/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 99 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 81 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 0(0(1(3(4(x1))))) 5(3(3(3(0(4(4(5(x1)))))))) -> 5(0(3(1(3(1(2(x1))))))) 0(3(1(2(4(4(0(5(2(x1))))))))) -> 0(1(3(3(3(1(0(5(2(x1))))))))) 0(2(5(0(3(0(5(3(0(4(x1)))))))))) -> 4(4(1(5(4(5(0(3(3(4(x1)))))))))) 0(3(0(5(4(4(5(4(4(4(2(x1))))))))))) -> 0(3(2(1(4(5(3(2(5(0(2(x1))))))))))) 0(5(3(2(3(4(3(4(2(1(1(2(5(x1))))))))))))) -> 0(5(3(1(0(4(0(1(5(5(0(1(x1)))))))))))) 0(4(0(2(0(0(4(4(2(3(0(1(1(3(x1)))))))))))))) -> 0(0(0(1(2(4(4(0(5(4(0(2(3(x1))))))))))))) 1(4(4(5(2(1(2(4(2(5(1(0(5(0(x1)))))))))))))) -> 1(0(3(0(4(3(3(4(5(2(2(4(1(4(x1)))))))))))))) 1(5(4(4(3(4(4(1(0(1(4(1(5(5(x1)))))))))))))) -> 0(1(4(2(5(0(3(5(4(4(0(4(1(x1))))))))))))) 2(0(4(5(0(0(4(4(3(2(1(5(1(4(x1)))))))))))))) -> 2(3(2(0(5(1(0(3(0(3(1(3(0(4(x1)))))))))))))) 4(5(3(5(0(5(1(4(3(2(5(2(2(2(x1)))))))))))))) -> 4(4(3(0(2(3(2(0(0(0(1(5(4(x1))))))))))))) 2(1(0(0(3(0(2(2(1(1(4(5(3(0(5(x1))))))))))))))) -> 4(3(5(3(1(4(5(2(1(3(0(3(3(0(4(x1))))))))))))))) 2(1(3(0(2(2(4(5(0(3(5(0(3(1(0(x1))))))))))))))) -> 2(5(5(0(1(5(1(0(4(0(0(1(4(0(x1)))))))))))))) 0(5(1(2(1(5(4(3(5(2(4(0(0(5(2(0(x1)))))))))))))))) -> 0(1(2(3(2(0(5(1(4(1(3(0(2(3(1(0(x1)))))))))))))))) 2(1(4(3(4(5(1(0(2(1(1(5(1(1(5(2(4(4(x1)))))))))))))))))) -> 4(4(1(0(5(1(5(5(0(2(1(4(1(3(1(2(1(4(x1)))))))))))))))))) 3(2(5(0(3(2(1(1(4(1(1(4(4(0(4(4(5(5(x1)))))))))))))))))) -> 3(3(2(4(1(3(4(3(5(1(5(3(5(4(4(1(2(2(x1)))))))))))))))))) 5(2(1(0(0(0(1(4(4(4(0(5(0(0(0(3(3(1(x1)))))))))))))))))) -> 5(4(2(4(1(3(1(2(2(5(4(5(2(5(4(4(0(5(5(x1))))))))))))))))))) 5(4(4(3(1(0(2(0(3(2(0(2(3(3(0(4(5(1(x1)))))))))))))))))) -> 4(3(2(0(2(4(5(2(0(3(4(5(5(0(2(2(1(4(x1)))))))))))))))))) 5(4(5(2(5(2(1(3(4(0(0(5(1(3(0(3(5(4(x1)))))))))))))))))) -> 4(1(3(2(4(4(5(1(5(3(2(1(2(5(3(5(2(4(x1)))))))))))))))))) 4(1(3(0(5(4(1(4(4(0(0(4(4(3(0(2(0(2(1(x1))))))))))))))))))) -> 5(4(3(0(4(1(0(0(4(5(1(0(5(3(1(4(2(4(1(x1))))))))))))))))))) 5(3(3(2(4(2(4(4(0(1(0(5(1(0(4(3(4(0(4(x1))))))))))))))))))) -> 5(5(2(1(5(2(4(5(5(3(4(0(0(3(5(2(4(4(x1)))))))))))))))))) 5(4(1(5(2(4(3(5(4(2(4(4(4(3(2(3(2(0(5(x1))))))))))))))))))) -> 0(4(4(0(2(0(1(2(5(3(0(1(4(3(3(2(5(5(x1)))))))))))))))))) 0(5(0(4(0(1(3(1(1(4(5(1(3(2(0(0(3(5(3(5(x1)))))))))))))))))))) -> 0(0(2(5(5(3(3(2(4(3(0(1(4(5(2(1(5(5(5(2(x1)))))))))))))))))))) 4(3(5(5(4(3(0(0(3(4(5(5(5(2(2(0(0(3(2(5(x1)))))))))))))))))))) -> 4(5(2(2(3(3(3(5(3(1(5(5(5(1(3(2(3(5(3(3(4(x1))))))))))))))))))))) 1(0(2(4(3(3(5(2(4(1(3(4(0(0(5(4(1(1(1(3(0(x1))))))))))))))))))))) -> 1(1(4(1(0(0(5(2(2(1(2(4(4(0(2(5(1(1(3(0(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 0(0(1(3(4(x1))))) 5(3(3(3(0(4(4(5(x1)))))))) -> 5(0(3(1(3(1(2(x1))))))) 0(3(1(2(4(4(0(5(2(x1))))))))) -> 0(1(3(3(3(1(0(5(2(x1))))))))) 0(2(5(0(3(0(5(3(0(4(x1)))))))))) -> 4(4(1(5(4(5(0(3(3(4(x1)))))))))) 0(3(0(5(4(4(5(4(4(4(2(x1))))))))))) -> 0(3(2(1(4(5(3(2(5(0(2(x1))))))))))) 0(5(3(2(3(4(3(4(2(1(1(2(5(x1))))))))))))) -> 0(5(3(1(0(4(0(1(5(5(0(1(x1)))))))))))) 0(4(0(2(0(0(4(4(2(3(0(1(1(3(x1)))))))))))))) -> 0(0(0(1(2(4(4(0(5(4(0(2(3(x1))))))))))))) 1(4(4(5(2(1(2(4(2(5(1(0(5(0(x1)))))))))))))) -> 1(0(3(0(4(3(3(4(5(2(2(4(1(4(x1)))))))))))))) 1(5(4(4(3(4(4(1(0(1(4(1(5(5(x1)))))))))))))) -> 0(1(4(2(5(0(3(5(4(4(0(4(1(x1))))))))))))) 2(0(4(5(0(0(4(4(3(2(1(5(1(4(x1)))))))))))))) -> 2(3(2(0(5(1(0(3(0(3(1(3(0(4(x1)))))))))))))) 4(5(3(5(0(5(1(4(3(2(5(2(2(2(x1)))))))))))))) -> 4(4(3(0(2(3(2(0(0(0(1(5(4(x1))))))))))))) 2(1(0(0(3(0(2(2(1(1(4(5(3(0(5(x1))))))))))))))) -> 4(3(5(3(1(4(5(2(1(3(0(3(3(0(4(x1))))))))))))))) 2(1(3(0(2(2(4(5(0(3(5(0(3(1(0(x1))))))))))))))) -> 2(5(5(0(1(5(1(0(4(0(0(1(4(0(x1)))))))))))))) 0(5(1(2(1(5(4(3(5(2(4(0(0(5(2(0(x1)))))))))))))))) -> 0(1(2(3(2(0(5(1(4(1(3(0(2(3(1(0(x1)))))))))))))))) 2(1(4(3(4(5(1(0(2(1(1(5(1(1(5(2(4(4(x1)))))))))))))))))) -> 4(4(1(0(5(1(5(5(0(2(1(4(1(3(1(2(1(4(x1)))))))))))))))))) 3(2(5(0(3(2(1(1(4(1(1(4(4(0(4(4(5(5(x1)))))))))))))))))) -> 3(3(2(4(1(3(4(3(5(1(5(3(5(4(4(1(2(2(x1)))))))))))))))))) 5(2(1(0(0(0(1(4(4(4(0(5(0(0(0(3(3(1(x1)))))))))))))))))) -> 5(4(2(4(1(3(1(2(2(5(4(5(2(5(4(4(0(5(5(x1))))))))))))))))))) 5(4(4(3(1(0(2(0(3(2(0(2(3(3(0(4(5(1(x1)))))))))))))))))) -> 4(3(2(0(2(4(5(2(0(3(4(5(5(0(2(2(1(4(x1)))))))))))))))))) 5(4(5(2(5(2(1(3(4(0(0(5(1(3(0(3(5(4(x1)))))))))))))))))) -> 4(1(3(2(4(4(5(1(5(3(2(1(2(5(3(5(2(4(x1)))))))))))))))))) 4(1(3(0(5(4(1(4(4(0(0(4(4(3(0(2(0(2(1(x1))))))))))))))))))) -> 5(4(3(0(4(1(0(0(4(5(1(0(5(3(1(4(2(4(1(x1))))))))))))))))))) 5(3(3(2(4(2(4(4(0(1(0(5(1(0(4(3(4(0(4(x1))))))))))))))))))) -> 5(5(2(1(5(2(4(5(5(3(4(0(0(3(5(2(4(4(x1)))))))))))))))))) 5(4(1(5(2(4(3(5(4(2(4(4(4(3(2(3(2(0(5(x1))))))))))))))))))) -> 0(4(4(0(2(0(1(2(5(3(0(1(4(3(3(2(5(5(x1)))))))))))))))))) 0(5(0(4(0(1(3(1(1(4(5(1(3(2(0(0(3(5(3(5(x1)))))))))))))))))))) -> 0(0(2(5(5(3(3(2(4(3(0(1(4(5(2(1(5(5(5(2(x1)))))))))))))))))))) 4(3(5(5(4(3(0(0(3(4(5(5(5(2(2(0(0(3(2(5(x1)))))))))))))))))))) -> 4(5(2(2(3(3(3(5(3(1(5(5(5(1(3(2(3(5(3(3(4(x1))))))))))))))))))))) 1(0(2(4(3(3(5(2(4(1(3(4(0(0(5(4(1(1(1(3(0(x1))))))))))))))))))))) -> 1(1(4(1(0(0(5(2(2(1(2(4(4(0(2(5(1(1(3(0(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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: 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(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 0(0(1(3(4(x1))))) 5(3(3(3(0(4(4(5(x1)))))))) -> 5(0(3(1(3(1(2(x1))))))) 0(3(1(2(4(4(0(5(2(x1))))))))) -> 0(1(3(3(3(1(0(5(2(x1))))))))) 0(2(5(0(3(0(5(3(0(4(x1)))))))))) -> 4(4(1(5(4(5(0(3(3(4(x1)))))))))) 0(3(0(5(4(4(5(4(4(4(2(x1))))))))))) -> 0(3(2(1(4(5(3(2(5(0(2(x1))))))))))) 0(5(3(2(3(4(3(4(2(1(1(2(5(x1))))))))))))) -> 0(5(3(1(0(4(0(1(5(5(0(1(x1)))))))))))) 0(4(0(2(0(0(4(4(2(3(0(1(1(3(x1)))))))))))))) -> 0(0(0(1(2(4(4(0(5(4(0(2(3(x1))))))))))))) 1(4(4(5(2(1(2(4(2(5(1(0(5(0(x1)))))))))))))) -> 1(0(3(0(4(3(3(4(5(2(2(4(1(4(x1)))))))))))))) 1(5(4(4(3(4(4(1(0(1(4(1(5(5(x1)))))))))))))) -> 0(1(4(2(5(0(3(5(4(4(0(4(1(x1))))))))))))) 2(0(4(5(0(0(4(4(3(2(1(5(1(4(x1)))))))))))))) -> 2(3(2(0(5(1(0(3(0(3(1(3(0(4(x1)))))))))))))) 4(5(3(5(0(5(1(4(3(2(5(2(2(2(x1)))))))))))))) -> 4(4(3(0(2(3(2(0(0(0(1(5(4(x1))))))))))))) 2(1(0(0(3(0(2(2(1(1(4(5(3(0(5(x1))))))))))))))) -> 4(3(5(3(1(4(5(2(1(3(0(3(3(0(4(x1))))))))))))))) 2(1(3(0(2(2(4(5(0(3(5(0(3(1(0(x1))))))))))))))) -> 2(5(5(0(1(5(1(0(4(0(0(1(4(0(x1)))))))))))))) 0(5(1(2(1(5(4(3(5(2(4(0(0(5(2(0(x1)))))))))))))))) -> 0(1(2(3(2(0(5(1(4(1(3(0(2(3(1(0(x1)))))))))))))))) 2(1(4(3(4(5(1(0(2(1(1(5(1(1(5(2(4(4(x1)))))))))))))))))) -> 4(4(1(0(5(1(5(5(0(2(1(4(1(3(1(2(1(4(x1)))))))))))))))))) 3(2(5(0(3(2(1(1(4(1(1(4(4(0(4(4(5(5(x1)))))))))))))))))) -> 3(3(2(4(1(3(4(3(5(1(5(3(5(4(4(1(2(2(x1)))))))))))))))))) 5(2(1(0(0(0(1(4(4(4(0(5(0(0(0(3(3(1(x1)))))))))))))))))) -> 5(4(2(4(1(3(1(2(2(5(4(5(2(5(4(4(0(5(5(x1))))))))))))))))))) 5(4(4(3(1(0(2(0(3(2(0(2(3(3(0(4(5(1(x1)))))))))))))))))) -> 4(3(2(0(2(4(5(2(0(3(4(5(5(0(2(2(1(4(x1)))))))))))))))))) 5(4(5(2(5(2(1(3(4(0(0(5(1(3(0(3(5(4(x1)))))))))))))))))) -> 4(1(3(2(4(4(5(1(5(3(2(1(2(5(3(5(2(4(x1)))))))))))))))))) 4(1(3(0(5(4(1(4(4(0(0(4(4(3(0(2(0(2(1(x1))))))))))))))))))) -> 5(4(3(0(4(1(0(0(4(5(1(0(5(3(1(4(2(4(1(x1))))))))))))))))))) 5(3(3(2(4(2(4(4(0(1(0(5(1(0(4(3(4(0(4(x1))))))))))))))))))) -> 5(5(2(1(5(2(4(5(5(3(4(0(0(3(5(2(4(4(x1)))))))))))))))))) 5(4(1(5(2(4(3(5(4(2(4(4(4(3(2(3(2(0(5(x1))))))))))))))))))) -> 0(4(4(0(2(0(1(2(5(3(0(1(4(3(3(2(5(5(x1)))))))))))))))))) 0(5(0(4(0(1(3(1(1(4(5(1(3(2(0(0(3(5(3(5(x1)))))))))))))))))))) -> 0(0(2(5(5(3(3(2(4(3(0(1(4(5(2(1(5(5(5(2(x1)))))))))))))))))))) 4(3(5(5(4(3(0(0(3(4(5(5(5(2(2(0(0(3(2(5(x1)))))))))))))))))))) -> 4(5(2(2(3(3(3(5(3(1(5(5(5(1(3(2(3(5(3(3(4(x1))))))))))))))))))))) 1(0(2(4(3(3(5(2(4(1(3(4(0(0(5(4(1(1(1(3(0(x1))))))))))))))))))))) -> 1(1(4(1(0(0(5(2(2(1(2(4(4(0(2(5(1(1(3(0(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 0(0(1(3(4(x1))))) 5(3(3(3(0(4(4(5(x1)))))))) -> 5(0(3(1(3(1(2(x1))))))) 0(3(1(2(4(4(0(5(2(x1))))))))) -> 0(1(3(3(3(1(0(5(2(x1))))))))) 0(2(5(0(3(0(5(3(0(4(x1)))))))))) -> 4(4(1(5(4(5(0(3(3(4(x1)))))))))) 0(3(0(5(4(4(5(4(4(4(2(x1))))))))))) -> 0(3(2(1(4(5(3(2(5(0(2(x1))))))))))) 0(5(3(2(3(4(3(4(2(1(1(2(5(x1))))))))))))) -> 0(5(3(1(0(4(0(1(5(5(0(1(x1)))))))))))) 0(4(0(2(0(0(4(4(2(3(0(1(1(3(x1)))))))))))))) -> 0(0(0(1(2(4(4(0(5(4(0(2(3(x1))))))))))))) 1(4(4(5(2(1(2(4(2(5(1(0(5(0(x1)))))))))))))) -> 1(0(3(0(4(3(3(4(5(2(2(4(1(4(x1)))))))))))))) 1(5(4(4(3(4(4(1(0(1(4(1(5(5(x1)))))))))))))) -> 0(1(4(2(5(0(3(5(4(4(0(4(1(x1))))))))))))) 2(0(4(5(0(0(4(4(3(2(1(5(1(4(x1)))))))))))))) -> 2(3(2(0(5(1(0(3(0(3(1(3(0(4(x1)))))))))))))) 4(5(3(5(0(5(1(4(3(2(5(2(2(2(x1)))))))))))))) -> 4(4(3(0(2(3(2(0(0(0(1(5(4(x1))))))))))))) 2(1(0(0(3(0(2(2(1(1(4(5(3(0(5(x1))))))))))))))) -> 4(3(5(3(1(4(5(2(1(3(0(3(3(0(4(x1))))))))))))))) 2(1(3(0(2(2(4(5(0(3(5(0(3(1(0(x1))))))))))))))) -> 2(5(5(0(1(5(1(0(4(0(0(1(4(0(x1)))))))))))))) 0(5(1(2(1(5(4(3(5(2(4(0(0(5(2(0(x1)))))))))))))))) -> 0(1(2(3(2(0(5(1(4(1(3(0(2(3(1(0(x1)))))))))))))))) 2(1(4(3(4(5(1(0(2(1(1(5(1(1(5(2(4(4(x1)))))))))))))))))) -> 4(4(1(0(5(1(5(5(0(2(1(4(1(3(1(2(1(4(x1)))))))))))))))))) 3(2(5(0(3(2(1(1(4(1(1(4(4(0(4(4(5(5(x1)))))))))))))))))) -> 3(3(2(4(1(3(4(3(5(1(5(3(5(4(4(1(2(2(x1)))))))))))))))))) 5(2(1(0(0(0(1(4(4(4(0(5(0(0(0(3(3(1(x1)))))))))))))))))) -> 5(4(2(4(1(3(1(2(2(5(4(5(2(5(4(4(0(5(5(x1))))))))))))))))))) 5(4(4(3(1(0(2(0(3(2(0(2(3(3(0(4(5(1(x1)))))))))))))))))) -> 4(3(2(0(2(4(5(2(0(3(4(5(5(0(2(2(1(4(x1)))))))))))))))))) 5(4(5(2(5(2(1(3(4(0(0(5(1(3(0(3(5(4(x1)))))))))))))))))) -> 4(1(3(2(4(4(5(1(5(3(2(1(2(5(3(5(2(4(x1)))))))))))))))))) 4(1(3(0(5(4(1(4(4(0(0(4(4(3(0(2(0(2(1(x1))))))))))))))))))) -> 5(4(3(0(4(1(0(0(4(5(1(0(5(3(1(4(2(4(1(x1))))))))))))))))))) 5(3(3(2(4(2(4(4(0(1(0(5(1(0(4(3(4(0(4(x1))))))))))))))))))) -> 5(5(2(1(5(2(4(5(5(3(4(0(0(3(5(2(4(4(x1)))))))))))))))))) 5(4(1(5(2(4(3(5(4(2(4(4(4(3(2(3(2(0(5(x1))))))))))))))))))) -> 0(4(4(0(2(0(1(2(5(3(0(1(4(3(3(2(5(5(x1)))))))))))))))))) 0(5(0(4(0(1(3(1(1(4(5(1(3(2(0(0(3(5(3(5(x1)))))))))))))))))))) -> 0(0(2(5(5(3(3(2(4(3(0(1(4(5(2(1(5(5(5(2(x1)))))))))))))))))))) 4(3(5(5(4(3(0(0(3(4(5(5(5(2(2(0(0(3(2(5(x1)))))))))))))))))))) -> 4(5(2(2(3(3(3(5(3(1(5(5(5(1(3(2(3(5(3(3(4(x1))))))))))))))))))))) 1(0(2(4(3(3(5(2(4(1(3(4(0(0(5(4(1(1(1(3(0(x1))))))))))))))))))))) -> 1(1(4(1(0(0(5(2(2(1(2(4(4(0(2(5(1(1(3(0(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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: FULL ---------------------------------------- (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 2. 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] {(151,152,[0_1|0, 5_1|0, 1_1|0, 2_1|0, 4_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, 5_1|1, 1_1|1, 2_1|1, 4_1|1, 3_1|1]), (151,154,[0_1|2]), (151,158,[0_1|2]), (151,166,[0_1|2]), (151,176,[4_1|2]), (151,185,[0_1|2]), (151,196,[0_1|2]), (151,211,[0_1|2]), (151,230,[0_1|2]), (151,242,[5_1|2]), (151,248,[5_1|2]), (151,265,[5_1|2]), (151,283,[4_1|2]), (151,300,[4_1|2]), (151,317,[0_1|2]), (151,334,[1_1|2]), (151,347,[0_1|2]), (151,359,[1_1|2]), (151,378,[2_1|2]), (151,391,[4_1|2]), (151,405,[2_1|2]), (151,418,[4_1|2]), (151,435,[4_1|2]), (151,447,[5_1|2]), (151,465,[4_1|2]), (151,485,[3_1|2]), (152,152,[cons_0_1|0, cons_5_1|0, cons_1_1|0, cons_2_1|0, cons_4_1|0, cons_3_1|0]), (153,152,[encArg_1|1]), (153,153,[0_1|1, 5_1|1, 1_1|1, 2_1|1, 4_1|1, 3_1|1]), (153,154,[0_1|2]), (153,158,[0_1|2]), (153,166,[0_1|2]), (153,176,[4_1|2]), (153,185,[0_1|2]), (153,196,[0_1|2]), (153,211,[0_1|2]), (153,230,[0_1|2]), (153,242,[5_1|2]), (153,248,[5_1|2]), (153,265,[5_1|2]), (153,283,[4_1|2]), (153,300,[4_1|2]), (153,317,[0_1|2]), (153,334,[1_1|2]), (153,347,[0_1|2]), (153,359,[1_1|2]), (153,378,[2_1|2]), (153,391,[4_1|2]), (153,405,[2_1|2]), (153,418,[4_1|2]), (153,435,[4_1|2]), (153,447,[5_1|2]), (153,465,[4_1|2]), (153,485,[3_1|2]), (154,155,[0_1|2]), (155,156,[1_1|2]), (156,157,[3_1|2]), (157,153,[4_1|2]), (157,378,[4_1|2]), (157,405,[4_1|2]), (157,435,[4_1|2]), (157,447,[5_1|2]), (157,465,[4_1|2]), (158,159,[1_1|2]), (159,160,[3_1|2]), (160,161,[3_1|2]), (161,162,[3_1|2]), (162,163,[1_1|2]), (163,164,[0_1|2]), (164,165,[5_1|2]), (164,265,[5_1|2]), (165,153,[2_1|2]), (165,378,[2_1|2]), (165,405,[2_1|2]), (165,391,[4_1|2]), (165,418,[4_1|2]), (166,167,[3_1|2]), (167,168,[2_1|2]), (168,169,[1_1|2]), (169,170,[4_1|2]), (170,171,[5_1|2]), (171,172,[3_1|2]), (172,173,[2_1|2]), (173,174,[5_1|2]), (174,175,[0_1|2]), (174,176,[4_1|2]), (175,153,[2_1|2]), (175,378,[2_1|2]), (175,405,[2_1|2]), (175,391,[4_1|2]), (175,418,[4_1|2]), (176,177,[4_1|2]), (177,178,[1_1|2]), (178,179,[5_1|2]), (179,180,[4_1|2]), (180,181,[5_1|2]), (181,182,[0_1|2]), (182,183,[3_1|2]), (183,184,[3_1|2]), (184,153,[4_1|2]), (184,176,[4_1|2]), (184,283,[4_1|2]), (184,300,[4_1|2]), (184,391,[4_1|2]), (184,418,[4_1|2]), (184,435,[4_1|2]), (184,465,[4_1|2]), (184,318,[4_1|2]), (184,447,[5_1|2]), (185,186,[5_1|2]), (186,187,[3_1|2]), (187,188,[1_1|2]), (188,189,[0_1|2]), (189,190,[4_1|2]), (190,191,[0_1|2]), (191,192,[1_1|2]), (192,193,[5_1|2]), (193,194,[5_1|2]), (194,195,[0_1|2]), (194,154,[0_1|2]), (195,153,[1_1|2]), (195,242,[1_1|2]), (195,248,[1_1|2]), (195,265,[1_1|2]), (195,447,[1_1|2]), (195,406,[1_1|2]), (195,334,[1_1|2]), (195,347,[0_1|2]), (195,359,[1_1|2]), (196,197,[1_1|2]), (197,198,[2_1|2]), (198,199,[3_1|2]), (199,200,[2_1|2]), (200,201,[0_1|2]), (201,202,[5_1|2]), (202,203,[1_1|2]), (203,204,[4_1|2]), (204,205,[1_1|2]), (205,206,[3_1|2]), (206,207,[0_1|2]), (207,208,[2_1|2]), (208,209,[3_1|2]), (209,210,[1_1|2]), (209,359,[1_1|2]), (210,153,[0_1|2]), (210,154,[0_1|2]), (210,158,[0_1|2]), (210,166,[0_1|2]), (210,185,[0_1|2]), (210,196,[0_1|2]), (210,211,[0_1|2]), (210,230,[0_1|2]), (210,317,[0_1|2]), (210,347,[0_1|2]), (210,176,[4_1|2]), (211,212,[0_1|2]), (212,213,[2_1|2]), (213,214,[5_1|2]), (214,215,[5_1|2]), (215,216,[3_1|2]), (216,217,[3_1|2]), (217,218,[2_1|2]), (218,219,[4_1|2]), (219,220,[3_1|2]), (220,221,[0_1|2]), (221,222,[1_1|2]), (222,223,[4_1|2]), (223,224,[5_1|2]), (224,225,[2_1|2]), (225,226,[1_1|2]), (226,227,[5_1|2]), (227,228,[5_1|2]), (228,229,[5_1|2]), (228,265,[5_1|2]), (229,153,[2_1|2]), (229,242,[2_1|2]), (229,248,[2_1|2]), (229,265,[2_1|2]), (229,447,[2_1|2]), (229,378,[2_1|2]), (229,391,[4_1|2]), (229,405,[2_1|2]), (229,418,[4_1|2]), (230,231,[0_1|2]), (231,232,[0_1|2]), (232,233,[1_1|2]), (233,234,[2_1|2]), (234,235,[4_1|2]), (235,236,[4_1|2]), (236,237,[0_1|2]), (237,238,[5_1|2]), (238,239,[4_1|2]), (239,240,[0_1|2]), (240,241,[2_1|2]), (241,153,[3_1|2]), (241,485,[3_1|2]), (242,243,[0_1|2]), (243,244,[3_1|2]), (244,245,[1_1|2]), (245,246,[3_1|2]), (246,247,[1_1|2]), (247,153,[2_1|2]), (247,242,[2_1|2]), (247,248,[2_1|2]), (247,265,[2_1|2]), (247,447,[2_1|2]), (247,466,[2_1|2]), (247,378,[2_1|2]), (247,391,[4_1|2]), (247,405,[2_1|2]), (247,418,[4_1|2]), (248,249,[5_1|2]), (249,250,[2_1|2]), (250,251,[1_1|2]), (251,252,[5_1|2]), (252,253,[2_1|2]), (253,254,[4_1|2]), (254,255,[5_1|2]), (255,256,[5_1|2]), (256,257,[3_1|2]), (257,258,[4_1|2]), (258,259,[0_1|2]), (259,260,[0_1|2]), (260,261,[3_1|2]), (261,262,[5_1|2]), (262,263,[2_1|2]), (263,264,[4_1|2]), (264,153,[4_1|2]), (264,176,[4_1|2]), (264,283,[4_1|2]), (264,300,[4_1|2]), (264,391,[4_1|2]), (264,418,[4_1|2]), (264,435,[4_1|2]), (264,465,[4_1|2]), (264,318,[4_1|2]), (264,447,[5_1|2]), (265,266,[4_1|2]), (266,267,[2_1|2]), (267,268,[4_1|2]), (268,269,[1_1|2]), (269,270,[3_1|2]), (270,271,[1_1|2]), (271,272,[2_1|2]), (272,273,[2_1|2]), (273,274,[5_1|2]), (274,275,[4_1|2]), (275,276,[5_1|2]), (276,277,[2_1|2]), (277,278,[5_1|2]), (278,279,[4_1|2]), (279,280,[4_1|2]), (280,281,[0_1|2]), (281,282,[5_1|2]), (282,153,[5_1|2]), (282,334,[5_1|2]), (282,359,[5_1|2]), (282,242,[5_1|2]), (282,248,[5_1|2]), (282,265,[5_1|2]), (282,283,[4_1|2]), (282,300,[4_1|2]), (282,317,[0_1|2]), (283,284,[3_1|2]), (284,285,[2_1|2]), (285,286,[0_1|2]), (286,287,[2_1|2]), (287,288,[4_1|2]), (288,289,[5_1|2]), (289,290,[2_1|2]), (290,291,[0_1|2]), (291,292,[3_1|2]), (292,293,[4_1|2]), (293,294,[5_1|2]), (294,295,[5_1|2]), (295,296,[0_1|2]), (296,297,[2_1|2]), (297,298,[2_1|2]), (297,418,[4_1|2]), (298,299,[1_1|2]), (298,334,[1_1|2]), (299,153,[4_1|2]), (299,334,[4_1|2]), (299,359,[4_1|2]), (299,435,[4_1|2]), (299,447,[5_1|2]), (299,465,[4_1|2]), (300,301,[1_1|2]), (301,302,[3_1|2]), (302,303,[2_1|2]), (303,304,[4_1|2]), (304,305,[4_1|2]), (305,306,[5_1|2]), (306,307,[1_1|2]), (307,308,[5_1|2]), (308,309,[3_1|2]), (309,310,[2_1|2]), (310,311,[1_1|2]), (311,312,[2_1|2]), (312,313,[5_1|2]), (313,314,[3_1|2]), (314,315,[5_1|2]), (315,316,[2_1|2]), (316,153,[4_1|2]), (316,176,[4_1|2]), (316,283,[4_1|2]), (316,300,[4_1|2]), (316,391,[4_1|2]), (316,418,[4_1|2]), (316,435,[4_1|2]), (316,465,[4_1|2]), (316,266,[4_1|2]), (316,448,[4_1|2]), (316,447,[5_1|2]), (317,318,[4_1|2]), (318,319,[4_1|2]), (319,320,[0_1|2]), (320,321,[2_1|2]), (321,322,[0_1|2]), (322,323,[1_1|2]), (323,324,[2_1|2]), (324,325,[5_1|2]), (325,326,[3_1|2]), (326,327,[0_1|2]), (327,328,[1_1|2]), (328,329,[4_1|2]), (329,330,[3_1|2]), (330,331,[3_1|2]), (331,332,[2_1|2]), (332,333,[5_1|2]), (333,153,[5_1|2]), (333,242,[5_1|2]), (333,248,[5_1|2]), (333,265,[5_1|2]), (333,447,[5_1|2]), (333,186,[5_1|2]), (333,382,[5_1|2]), (333,283,[4_1|2]), (333,300,[4_1|2]), (333,317,[0_1|2]), (334,335,[0_1|2]), (335,336,[3_1|2]), (336,337,[0_1|2]), (337,338,[4_1|2]), (338,339,[3_1|2]), (339,340,[3_1|2]), (340,341,[4_1|2]), (341,342,[5_1|2]), (342,343,[2_1|2]), (343,344,[2_1|2]), (344,345,[4_1|2]), (345,346,[1_1|2]), (345,334,[1_1|2]), (346,153,[4_1|2]), (346,154,[4_1|2]), (346,158,[4_1|2]), (346,166,[4_1|2]), (346,185,[4_1|2]), (346,196,[4_1|2]), (346,211,[4_1|2]), (346,230,[4_1|2]), (346,317,[4_1|2]), (346,347,[4_1|2]), (346,243,[4_1|2]), (346,435,[4_1|2]), (346,447,[5_1|2]), (346,465,[4_1|2]), (347,348,[1_1|2]), (348,349,[4_1|2]), (349,350,[2_1|2]), (350,351,[5_1|2]), (351,352,[0_1|2]), (352,353,[3_1|2]), (353,354,[5_1|2]), (354,355,[4_1|2]), (355,356,[4_1|2]), (356,357,[0_1|2]), (357,358,[4_1|2]), (357,447,[5_1|2]), (358,153,[1_1|2]), (358,242,[1_1|2]), (358,248,[1_1|2]), (358,265,[1_1|2]), (358,447,[1_1|2]), (358,249,[1_1|2]), (358,334,[1_1|2]), (358,347,[0_1|2]), (358,359,[1_1|2]), (359,360,[1_1|2]), (360,361,[4_1|2]), (361,362,[1_1|2]), (362,363,[0_1|2]), (363,364,[0_1|2]), (364,365,[5_1|2]), (365,366,[2_1|2]), (366,367,[2_1|2]), (367,368,[1_1|2]), (368,369,[2_1|2]), (369,370,[4_1|2]), (370,371,[4_1|2]), (371,372,[0_1|2]), (372,373,[2_1|2]), (373,374,[5_1|2]), (374,375,[1_1|2]), (375,376,[1_1|2]), (376,377,[3_1|2]), (377,153,[0_1|2]), (377,154,[0_1|2]), (377,158,[0_1|2]), (377,166,[0_1|2]), (377,185,[0_1|2]), (377,196,[0_1|2]), (377,211,[0_1|2]), (377,230,[0_1|2]), (377,317,[0_1|2]), (377,347,[0_1|2]), (377,176,[4_1|2]), (378,379,[3_1|2]), (379,380,[2_1|2]), (380,381,[0_1|2]), (381,382,[5_1|2]), (382,383,[1_1|2]), (383,384,[0_1|2]), (384,385,[3_1|2]), (385,386,[0_1|2]), (386,387,[3_1|2]), (387,388,[1_1|2]), (388,389,[3_1|2]), (389,390,[0_1|2]), (389,230,[0_1|2]), (390,153,[4_1|2]), (390,176,[4_1|2]), (390,283,[4_1|2]), (390,300,[4_1|2]), (390,391,[4_1|2]), (390,418,[4_1|2]), (390,435,[4_1|2]), (390,465,[4_1|2]), (390,447,[5_1|2]), (391,392,[3_1|2]), (392,393,[5_1|2]), (393,394,[3_1|2]), (394,395,[1_1|2]), (395,396,[4_1|2]), (396,397,[5_1|2]), (397,398,[2_1|2]), (398,399,[1_1|2]), (399,400,[3_1|2]), (400,401,[0_1|2]), (401,402,[3_1|2]), (402,403,[3_1|2]), (403,404,[0_1|2]), (403,230,[0_1|2]), (404,153,[4_1|2]), (404,242,[4_1|2]), (404,248,[4_1|2]), (404,265,[4_1|2]), (404,447,[4_1|2, 5_1|2]), (404,186,[4_1|2]), (404,435,[4_1|2]), (404,465,[4_1|2]), (405,406,[5_1|2]), (406,407,[5_1|2]), (407,408,[0_1|2]), (408,409,[1_1|2]), (409,410,[5_1|2]), (410,411,[1_1|2]), (411,412,[0_1|2]), (412,413,[4_1|2]), (413,414,[0_1|2]), (414,415,[0_1|2]), (415,416,[1_1|2]), (416,417,[4_1|2]), (417,153,[0_1|2]), (417,154,[0_1|2]), (417,158,[0_1|2]), (417,166,[0_1|2]), (417,185,[0_1|2]), (417,196,[0_1|2]), (417,211,[0_1|2]), (417,230,[0_1|2]), (417,317,[0_1|2]), (417,347,[0_1|2]), (417,335,[0_1|2]), (417,176,[4_1|2]), (418,419,[4_1|2]), (419,420,[1_1|2]), (420,421,[0_1|2]), (421,422,[5_1|2]), (422,423,[1_1|2]), (423,424,[5_1|2]), (424,425,[5_1|2]), (425,426,[0_1|2]), (426,427,[2_1|2]), (427,428,[1_1|2]), (428,429,[4_1|2]), (429,430,[1_1|2]), (430,431,[3_1|2]), (431,432,[1_1|2]), (432,433,[2_1|2]), (432,418,[4_1|2]), (433,434,[1_1|2]), (433,334,[1_1|2]), (434,153,[4_1|2]), (434,176,[4_1|2]), (434,283,[4_1|2]), (434,300,[4_1|2]), (434,391,[4_1|2]), (434,418,[4_1|2]), (434,435,[4_1|2]), (434,465,[4_1|2]), (434,177,[4_1|2]), (434,419,[4_1|2]), (434,436,[4_1|2]), (434,447,[5_1|2]), (435,436,[4_1|2]), (436,437,[3_1|2]), (437,438,[0_1|2]), (438,439,[2_1|2]), (439,440,[3_1|2]), (440,441,[2_1|2]), (441,442,[0_1|2]), (442,443,[0_1|2]), (443,444,[0_1|2]), (444,445,[1_1|2]), (444,347,[0_1|2]), (445,446,[5_1|2]), (445,283,[4_1|2]), (445,300,[4_1|2]), (445,317,[0_1|2]), (446,153,[4_1|2]), (446,378,[4_1|2]), (446,405,[4_1|2]), (446,435,[4_1|2]), (446,447,[5_1|2]), (446,465,[4_1|2]), (447,448,[4_1|2]), (448,449,[3_1|2]), (449,450,[0_1|2]), (450,451,[4_1|2]), (451,452,[1_1|2]), (452,453,[0_1|2]), (453,454,[0_1|2]), (454,455,[4_1|2]), (455,456,[5_1|2]), (456,457,[1_1|2]), (457,458,[0_1|2]), (458,459,[5_1|2]), (459,460,[3_1|2]), (460,461,[1_1|2]), (461,462,[4_1|2]), (462,463,[2_1|2]), (463,464,[4_1|2]), (463,447,[5_1|2]), (464,153,[1_1|2]), (464,334,[1_1|2]), (464,359,[1_1|2]), (464,347,[0_1|2]), (465,466,[5_1|2]), (466,467,[2_1|2]), (467,468,[2_1|2]), (468,469,[3_1|2]), (469,470,[3_1|2]), (470,471,[3_1|2]), (471,472,[5_1|2]), (472,473,[3_1|2]), (473,474,[1_1|2]), (474,475,[5_1|2]), (475,476,[5_1|2]), (476,477,[5_1|2]), (477,478,[1_1|2]), (478,479,[3_1|2]), (479,480,[2_1|2]), (480,481,[3_1|2]), (481,482,[5_1|2]), (482,483,[3_1|2]), (483,484,[3_1|2]), (484,153,[4_1|2]), (484,242,[4_1|2]), (484,248,[4_1|2]), (484,265,[4_1|2]), (484,447,[4_1|2, 5_1|2]), (484,406,[4_1|2]), (484,435,[4_1|2]), (484,465,[4_1|2]), (485,486,[3_1|2]), (486,487,[2_1|2]), (487,488,[4_1|2]), (488,489,[1_1|2]), (489,490,[3_1|2]), (490,491,[4_1|2]), (491,492,[3_1|2]), (492,493,[5_1|2]), (493,494,[1_1|2]), (494,495,[5_1|2]), (495,496,[3_1|2]), (496,497,[5_1|2]), (497,498,[4_1|2]), (498,499,[4_1|2]), (499,500,[1_1|2]), (500,501,[2_1|2]), (501,153,[2_1|2]), (501,242,[2_1|2]), (501,248,[2_1|2]), (501,265,[2_1|2]), (501,447,[2_1|2]), (501,249,[2_1|2]), (501,378,[2_1|2]), (501,391,[4_1|2]), (501,405,[2_1|2]), (501,418,[4_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)