/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 86 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 121 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(0(x1)))) -> 0(1(0(x1))) 2(3(4(4(x1)))) -> 0(3(4(x1))) 2(2(4(3(2(5(2(x1))))))) -> 0(3(0(0(0(x1))))) 3(2(2(0(4(1(3(0(x1)))))))) -> 3(4(2(0(1(0(5(0(x1)))))))) 2(2(2(3(0(1(2(2(5(x1))))))))) -> 0(1(1(1(3(5(2(5(x1)))))))) 3(0(4(3(1(1(0(1(1(x1))))))))) -> 3(4(1(2(3(3(4(3(x1)))))))) 2(4(2(0(3(5(0(3(2(4(x1)))))))))) -> 0(0(0(1(0(1(3(2(x1)))))))) 4(2(3(3(1(0(5(2(5(3(x1)))))))))) -> 5(4(4(0(2(0(0(4(3(x1))))))))) 4(3(2(0(1(5(0(3(0(5(x1)))))))))) -> 4(4(1(2(5(5(2(4(2(1(x1)))))))))) 0(3(3(1(3(0(3(4(5(0(5(x1))))))))))) -> 0(3(4(1(5(2(0(2(1(3(x1)))))))))) 5(3(4(0(4(4(0(5(2(2(0(x1))))))))))) -> 5(4(5(3(1(0(0(4(5(2(x1)))))))))) 2(1(5(1(2(4(2(2(2(1(5(1(x1)))))))))))) -> 2(1(3(2(5(1(2(0(2(3(4(x1))))))))))) 2(2(5(5(1(2(1(5(5(5(3(0(2(x1))))))))))))) -> 4(4(4(1(1(4(1(5(5(1(2(x1))))))))))) 3(3(5(1(5(4(2(3(2(1(0(0(2(0(x1)))))))))))))) -> 4(3(0(3(5(1(1(3(1(5(4(3(1(x1))))))))))))) 2(3(3(0(3(3(3(1(4(2(0(5(4(1(2(2(x1)))))))))))))))) -> 3(5(1(0(3(0(3(3(4(5(4(2(2(5(1(x1))))))))))))))) 3(5(4(3(2(3(0(3(5(2(4(4(1(0(5(4(x1)))))))))))))))) -> 3(5(4(5(0(5(1(0(2(3(3(4(5(2(2(2(4(x1))))))))))))))))) 4(1(1(4(5(2(4(1(4(2(5(3(5(2(0(0(x1)))))))))))))))) -> 4(3(0(2(3(3(5(4(4(0(0(0(5(4(2(3(x1)))))))))))))))) 4(4(1(2(0(2(0(5(4(3(0(0(1(2(2(2(x1)))))))))))))))) -> 4(5(4(2(4(1(0(4(0(1(1(2(4(4(4(x1))))))))))))))) 0(2(1(1(5(2(5(0(3(5(5(0(3(5(4(2(2(x1))))))))))))))))) -> 4(2(2(5(1(5(1(5(4(4(4(4(0(2(5(3(0(x1))))))))))))))))) 2(5(5(1(0(0(2(1(2(3(5(1(1(1(1(2(2(4(x1)))))))))))))))))) -> 4(3(2(1(5(4(0(3(2(3(5(5(2(4(1(4(x1)))))))))))))))) 2(5(5(3(4(2(0(0(2(0(4(0(4(4(2(5(4(0(x1)))))))))))))))))) -> 5(1(1(4(0(5(5(2(5(0(1(2(4(2(4(4(x1)))))))))))))))) 3(0(1(1(5(1(2(1(0(3(2(3(2(0(3(4(3(4(0(x1))))))))))))))))))) -> 1(5(3(0(5(4(4(0(0(4(1(3(4(3(1(4(1(x1))))))))))))))))) 3(3(5(4(4(1(5(5(2(1(5(4(4(5(1(0(5(3(1(x1))))))))))))))))))) -> 1(0(0(0(1(0(4(5(4(0(2(3(5(3(5(4(4(0(3(x1))))))))))))))))))) 3(5(2(2(4(2(0(3(4(4(3(5(4(4(1(4(5(2(4(x1))))))))))))))))))) -> 4(2(1(2(0(3(2(3(0(3(2(5(1(4(1(2(3(4(x1)))))))))))))))))) 4(0(4(4(5(2(2(0(4(4(0(3(5(3(5(5(0(5(2(x1))))))))))))))))))) -> 5(2(3(3(2(2(0(4(4(1(5(5(3(4(2(3(5(2(x1)))))))))))))))))) 4(2(2(4(3(4(4(3(1(3(2(2(2(0(1(2(5(4(1(x1))))))))))))))))))) -> 5(4(4(5(4(2(0(2(3(2(3(4(2(1(2(1(4(1(x1)))))))))))))))))) 3(3(4(5(1(3(1(3(4(1(4(0(2(2(0(4(2(4(2(2(x1)))))))))))))))))))) -> 1(0(1(4(3(3(4(4(4(3(5(0(1(2(0(2(3(3(x1)))))))))))))))))) 3(5(2(0(2(0(4(5(2(1(4(2(5(0(5(2(5(3(2(5(x1)))))))))))))))))))) -> 1(2(5(4(5(3(3(0(0(5(5(4(4(0(3(0(3(1(x1)))))))))))))))))) 2(3(1(5(2(3(0(3(3(1(3(2(1(3(1(4(3(5(1(5(4(x1))))))))))))))))))))) -> 3(1(1(4(5(4(5(3(4(2(4(5(3(5(4(1(3(4(0(4(x1)))))))))))))))))))) 2(4(1(3(0(5(2(2(2(4(4(5(2(0(3(4(5(1(5(5(0(x1))))))))))))))))))))) -> 4(3(5(4(2(4(5(1(0(2(3(5(5(0(4(4(5(3(3(5(1(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(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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 (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(0(x1)))) -> 0(1(0(x1))) 2(3(4(4(x1)))) -> 0(3(4(x1))) 2(2(4(3(2(5(2(x1))))))) -> 0(3(0(0(0(x1))))) 3(2(2(0(4(1(3(0(x1)))))))) -> 3(4(2(0(1(0(5(0(x1)))))))) 2(2(2(3(0(1(2(2(5(x1))))))))) -> 0(1(1(1(3(5(2(5(x1)))))))) 3(0(4(3(1(1(0(1(1(x1))))))))) -> 3(4(1(2(3(3(4(3(x1)))))))) 2(4(2(0(3(5(0(3(2(4(x1)))))))))) -> 0(0(0(1(0(1(3(2(x1)))))))) 4(2(3(3(1(0(5(2(5(3(x1)))))))))) -> 5(4(4(0(2(0(0(4(3(x1))))))))) 4(3(2(0(1(5(0(3(0(5(x1)))))))))) -> 4(4(1(2(5(5(2(4(2(1(x1)))))))))) 0(3(3(1(3(0(3(4(5(0(5(x1))))))))))) -> 0(3(4(1(5(2(0(2(1(3(x1)))))))))) 5(3(4(0(4(4(0(5(2(2(0(x1))))))))))) -> 5(4(5(3(1(0(0(4(5(2(x1)))))))))) 2(1(5(1(2(4(2(2(2(1(5(1(x1)))))))))))) -> 2(1(3(2(5(1(2(0(2(3(4(x1))))))))))) 2(2(5(5(1(2(1(5(5(5(3(0(2(x1))))))))))))) -> 4(4(4(1(1(4(1(5(5(1(2(x1))))))))))) 3(3(5(1(5(4(2(3(2(1(0(0(2(0(x1)))))))))))))) -> 4(3(0(3(5(1(1(3(1(5(4(3(1(x1))))))))))))) 2(3(3(0(3(3(3(1(4(2(0(5(4(1(2(2(x1)))))))))))))))) -> 3(5(1(0(3(0(3(3(4(5(4(2(2(5(1(x1))))))))))))))) 3(5(4(3(2(3(0(3(5(2(4(4(1(0(5(4(x1)))))))))))))))) -> 3(5(4(5(0(5(1(0(2(3(3(4(5(2(2(2(4(x1))))))))))))))))) 4(1(1(4(5(2(4(1(4(2(5(3(5(2(0(0(x1)))))))))))))))) -> 4(3(0(2(3(3(5(4(4(0(0(0(5(4(2(3(x1)))))))))))))))) 4(4(1(2(0(2(0(5(4(3(0(0(1(2(2(2(x1)))))))))))))))) -> 4(5(4(2(4(1(0(4(0(1(1(2(4(4(4(x1))))))))))))))) 0(2(1(1(5(2(5(0(3(5(5(0(3(5(4(2(2(x1))))))))))))))))) -> 4(2(2(5(1(5(1(5(4(4(4(4(0(2(5(3(0(x1))))))))))))))))) 2(5(5(1(0(0(2(1(2(3(5(1(1(1(1(2(2(4(x1)))))))))))))))))) -> 4(3(2(1(5(4(0(3(2(3(5(5(2(4(1(4(x1)))))))))))))))) 2(5(5(3(4(2(0(0(2(0(4(0(4(4(2(5(4(0(x1)))))))))))))))))) -> 5(1(1(4(0(5(5(2(5(0(1(2(4(2(4(4(x1)))))))))))))))) 3(0(1(1(5(1(2(1(0(3(2(3(2(0(3(4(3(4(0(x1))))))))))))))))))) -> 1(5(3(0(5(4(4(0(0(4(1(3(4(3(1(4(1(x1))))))))))))))))) 3(3(5(4(4(1(5(5(2(1(5(4(4(5(1(0(5(3(1(x1))))))))))))))))))) -> 1(0(0(0(1(0(4(5(4(0(2(3(5(3(5(4(4(0(3(x1))))))))))))))))))) 3(5(2(2(4(2(0(3(4(4(3(5(4(4(1(4(5(2(4(x1))))))))))))))))))) -> 4(2(1(2(0(3(2(3(0(3(2(5(1(4(1(2(3(4(x1)))))))))))))))))) 4(0(4(4(5(2(2(0(4(4(0(3(5(3(5(5(0(5(2(x1))))))))))))))))))) -> 5(2(3(3(2(2(0(4(4(1(5(5(3(4(2(3(5(2(x1)))))))))))))))))) 4(2(2(4(3(4(4(3(1(3(2(2(2(0(1(2(5(4(1(x1))))))))))))))))))) -> 5(4(4(5(4(2(0(2(3(2(3(4(2(1(2(1(4(1(x1)))))))))))))))))) 3(3(4(5(1(3(1(3(4(1(4(0(2(2(0(4(2(4(2(2(x1)))))))))))))))))))) -> 1(0(1(4(3(3(4(4(4(3(5(0(1(2(0(2(3(3(x1)))))))))))))))))) 3(5(2(0(2(0(4(5(2(1(4(2(5(0(5(2(5(3(2(5(x1)))))))))))))))))))) -> 1(2(5(4(5(3(3(0(0(5(5(4(4(0(3(0(3(1(x1)))))))))))))))))) 2(3(1(5(2(3(0(3(3(1(3(2(1(3(1(4(3(5(1(5(4(x1))))))))))))))))))))) -> 3(1(1(4(5(4(5(3(4(2(4(5(3(5(4(1(3(4(0(4(x1)))))))))))))))))))) 2(4(1(3(0(5(2(2(2(4(4(5(2(0(3(4(5(1(5(5(0(x1))))))))))))))))))))) -> 4(3(5(4(2(4(5(1(0(2(3(5(5(0(4(4(5(3(3(5(1(x1))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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: 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(0(x1)))) -> 0(1(0(x1))) 2(3(4(4(x1)))) -> 0(3(4(x1))) 2(2(4(3(2(5(2(x1))))))) -> 0(3(0(0(0(x1))))) 3(2(2(0(4(1(3(0(x1)))))))) -> 3(4(2(0(1(0(5(0(x1)))))))) 2(2(2(3(0(1(2(2(5(x1))))))))) -> 0(1(1(1(3(5(2(5(x1)))))))) 3(0(4(3(1(1(0(1(1(x1))))))))) -> 3(4(1(2(3(3(4(3(x1)))))))) 2(4(2(0(3(5(0(3(2(4(x1)))))))))) -> 0(0(0(1(0(1(3(2(x1)))))))) 4(2(3(3(1(0(5(2(5(3(x1)))))))))) -> 5(4(4(0(2(0(0(4(3(x1))))))))) 4(3(2(0(1(5(0(3(0(5(x1)))))))))) -> 4(4(1(2(5(5(2(4(2(1(x1)))))))))) 0(3(3(1(3(0(3(4(5(0(5(x1))))))))))) -> 0(3(4(1(5(2(0(2(1(3(x1)))))))))) 5(3(4(0(4(4(0(5(2(2(0(x1))))))))))) -> 5(4(5(3(1(0(0(4(5(2(x1)))))))))) 2(1(5(1(2(4(2(2(2(1(5(1(x1)))))))))))) -> 2(1(3(2(5(1(2(0(2(3(4(x1))))))))))) 2(2(5(5(1(2(1(5(5(5(3(0(2(x1))))))))))))) -> 4(4(4(1(1(4(1(5(5(1(2(x1))))))))))) 3(3(5(1(5(4(2(3(2(1(0(0(2(0(x1)))))))))))))) -> 4(3(0(3(5(1(1(3(1(5(4(3(1(x1))))))))))))) 2(3(3(0(3(3(3(1(4(2(0(5(4(1(2(2(x1)))))))))))))))) -> 3(5(1(0(3(0(3(3(4(5(4(2(2(5(1(x1))))))))))))))) 3(5(4(3(2(3(0(3(5(2(4(4(1(0(5(4(x1)))))))))))))))) -> 3(5(4(5(0(5(1(0(2(3(3(4(5(2(2(2(4(x1))))))))))))))))) 4(1(1(4(5(2(4(1(4(2(5(3(5(2(0(0(x1)))))))))))))))) -> 4(3(0(2(3(3(5(4(4(0(0(0(5(4(2(3(x1)))))))))))))))) 4(4(1(2(0(2(0(5(4(3(0(0(1(2(2(2(x1)))))))))))))))) -> 4(5(4(2(4(1(0(4(0(1(1(2(4(4(4(x1))))))))))))))) 0(2(1(1(5(2(5(0(3(5(5(0(3(5(4(2(2(x1))))))))))))))))) -> 4(2(2(5(1(5(1(5(4(4(4(4(0(2(5(3(0(x1))))))))))))))))) 2(5(5(1(0(0(2(1(2(3(5(1(1(1(1(2(2(4(x1)))))))))))))))))) -> 4(3(2(1(5(4(0(3(2(3(5(5(2(4(1(4(x1)))))))))))))))) 2(5(5(3(4(2(0(0(2(0(4(0(4(4(2(5(4(0(x1)))))))))))))))))) -> 5(1(1(4(0(5(5(2(5(0(1(2(4(2(4(4(x1)))))))))))))))) 3(0(1(1(5(1(2(1(0(3(2(3(2(0(3(4(3(4(0(x1))))))))))))))))))) -> 1(5(3(0(5(4(4(0(0(4(1(3(4(3(1(4(1(x1))))))))))))))))) 3(3(5(4(4(1(5(5(2(1(5(4(4(5(1(0(5(3(1(x1))))))))))))))))))) -> 1(0(0(0(1(0(4(5(4(0(2(3(5(3(5(4(4(0(3(x1))))))))))))))))))) 3(5(2(2(4(2(0(3(4(4(3(5(4(4(1(4(5(2(4(x1))))))))))))))))))) -> 4(2(1(2(0(3(2(3(0(3(2(5(1(4(1(2(3(4(x1)))))))))))))))))) 4(0(4(4(5(2(2(0(4(4(0(3(5(3(5(5(0(5(2(x1))))))))))))))))))) -> 5(2(3(3(2(2(0(4(4(1(5(5(3(4(2(3(5(2(x1)))))))))))))))))) 4(2(2(4(3(4(4(3(1(3(2(2(2(0(1(2(5(4(1(x1))))))))))))))))))) -> 5(4(4(5(4(2(0(2(3(2(3(4(2(1(2(1(4(1(x1)))))))))))))))))) 3(3(4(5(1(3(1(3(4(1(4(0(2(2(0(4(2(4(2(2(x1)))))))))))))))))))) -> 1(0(1(4(3(3(4(4(4(3(5(0(1(2(0(2(3(3(x1)))))))))))))))))) 3(5(2(0(2(0(4(5(2(1(4(2(5(0(5(2(5(3(2(5(x1)))))))))))))))))))) -> 1(2(5(4(5(3(3(0(0(5(5(4(4(0(3(0(3(1(x1)))))))))))))))))) 2(3(1(5(2(3(0(3(3(1(3(2(1(3(1(4(3(5(1(5(4(x1))))))))))))))))))))) -> 3(1(1(4(5(4(5(3(4(2(4(5(3(5(4(1(3(4(0(4(x1)))))))))))))))))))) 2(4(1(3(0(5(2(2(2(4(4(5(2(0(3(4(5(1(5(5(0(x1))))))))))))))))))))) -> 4(3(5(4(2(4(5(1(0(2(3(5(5(0(4(4(5(3(3(5(1(x1))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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: 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(0(x1)))) -> 0(1(0(x1))) 2(3(4(4(x1)))) -> 0(3(4(x1))) 2(2(4(3(2(5(2(x1))))))) -> 0(3(0(0(0(x1))))) 3(2(2(0(4(1(3(0(x1)))))))) -> 3(4(2(0(1(0(5(0(x1)))))))) 2(2(2(3(0(1(2(2(5(x1))))))))) -> 0(1(1(1(3(5(2(5(x1)))))))) 3(0(4(3(1(1(0(1(1(x1))))))))) -> 3(4(1(2(3(3(4(3(x1)))))))) 2(4(2(0(3(5(0(3(2(4(x1)))))))))) -> 0(0(0(1(0(1(3(2(x1)))))))) 4(2(3(3(1(0(5(2(5(3(x1)))))))))) -> 5(4(4(0(2(0(0(4(3(x1))))))))) 4(3(2(0(1(5(0(3(0(5(x1)))))))))) -> 4(4(1(2(5(5(2(4(2(1(x1)))))))))) 0(3(3(1(3(0(3(4(5(0(5(x1))))))))))) -> 0(3(4(1(5(2(0(2(1(3(x1)))))))))) 5(3(4(0(4(4(0(5(2(2(0(x1))))))))))) -> 5(4(5(3(1(0(0(4(5(2(x1)))))))))) 2(1(5(1(2(4(2(2(2(1(5(1(x1)))))))))))) -> 2(1(3(2(5(1(2(0(2(3(4(x1))))))))))) 2(2(5(5(1(2(1(5(5(5(3(0(2(x1))))))))))))) -> 4(4(4(1(1(4(1(5(5(1(2(x1))))))))))) 3(3(5(1(5(4(2(3(2(1(0(0(2(0(x1)))))))))))))) -> 4(3(0(3(5(1(1(3(1(5(4(3(1(x1))))))))))))) 2(3(3(0(3(3(3(1(4(2(0(5(4(1(2(2(x1)))))))))))))))) -> 3(5(1(0(3(0(3(3(4(5(4(2(2(5(1(x1))))))))))))))) 3(5(4(3(2(3(0(3(5(2(4(4(1(0(5(4(x1)))))))))))))))) -> 3(5(4(5(0(5(1(0(2(3(3(4(5(2(2(2(4(x1))))))))))))))))) 4(1(1(4(5(2(4(1(4(2(5(3(5(2(0(0(x1)))))))))))))))) -> 4(3(0(2(3(3(5(4(4(0(0(0(5(4(2(3(x1)))))))))))))))) 4(4(1(2(0(2(0(5(4(3(0(0(1(2(2(2(x1)))))))))))))))) -> 4(5(4(2(4(1(0(4(0(1(1(2(4(4(4(x1))))))))))))))) 0(2(1(1(5(2(5(0(3(5(5(0(3(5(4(2(2(x1))))))))))))))))) -> 4(2(2(5(1(5(1(5(4(4(4(4(0(2(5(3(0(x1))))))))))))))))) 2(5(5(1(0(0(2(1(2(3(5(1(1(1(1(2(2(4(x1)))))))))))))))))) -> 4(3(2(1(5(4(0(3(2(3(5(5(2(4(1(4(x1)))))))))))))))) 2(5(5(3(4(2(0(0(2(0(4(0(4(4(2(5(4(0(x1)))))))))))))))))) -> 5(1(1(4(0(5(5(2(5(0(1(2(4(2(4(4(x1)))))))))))))))) 3(0(1(1(5(1(2(1(0(3(2(3(2(0(3(4(3(4(0(x1))))))))))))))))))) -> 1(5(3(0(5(4(4(0(0(4(1(3(4(3(1(4(1(x1))))))))))))))))) 3(3(5(4(4(1(5(5(2(1(5(4(4(5(1(0(5(3(1(x1))))))))))))))))))) -> 1(0(0(0(1(0(4(5(4(0(2(3(5(3(5(4(4(0(3(x1))))))))))))))))))) 3(5(2(2(4(2(0(3(4(4(3(5(4(4(1(4(5(2(4(x1))))))))))))))))))) -> 4(2(1(2(0(3(2(3(0(3(2(5(1(4(1(2(3(4(x1)))))))))))))))))) 4(0(4(4(5(2(2(0(4(4(0(3(5(3(5(5(0(5(2(x1))))))))))))))))))) -> 5(2(3(3(2(2(0(4(4(1(5(5(3(4(2(3(5(2(x1)))))))))))))))))) 4(2(2(4(3(4(4(3(1(3(2(2(2(0(1(2(5(4(1(x1))))))))))))))))))) -> 5(4(4(5(4(2(0(2(3(2(3(4(2(1(2(1(4(1(x1)))))))))))))))))) 3(3(4(5(1(3(1(3(4(1(4(0(2(2(0(4(2(4(2(2(x1)))))))))))))))))))) -> 1(0(1(4(3(3(4(4(4(3(5(0(1(2(0(2(3(3(x1)))))))))))))))))) 3(5(2(0(2(0(4(5(2(1(4(2(5(0(5(2(5(3(2(5(x1)))))))))))))))))))) -> 1(2(5(4(5(3(3(0(0(5(5(4(4(0(3(0(3(1(x1)))))))))))))))))) 2(3(1(5(2(3(0(3(3(1(3(2(1(3(1(4(3(5(1(5(4(x1))))))))))))))))))))) -> 3(1(1(4(5(4(5(3(4(2(4(5(3(5(4(1(3(4(0(4(x1)))))))))))))))))))) 2(4(1(3(0(5(2(2(2(4(4(5(2(0(3(4(5(1(5(5(0(x1))))))))))))))))))))) -> 4(3(5(4(2(4(5(1(0(2(3(5(5(0(4(4(5(3(3(5(1(x1))))))))))))))))))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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)) 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. 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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] {(115,116,[0_1|0, 2_1|0, 3_1|0, 4_1|0, 5_1|0, encArg_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_3_1|0, 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(380,381,[4_1|2]), (381,382,[5_1|2]), (382,383,[3_1|2]), (383,384,[3_1|2]), (384,385,[0_1|2]), (385,386,[0_1|2]), (386,387,[5_1|2]), (387,388,[5_1|2]), (388,389,[4_1|2]), (389,390,[4_1|2]), (390,391,[0_1|2]), (391,392,[3_1|2]), (392,393,[0_1|2]), (393,394,[3_1|2]), (394,117,[1_1|2]), (394,253,[1_1|2]), (394,395,[1_1|2]), (394,403,[1_1|2]), (394,458,[1_1|2]), (394,475,[1_1|2]), (395,396,[4_1|2]), (396,397,[4_1|2]), (397,398,[0_1|2]), (398,399,[2_1|2]), (399,400,[0_1|2]), (400,401,[0_1|2]), (401,402,[4_1|2]), (401,420,[4_1|2]), (402,117,[3_1|2]), (402,147,[3_1|2]), (402,161,[3_1|2]), (402,268,[3_1|2]), (402,275,[3_1|2]), (402,345,[3_1|2]), (402,282,[1_1|2]), (402,298,[4_1|2]), (402,310,[1_1|2]), (402,328,[1_1|2]), (402,361,[4_1|2]), (402,378,[1_1|2]), (403,404,[4_1|2]), (404,405,[4_1|2]), (405,406,[5_1|2]), (406,407,[4_1|2]), (407,408,[2_1|2]), (408,409,[0_1|2]), (409,410,[2_1|2]), (410,411,[3_1|2]), (411,412,[2_1|2]), (412,413,[3_1|2]), (413,414,[4_1|2]), (414,415,[2_1|2]), (415,416,[1_1|2]), (416,417,[2_1|2]), (417,418,[1_1|2]), (418,419,[4_1|2]), (418,429,[4_1|2]), (419,117,[1_1|2]), (419,282,[1_1|2]), (419,310,[1_1|2]), (419,328,[1_1|2]), (419,378,[1_1|2]), (420,421,[4_1|2]), (421,422,[1_1|2]), (422,423,[2_1|2]), (423,424,[5_1|2]), (424,425,[5_1|2]), (425,426,[2_1|2]), (426,427,[4_1|2]), (427,428,[2_1|2]), (427,228,[2_1|2]), (428,117,[1_1|2]), (428,253,[1_1|2]), (428,395,[1_1|2]), (428,403,[1_1|2]), (428,458,[1_1|2]), (428,475,[1_1|2]), (429,430,[3_1|2]), (430,431,[0_1|2]), (431,432,[2_1|2]), (432,433,[3_1|2]), (433,434,[3_1|2]), (434,435,[5_1|2]), (435,436,[4_1|2]), (436,437,[4_1|2]), (437,438,[0_1|2]), (438,439,[0_1|2]), (439,440,[0_1|2]), (440,441,[5_1|2]), (441,442,[4_1|2]), (441,395,[5_1|2]), (442,443,[2_1|2]), (442,145,[0_1|2]), (442,147,[3_1|2]), (442,161,[3_1|2]), (442,488,[0_1|3]), (443,117,[3_1|2]), (443,118,[3_1|2]), (443,120,[3_1|2]), (443,145,[3_1|2]), (443,180,[3_1|2]), (443,184,[3_1|2]), (443,201,[3_1|2]), (443,202,[3_1|2]), (443,268,[3_1|2]), (443,275,[3_1|2]), (443,282,[1_1|2]), (443,298,[4_1|2]), (443,310,[1_1|2]), (443,328,[1_1|2]), (443,345,[3_1|2]), (443,361,[4_1|2]), (443,378,[1_1|2]), (444,445,[5_1|2]), (445,446,[4_1|2]), (446,447,[2_1|2]), (447,448,[4_1|2]), (448,449,[1_1|2]), (449,450,[0_1|2]), (450,451,[4_1|2]), (451,452,[0_1|2]), (452,453,[1_1|2]), (453,454,[1_1|2]), (454,455,[2_1|2]), (455,456,[4_1|2]), (456,457,[4_1|2]), (456,444,[4_1|2]), (457,117,[4_1|2]), (457,228,[4_1|2]), (457,395,[5_1|2]), (457,403,[5_1|2]), (457,420,[4_1|2]), (457,429,[4_1|2]), (457,444,[4_1|2]), (457,458,[5_1|2]), (458,459,[2_1|2]), (459,460,[3_1|2]), (460,461,[3_1|2]), (461,462,[2_1|2]), (462,463,[2_1|2]), (463,464,[0_1|2]), (464,465,[4_1|2]), (465,466,[4_1|2]), (466,467,[1_1|2]), (467,468,[5_1|2]), (468,469,[5_1|2]), (469,470,[3_1|2]), (470,471,[4_1|2]), (471,472,[2_1|2]), (472,473,[3_1|2]), (472,361,[4_1|2]), (472,378,[1_1|2]), (473,474,[5_1|2]), (474,117,[2_1|2]), (474,228,[2_1|2]), (474,459,[2_1|2]), (474,145,[0_1|2]), (474,147,[3_1|2]), (474,161,[3_1|2]), (474,180,[0_1|2]), (474,184,[0_1|2]), (474,191,[4_1|2]), (474,201,[0_1|2]), (474,208,[4_1|2]), (474,238,[4_1|2]), (474,253,[5_1|2]), (475,476,[4_1|2]), (476,477,[5_1|2]), (477,478,[3_1|2]), (478,479,[1_1|2]), (479,480,[0_1|2]), (480,481,[0_1|2]), (481,482,[4_1|2]), (482,483,[5_1|2]), (483,117,[2_1|2]), (483,118,[2_1|2]), (483,120,[2_1|2]), (483,145,[2_1|2, 0_1|2]), (483,180,[2_1|2, 0_1|2]), (483,184,[2_1|2, 0_1|2]), (483,201,[2_1|2, 0_1|2]), (483,147,[3_1|2]), (483,161,[3_1|2]), (483,191,[4_1|2]), (483,208,[4_1|2]), (483,228,[2_1|2]), (483,238,[4_1|2]), (483,253,[5_1|2]), (483,484,[0_1|3]), (484,485,[3_1|3]), (485,129,[4_1|3]), (485,191,[4_1|3]), (485,208,[4_1|3]), (485,238,[4_1|3]), (485,298,[4_1|3]), (485,361,[4_1|3]), (485,420,[4_1|3]), (485,429,[4_1|3]), (485,444,[4_1|3]), (485,421,[4_1|3]), (485,192,[4_1|3]), (485,193,[4_1|3]), (486,487,[1_1|3]), (487,342,[0_1|3]), (488,489,[3_1|3]), (489,192,[4_1|3]), (489,421,[4_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)