(x1))))))))))))))))) 3(0(0(2(1(1(3(5(1(2(2(2(5(1(0(0(0(1(x1)))))))))))))))))) -> 3(5(1(0(4(0(1(2(2(5(0(3(4(3(5(5(4(3(x1)))))))))))))))))) 4(5(5(1(5(3(5(3(2(0(4(4(2(1(0(3(5(3(x1)))))))))))))))))) -> 4(4(0(5(1(3(5(5(3(4(4(0(0(4(3(0(0(0(x1)))))))))))))))))) 4(5(5(2(5(1(0(2(1(0(1(4(4(4(2(1(5(1(x1)))))))))))))))))) -> 4(3(2(3(0(2(5(3(4(1(4(4(5(1(1(4(0(x1))))))))))))))))) 0(2(0(2(2(0(1(1(2(4(1(1(0(3(3(2(1(4(1(4(x1)))))))))))))))))))) -> 2(2(3(3(0(2(1(3(5(3(4(4(1(2(4(4(4(4(4(0(x1)))))))))))))))))))) 3(4(1(1(0(3(4(0(5(5(5(5(3(5(2(3(2(3(1(3(x1)))))))))))))))))))) -> 3(4(3(4(3(1(0(1(1(4(5(5(2(3(2(3(0(2(3(x1))))))))))))))))))) 4(5(2(3(5(4(5(0(5(1(2(3(0(1(1(0(3(5(0(3(0(x1))))))))))))))))))))) -> 4(4(3(1(4(4(0(5(3(5(2(1(4(2(4(1(0(2(4(5(0(x1))))))))))))))))))))) 5(1(4(0(0(3(4(2(3(0(3(5(4(0(4(2(4(0(0(5(0(x1))))))))))))))))))))) -> 5(0(3(2(2(0(4(1(1(5(3(0(1(5(0(1(3(2(2(3(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(2(1(x1)))) -> 3(3(2(x1))) 1(2(3(2(x1)))) -> 3(4(4(2(x1)))) 0(5(1(4(1(x1))))) -> 4(1(4(3(x1)))) 4(3(0(2(2(x1))))) -> 4(1(1(4(5(x1))))) 5(5(1(5(2(x1))))) -> 4(3(5(2(x1)))) 0(4(3(3(4(4(1(x1))))))) -> 4(4(2(5(0(2(2(x1))))))) 2(0(4(1(2(2(1(3(x1)))))))) -> 2(3(0(4(5(5(1(1(x1)))))))) 1(2(2(1(5(2(1(2(1(x1))))))))) -> 1(2(2(2(0(2(4(4(3(x1))))))))) 4(5(1(4(3(4(3(5(4(3(x1)))))))))) -> 4(3(2(0(2(4(3(2(3(x1))))))))) 1(4(1(2(5(3(4(3(3(2(2(x1))))))))))) -> 0(0(1(4(0(4(5(2(3(0(4(x1))))))))))) 4(5(1(3(2(2(5(4(3(5(4(x1))))))))))) -> 4(0(1(1(5(3(5(4(2(2(4(x1))))))))))) 5(1(4(0(1(5(5(3(3(0(3(2(x1)))))))))))) -> 3(0(5(3(2(0(1(0(4(1(2(x1))))))))))) 0(4(1(1(3(3(2(5(4(2(2(1(3(x1))))))))))))) -> 1(0(1(4(3(4(4(2(3(4(2(2(1(x1))))))))))))) 5(1(3(3(5(3(1(3(2(1(2(0(4(x1))))))))))))) -> 2(2(3(2(3(2(3(5(2(5(1(4(x1)))))))))))) 0(5(3(5(3(3(3(3(4(5(5(5(4(4(x1)))))))))))))) -> 0(1(0(3(2(3(4(0(5(5(2(4(0(x1))))))))))))) 1(5(0(1(0(4(4(2(2(3(4(1(4(1(x1)))))))))))))) -> 3(4(5(0(0(4(0(3(5(0(4(1(5(4(x1)))))))))))))) 5(1(0(5(2(2(2(3(3(2(5(1(5(1(x1)))))))))))))) -> 5(4(0(1(4(3(2(2(3(3(3(5(5(1(x1)))))))))))))) 5(1(2(2(4(0(2(4(2(5(2(1(4(0(5(x1))))))))))))))) -> 0(1(4(3(0(5(3(4(3(3(1(4(1(5(x1)))))))))))))) 0(4(0(3(2(0(2(1(2(0(0(2(4(2(3(4(x1)))))))))))))))) -> 3(2(1(3(3(4(5(5(4(0(3(2(1(2(3(x1))))))))))))))) 1(5(1(3(3(3(0(4(0(2(3(1(5(1(4(2(x1)))))))))))))))) -> 3(4(0(5(0(4(4(0(2(1(3(1(4(0(4(2(x1)))))))))))))))) 5(1(1(0(0(3(2(5(0(3(4(2(1(2(5(1(x1)))))))))))))))) -> 5(1(3(2(1(0(1(0(5(5(3(1(1(4(1(0(x1)))))))))))))))) 3(2(1(2(4(2(1(1(3(3(3(5(2(2(0(4(4(x1))))))))))))))))) -> 3(2(3(1(0(2(0(2(4(5(5(4(1(0(0(2(5(0(x1)))))))))))))))))) 4(1(0(3(0(4(3(2(2(1(3(2(2(4(0(2(4(x1))))))))))))))))) -> 4(2(5(3(3(3(1(2(4(5(3(5(3(5(1(3(4(x1))))))))))))))))) 3(0(0(2(1(1(3(5(1(2(2(2(5(1(0(0(0(1(x1)))))))))))))))))) -> 3(5(1(0(4(0(1(2(2(5(0(3(4(3(5(5(4(3(x1)))))))))))))))))) 4(5(5(1(5(3(5(3(2(0(4(4(2(1(0(3(5(3(x1)))))))))))))))))) -> 4(4(0(5(1(3(5(5(3(4(4(0(0(4(3(0(0(0(x1)))))))))))))))))) 4(5(5(2(5(1(0(2(1(0(1(4(4(4(2(1(5(1(x1)))))))))))))))))) -> 4(3(2(3(0(2(5(3(4(1(4(4(5(1(1(4(0(x1))))))))))))))))) 0(2(0(2(2(0(1(1(2(4(1(1(0(3(3(2(1(4(1(4(x1)))))))))))))))))))) -> 2(2(3(3(0(2(1(3(5(3(4(4(1(2(4(4(4(4(4(0(x1)))))))))))))))))))) 3(4(1(1(0(3(4(0(5(5(5(5(3(5(2(3(2(3(1(3(x1)))))))))))))))))))) -> 3(4(3(4(3(1(0(1(1(4(5(5(2(3(2(3(0(2(3(x1))))))))))))))))))) 4(5(2(3(5(4(5(0(5(1(2(3(0(1(1(0(3(5(0(3(0(x1))))))))))))))))))))) -> 4(4(3(1(4(4(0(5(3(5(2(1(4(2(4(1(0(2(4(5(0(x1))))))))))))))))))))) 5(1(4(0(0(3(4(2(3(0(3(5(4(0(4(2(4(0(0(5(0(x1))))))))))))))))))))) -> 5(0(3(2(2(0(4(1(1(5(3(0(1(5(0(1(3(2(2(3(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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. "[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] {(151,152,[0_1|0, 1_1|0, 4_1|0, 5_1|0, 2_1|0, 3_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]), (151,153,[0_1|1, 1_1|1, 4_1|1, 5_1|1, 2_1|1, 3_1|1]), (151,154,[3_1|2]), (151,156,[4_1|2]), (151,159,[0_1|2]), (151,171,[4_1|2]), (151,177,[1_1|2]), (151,189,[3_1|2]), (151,203,[2_1|2]), (151,222,[3_1|2]), (151,225,[1_1|2]), (151,233,[0_1|2]), (151,243,[3_1|2]), (151,256,[3_1|2]), (151,271,[4_1|2]), (151,275,[4_1|2]), (151,283,[4_1|2]), (151,293,[4_1|2]), (151,310,[4_1|2]), (151,326,[4_1|2]), (151,346,[4_1|2]), (151,362,[4_1|2]), (151,365,[3_1|2]), (151,375,[5_1|2]), (151,394,[2_1|2]), (151,405,[5_1|2]), (151,418,[0_1|2]), (151,431,[5_1|2]), (151,446,[2_1|2]), (151,453,[3_1|2]), (151,470,[3_1|2]), (151,487,[3_1|2]), (152,152,[cons_0_1|0, cons_1_1|0, cons_4_1|0, cons_5_1|0, cons_2_1|0, cons_3_1|0]), (153,152,[encArg_1|1]), (153,153,[0_1|1, 1_1|1, 4_1|1, 5_1|1, 2_1|1, 3_1|1]), (153,154,[3_1|2]), (153,156,[4_1|2]), (153,159,[0_1|2]), (153,171,[4_1|2]), (153,177,[1_1|2]), (153,189,[3_1|2]), (153,203,[2_1|2]), (153,222,[3_1|2]), (153,225,[1_1|2]), (153,233,[0_1|2]), (153,243,[3_1|2]), (153,256,[3_1|2]), (153,271,[4_1|2]), 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(415,362,[4_1|2]), (416,417,[5_1|2]), (416,365,[3_1|2]), (416,375,[5_1|2]), (416,394,[2_1|2]), (416,405,[5_1|2]), (416,418,[0_1|2]), (416,431,[5_1|2]), (417,153,[1_1|2]), (417,177,[1_1|2]), (417,225,[1_1|2]), (417,432,[1_1|2]), (417,222,[3_1|2]), (417,233,[0_1|2]), (417,243,[3_1|2]), (417,256,[3_1|2]), (418,419,[1_1|2]), (419,420,[4_1|2]), (420,421,[3_1|2]), (421,422,[0_1|2]), (422,423,[5_1|2]), (423,424,[3_1|2]), (424,425,[4_1|2]), (425,426,[3_1|2]), (426,427,[3_1|2]), (427,428,[1_1|2]), (428,429,[4_1|2]), (429,430,[1_1|2]), (429,243,[3_1|2]), (429,256,[3_1|2]), (430,153,[5_1|2]), (430,375,[5_1|2]), (430,405,[5_1|2]), (430,431,[5_1|2]), (430,362,[4_1|2]), (430,365,[3_1|2]), (430,394,[2_1|2]), (430,418,[0_1|2]), (431,432,[1_1|2]), (432,433,[3_1|2]), (433,434,[2_1|2]), (434,435,[1_1|2]), (435,436,[0_1|2]), (436,437,[1_1|2]), (437,438,[0_1|2]), (438,439,[5_1|2]), (439,440,[5_1|2]), (440,441,[3_1|2]), (441,442,[1_1|2]), (442,443,[1_1|2]), (443,444,[4_1|2]), (443,346,[4_1|2]), (444,445,[1_1|2]), (445,153,[0_1|2]), (445,177,[0_1|2, 1_1|2]), (445,225,[0_1|2]), (445,432,[0_1|2]), (445,154,[3_1|2]), (445,156,[4_1|2]), (445,159,[0_1|2]), (445,171,[4_1|2]), (445,189,[3_1|2]), (445,203,[2_1|2]), (446,447,[3_1|2]), (447,448,[0_1|2]), (448,449,[4_1|2]), (449,450,[5_1|2]), (450,451,[5_1|2]), (450,431,[5_1|2]), (451,452,[1_1|2]), (452,153,[1_1|2]), (452,154,[1_1|2]), (452,189,[1_1|2]), (452,222,[1_1|2, 3_1|2]), (452,243,[1_1|2, 3_1|2]), (452,256,[1_1|2, 3_1|2]), (452,365,[1_1|2]), (452,453,[1_1|2]), (452,470,[1_1|2]), (452,487,[1_1|2]), (452,225,[1_1|2]), (452,233,[0_1|2]), (453,454,[2_1|2]), (454,455,[3_1|2]), (455,456,[1_1|2]), (456,457,[0_1|2]), (457,458,[2_1|2]), (458,459,[0_1|2]), (459,460,[2_1|2]), (460,461,[4_1|2]), (461,462,[5_1|2]), (462,463,[5_1|2]), (463,464,[4_1|2]), (464,465,[1_1|2]), (465,466,[0_1|2]), (466,467,[0_1|2]), (467,468,[2_1|2]), (468,469,[5_1|2]), (469,153,[0_1|2]), (469,156,[0_1|2, 4_1|2]), (469,171,[0_1|2, 4_1|2]), (469,271,[0_1|2]), (469,275,[0_1|2]), (469,283,[0_1|2]), (469,293,[0_1|2]), (469,310,[0_1|2]), (469,326,[0_1|2]), (469,346,[0_1|2]), (469,362,[0_1|2]), (469,172,[0_1|2]), (469,294,[0_1|2]), (469,327,[0_1|2]), (469,154,[3_1|2]), (469,159,[0_1|2]), (469,177,[1_1|2]), (469,189,[3_1|2]), (469,203,[2_1|2]), (470,471,[5_1|2]), (471,472,[1_1|2]), (472,473,[0_1|2]), (473,474,[4_1|2]), (474,475,[0_1|2]), (475,476,[1_1|2]), (476,477,[2_1|2]), (477,478,[2_1|2]), (478,479,[5_1|2]), (479,480,[0_1|2]), (480,481,[3_1|2]), (481,482,[4_1|2]), (482,483,[3_1|2]), (483,484,[5_1|2]), (484,485,[5_1|2]), (485,486,[4_1|2]), (485,271,[4_1|2]), (486,153,[3_1|2]), (486,177,[3_1|2]), (486,225,[3_1|2]), (486,160,[3_1|2]), (486,419,[3_1|2]), (486,235,[3_1|2]), (486,453,[3_1|2]), (486,470,[3_1|2]), (486,487,[3_1|2]), (487,488,[4_1|2]), (488,489,[3_1|2]), (489,490,[4_1|2]), (490,491,[3_1|2]), (491,492,[1_1|2]), (492,493,[0_1|2]), (493,494,[1_1|2]), (494,495,[1_1|2]), (495,496,[4_1|2]), (496,497,[5_1|2]), (497,498,[5_1|2]), (498,499,[2_1|2]), (499,500,[3_1|2]), (500,501,[2_1|2]), (501,502,[3_1|2]), (502,503,[0_1|2]), (503,504,[2_1|2]), (504,153,[3_1|2]), (504,154,[3_1|2]), (504,189,[3_1|2]), (504,222,[3_1|2]), (504,243,[3_1|2]), (504,256,[3_1|2]), (504,365,[3_1|2]), (504,453,[3_1|2]), (504,470,[3_1|2]), (504,487,[3_1|2]), (505,506,[4_1|3]), (506,507,[4_1|3]), (507,203,[2_1|3]), (507,394,[2_1|3]), (507,446,[2_1|3]), (507,347,[2_1|3]), (507,454,[2_1|3]), (508,509,[4_1|3]), (509,510,[4_1|3]), (510,190,[2_1|3]), (510,454,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)