/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 47 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 145 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(0(0(1(x1)))) -> 0(1(2(x1))) 0(3(4(0(1(1(4(3(4(x1))))))))) -> 0(3(4(1(0(5(1(4(x1)))))))) 2(1(5(5(3(5(5(3(2(x1))))))))) -> 2(2(0(2(4(0(1(2(x1)))))))) 3(3(2(4(4(4(5(0(0(x1))))))))) -> 3(5(0(1(4(5(4(0(0(x1))))))))) 3(4(3(3(5(5(3(4(4(x1))))))))) -> 5(0(2(0(5(4(0(x1))))))) 1(1(0(5(4(1(1(3(5(1(x1)))))))))) -> 1(1(5(4(5(4(4(4(1(x1))))))))) 3(0(1(1(1(3(5(0(0(4(x1)))))))))) -> 3(5(3(3(0(4(3(4(0(x1))))))))) 5(0(1(4(5(4(3(0(0(1(x1)))))))))) -> 5(1(4(0(5(2(5(0(1(x1))))))))) 3(4(3(4(5(2(3(2(4(1(3(x1))))))))))) -> 2(4(4(2(2(2(3(5(5(3(x1)))))))))) 2(1(2(5(2(4(2(1(4(1(0(0(x1)))))))))))) -> 0(1(2(5(2(5(5(0(2(3(0(x1))))))))))) 2(5(4(0(3(1(2(5(1(1(4(5(x1)))))))))))) -> 1(0(3(4(4(1(3(5(1(1(4(5(x1)))))))))))) 1(5(2(1(5(2(3(3(2(1(5(1(1(x1))))))))))))) -> 1(5(0(3(2(5(3(5(2(2(0(x1))))))))))) 5(4(4(1(1(4(2(0(5(1(4(4(4(x1))))))))))))) -> 5(4(0(1(3(4(0(4(2(1(2(5(x1)))))))))))) 4(5(5(2(1(3(4(2(0(5(1(5(1(1(x1)))))))))))))) -> 4(5(2(3(2(0(5(4(3(5(4(2(x1)))))))))))) 3(0(5(2(2(0(4(5(3(4(3(5(2(5(2(x1))))))))))))))) -> 2(0(4(2(0(2(2(1(2(4(5(0(4(3(2(x1))))))))))))))) 4(2(5(0(5(4(2(5(0(3(2(1(5(5(4(x1))))))))))))))) -> 4(4(0(5(2(1(2(1(1(5(1(4(3(5(1(3(x1)))))))))))))))) 2(0(5(2(3(1(3(3(2(1(0(4(2(0(3(0(x1)))))))))))))))) -> 3(2(2(3(2(0(1(2(0(1(2(0(5(5(4(0(x1)))))))))))))))) 4(4(5(1(0(1(5(0(1(0(4(5(3(5(4(4(x1)))))))))))))))) -> 4(1(2(1(2(2(0(1(1(1(0(3(3(2(3(3(4(x1))))))))))))))))) 2(2(0(4(3(1(5(1(1(4(2(3(0(4(1(1(3(x1))))))))))))))))) -> 0(3(5(0(3(1(2(2(5(2(2(2(5(4(4(x1))))))))))))))) 5(5(1(0(5(0(5(0(5(0(3(0(2(5(5(4(1(x1))))))))))))))))) -> 2(1(1(3(1(5(0(2(0(0(4(5(5(5(5(2(5(1(x1)))))))))))))))))) 1(0(0(0(2(5(4(1(0(1(2(2(5(0(4(3(4(4(x1)))))))))))))))))) -> 1(2(2(5(1(5(2(0(2(4(3(4(3(0(4(2(0(x1))))))))))))))))) 2(2(3(0(5(3(0(4(0(3(1(1(1(5(1(3(1(0(x1)))))))))))))))))) -> 2(4(1(5(5(0(1(2(0(2(5(1(4(1(0(0(x1)))))))))))))))) 2(1(1(4(4(2(1(0(3(0(0(3(3(4(3(1(0(4(4(x1))))))))))))))))))) -> 2(1(0(3(4(1(5(4(0(1(2(1(5(2(1(1(2(3(4(x1))))))))))))))))))) 4(1(4(0(2(0(5(5(2(0(2(2(0(1(2(1(5(5(2(4(x1)))))))))))))))))))) -> 4(3(1(4(5(0(0(5(2(5(3(0(5(0(1(1(0(3(5(x1))))))))))))))))))) 4(3(3(5(3(5(1(3(1(0(3(1(5(4(1(3(4(4(5(2(x1)))))))))))))))))))) -> 4(3(5(5(5(0(5(2(5(5(3(3(2(4(3(4(4(2(x1)))))))))))))))))) 5(4(4(0(5(0(4(0(3(4(4(4(0(5(3(3(3(5(2(4(x1)))))))))))))))))))) -> 3(2(4(1(5(2(4(5(5(3(5(0(3(4(1(4(3(1(2(2(2(x1))))))))))))))))))))) 0(0(2(4(0(2(0(0(1(5(4(5(5(1(3(0(1(5(1(5(4(x1))))))))))))))))))))) -> 0(2(3(2(4(0(4(4(1(5(2(2(2(0(0(1(3(4(2(5(x1)))))))))))))))))))) 1(1(1(2(5(4(2(3(5(1(3(3(2(2(1(4(0(2(5(4(1(x1))))))))))))))))))))) -> 4(3(2(0(1(1(5(1(3(2(4(3(3(4(3(4(4(3(0(2(x1)))))))))))))))))))) 2(2(2(2(2(1(2(0(0(5(3(0(1(0(2(0(1(3(1(1(2(x1))))))))))))))))))))) -> 0(3(3(4(2(2(3(0(0(3(1(2(2(4(4(4(1(5(0(x1))))))))))))))))))) 2(3(3(2(1(4(0(4(5(1(1(2(5(2(2(4(1(2(1(0(2(x1))))))))))))))))))))) -> 5(5(4(2(3(5(2(0(4(2(0(2(3(3(0(5(1(1(4(2(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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(0(1(x1)))) -> 0(1(2(x1))) 0(3(4(0(1(1(4(3(4(x1))))))))) -> 0(3(4(1(0(5(1(4(x1)))))))) 2(1(5(5(3(5(5(3(2(x1))))))))) -> 2(2(0(2(4(0(1(2(x1)))))))) 3(3(2(4(4(4(5(0(0(x1))))))))) -> 3(5(0(1(4(5(4(0(0(x1))))))))) 3(4(3(3(5(5(3(4(4(x1))))))))) -> 5(0(2(0(5(4(0(x1))))))) 1(1(0(5(4(1(1(3(5(1(x1)))))))))) -> 1(1(5(4(5(4(4(4(1(x1))))))))) 3(0(1(1(1(3(5(0(0(4(x1)))))))))) -> 3(5(3(3(0(4(3(4(0(x1))))))))) 5(0(1(4(5(4(3(0(0(1(x1)))))))))) -> 5(1(4(0(5(2(5(0(1(x1))))))))) 3(4(3(4(5(2(3(2(4(1(3(x1))))))))))) -> 2(4(4(2(2(2(3(5(5(3(x1)))))))))) 2(1(2(5(2(4(2(1(4(1(0(0(x1)))))))))))) -> 0(1(2(5(2(5(5(0(2(3(0(x1))))))))))) 2(5(4(0(3(1(2(5(1(1(4(5(x1)))))))))))) -> 1(0(3(4(4(1(3(5(1(1(4(5(x1)))))))))))) 1(5(2(1(5(2(3(3(2(1(5(1(1(x1))))))))))))) -> 1(5(0(3(2(5(3(5(2(2(0(x1))))))))))) 5(4(4(1(1(4(2(0(5(1(4(4(4(x1))))))))))))) -> 5(4(0(1(3(4(0(4(2(1(2(5(x1)))))))))))) 4(5(5(2(1(3(4(2(0(5(1(5(1(1(x1)))))))))))))) -> 4(5(2(3(2(0(5(4(3(5(4(2(x1)))))))))))) 3(0(5(2(2(0(4(5(3(4(3(5(2(5(2(x1))))))))))))))) -> 2(0(4(2(0(2(2(1(2(4(5(0(4(3(2(x1))))))))))))))) 4(2(5(0(5(4(2(5(0(3(2(1(5(5(4(x1))))))))))))))) -> 4(4(0(5(2(1(2(1(1(5(1(4(3(5(1(3(x1)))))))))))))))) 2(0(5(2(3(1(3(3(2(1(0(4(2(0(3(0(x1)))))))))))))))) -> 3(2(2(3(2(0(1(2(0(1(2(0(5(5(4(0(x1)))))))))))))))) 4(4(5(1(0(1(5(0(1(0(4(5(3(5(4(4(x1)))))))))))))))) -> 4(1(2(1(2(2(0(1(1(1(0(3(3(2(3(3(4(x1))))))))))))))))) 2(2(0(4(3(1(5(1(1(4(2(3(0(4(1(1(3(x1))))))))))))))))) -> 0(3(5(0(3(1(2(2(5(2(2(2(5(4(4(x1))))))))))))))) 5(5(1(0(5(0(5(0(5(0(3(0(2(5(5(4(1(x1))))))))))))))))) -> 2(1(1(3(1(5(0(2(0(0(4(5(5(5(5(2(5(1(x1)))))))))))))))))) 1(0(0(0(2(5(4(1(0(1(2(2(5(0(4(3(4(4(x1)))))))))))))))))) -> 1(2(2(5(1(5(2(0(2(4(3(4(3(0(4(2(0(x1))))))))))))))))) 2(2(3(0(5(3(0(4(0(3(1(1(1(5(1(3(1(0(x1)))))))))))))))))) -> 2(4(1(5(5(0(1(2(0(2(5(1(4(1(0(0(x1)))))))))))))))) 2(1(1(4(4(2(1(0(3(0(0(3(3(4(3(1(0(4(4(x1))))))))))))))))))) -> 2(1(0(3(4(1(5(4(0(1(2(1(5(2(1(1(2(3(4(x1))))))))))))))))))) 4(1(4(0(2(0(5(5(2(0(2(2(0(1(2(1(5(5(2(4(x1)))))))))))))))))))) -> 4(3(1(4(5(0(0(5(2(5(3(0(5(0(1(1(0(3(5(x1))))))))))))))))))) 4(3(3(5(3(5(1(3(1(0(3(1(5(4(1(3(4(4(5(2(x1)))))))))))))))))))) -> 4(3(5(5(5(0(5(2(5(5(3(3(2(4(3(4(4(2(x1)))))))))))))))))) 5(4(4(0(5(0(4(0(3(4(4(4(0(5(3(3(3(5(2(4(x1)))))))))))))))))))) -> 3(2(4(1(5(2(4(5(5(3(5(0(3(4(1(4(3(1(2(2(2(x1))))))))))))))))))))) 0(0(2(4(0(2(0(0(1(5(4(5(5(1(3(0(1(5(1(5(4(x1))))))))))))))))))))) -> 0(2(3(2(4(0(4(4(1(5(2(2(2(0(0(1(3(4(2(5(x1)))))))))))))))))))) 1(1(1(2(5(4(2(3(5(1(3(3(2(2(1(4(0(2(5(4(1(x1))))))))))))))))))))) -> 4(3(2(0(1(1(5(1(3(2(4(3(3(4(3(4(4(3(0(2(x1)))))))))))))))))))) 2(2(2(2(2(1(2(0(0(5(3(0(1(0(2(0(1(3(1(1(2(x1))))))))))))))))))))) -> 0(3(3(4(2(2(3(0(0(3(1(2(2(4(4(4(1(5(0(x1))))))))))))))))))) 2(3(3(2(1(4(0(4(5(1(1(2(5(2(2(4(1(2(1(0(2(x1))))))))))))))))))))) -> 5(5(4(2(3(5(2(0(4(2(0(2(3(3(0(5(1(1(4(2(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: 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_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(0(1(x1)))) -> 0(1(2(x1))) 0(3(4(0(1(1(4(3(4(x1))))))))) -> 0(3(4(1(0(5(1(4(x1)))))))) 2(1(5(5(3(5(5(3(2(x1))))))))) -> 2(2(0(2(4(0(1(2(x1)))))))) 3(3(2(4(4(4(5(0(0(x1))))))))) -> 3(5(0(1(4(5(4(0(0(x1))))))))) 3(4(3(3(5(5(3(4(4(x1))))))))) -> 5(0(2(0(5(4(0(x1))))))) 1(1(0(5(4(1(1(3(5(1(x1)))))))))) -> 1(1(5(4(5(4(4(4(1(x1))))))))) 3(0(1(1(1(3(5(0(0(4(x1)))))))))) -> 3(5(3(3(0(4(3(4(0(x1))))))))) 5(0(1(4(5(4(3(0(0(1(x1)))))))))) -> 5(1(4(0(5(2(5(0(1(x1))))))))) 3(4(3(4(5(2(3(2(4(1(3(x1))))))))))) -> 2(4(4(2(2(2(3(5(5(3(x1)))))))))) 2(1(2(5(2(4(2(1(4(1(0(0(x1)))))))))))) -> 0(1(2(5(2(5(5(0(2(3(0(x1))))))))))) 2(5(4(0(3(1(2(5(1(1(4(5(x1)))))))))))) -> 1(0(3(4(4(1(3(5(1(1(4(5(x1)))))))))))) 1(5(2(1(5(2(3(3(2(1(5(1(1(x1))))))))))))) -> 1(5(0(3(2(5(3(5(2(2(0(x1))))))))))) 5(4(4(1(1(4(2(0(5(1(4(4(4(x1))))))))))))) -> 5(4(0(1(3(4(0(4(2(1(2(5(x1)))))))))))) 4(5(5(2(1(3(4(2(0(5(1(5(1(1(x1)))))))))))))) -> 4(5(2(3(2(0(5(4(3(5(4(2(x1)))))))))))) 3(0(5(2(2(0(4(5(3(4(3(5(2(5(2(x1))))))))))))))) -> 2(0(4(2(0(2(2(1(2(4(5(0(4(3(2(x1))))))))))))))) 4(2(5(0(5(4(2(5(0(3(2(1(5(5(4(x1))))))))))))))) -> 4(4(0(5(2(1(2(1(1(5(1(4(3(5(1(3(x1)))))))))))))))) 2(0(5(2(3(1(3(3(2(1(0(4(2(0(3(0(x1)))))))))))))))) -> 3(2(2(3(2(0(1(2(0(1(2(0(5(5(4(0(x1)))))))))))))))) 4(4(5(1(0(1(5(0(1(0(4(5(3(5(4(4(x1)))))))))))))))) -> 4(1(2(1(2(2(0(1(1(1(0(3(3(2(3(3(4(x1))))))))))))))))) 2(2(0(4(3(1(5(1(1(4(2(3(0(4(1(1(3(x1))))))))))))))))) -> 0(3(5(0(3(1(2(2(5(2(2(2(5(4(4(x1))))))))))))))) 5(5(1(0(5(0(5(0(5(0(3(0(2(5(5(4(1(x1))))))))))))))))) -> 2(1(1(3(1(5(0(2(0(0(4(5(5(5(5(2(5(1(x1)))))))))))))))))) 1(0(0(0(2(5(4(1(0(1(2(2(5(0(4(3(4(4(x1)))))))))))))))))) -> 1(2(2(5(1(5(2(0(2(4(3(4(3(0(4(2(0(x1))))))))))))))))) 2(2(3(0(5(3(0(4(0(3(1(1(1(5(1(3(1(0(x1)))))))))))))))))) -> 2(4(1(5(5(0(1(2(0(2(5(1(4(1(0(0(x1)))))))))))))))) 2(1(1(4(4(2(1(0(3(0(0(3(3(4(3(1(0(4(4(x1))))))))))))))))))) -> 2(1(0(3(4(1(5(4(0(1(2(1(5(2(1(1(2(3(4(x1))))))))))))))))))) 4(1(4(0(2(0(5(5(2(0(2(2(0(1(2(1(5(5(2(4(x1)))))))))))))))))))) -> 4(3(1(4(5(0(0(5(2(5(3(0(5(0(1(1(0(3(5(x1))))))))))))))))))) 4(3(3(5(3(5(1(3(1(0(3(1(5(4(1(3(4(4(5(2(x1)))))))))))))))))))) -> 4(3(5(5(5(0(5(2(5(5(3(3(2(4(3(4(4(2(x1)))))))))))))))))) 5(4(4(0(5(0(4(0(3(4(4(4(0(5(3(3(3(5(2(4(x1)))))))))))))))))))) -> 3(2(4(1(5(2(4(5(5(3(5(0(3(4(1(4(3(1(2(2(2(x1))))))))))))))))))))) 0(0(2(4(0(2(0(0(1(5(4(5(5(1(3(0(1(5(1(5(4(x1))))))))))))))))))))) -> 0(2(3(2(4(0(4(4(1(5(2(2(2(0(0(1(3(4(2(5(x1)))))))))))))))))))) 1(1(1(2(5(4(2(3(5(1(3(3(2(2(1(4(0(2(5(4(1(x1))))))))))))))))))))) -> 4(3(2(0(1(1(5(1(3(2(4(3(3(4(3(4(4(3(0(2(x1)))))))))))))))))))) 2(2(2(2(2(1(2(0(0(5(3(0(1(0(2(0(1(3(1(1(2(x1))))))))))))))))))))) -> 0(3(3(4(2(2(3(0(0(3(1(2(2(4(4(4(1(5(0(x1))))))))))))))))))) 2(3(3(2(1(4(0(4(5(1(1(2(5(2(2(4(1(2(1(0(2(x1))))))))))))))))))))) -> 5(5(4(2(3(5(2(0(4(2(0(2(3(3(0(5(1(1(4(2(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: 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_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(0(1(x1)))) -> 0(1(2(x1))) 0(3(4(0(1(1(4(3(4(x1))))))))) -> 0(3(4(1(0(5(1(4(x1)))))))) 2(1(5(5(3(5(5(3(2(x1))))))))) -> 2(2(0(2(4(0(1(2(x1)))))))) 3(3(2(4(4(4(5(0(0(x1))))))))) -> 3(5(0(1(4(5(4(0(0(x1))))))))) 3(4(3(3(5(5(3(4(4(x1))))))))) -> 5(0(2(0(5(4(0(x1))))))) 1(1(0(5(4(1(1(3(5(1(x1)))))))))) -> 1(1(5(4(5(4(4(4(1(x1))))))))) 3(0(1(1(1(3(5(0(0(4(x1)))))))))) -> 3(5(3(3(0(4(3(4(0(x1))))))))) 5(0(1(4(5(4(3(0(0(1(x1)))))))))) -> 5(1(4(0(5(2(5(0(1(x1))))))))) 3(4(3(4(5(2(3(2(4(1(3(x1))))))))))) -> 2(4(4(2(2(2(3(5(5(3(x1)))))))))) 2(1(2(5(2(4(2(1(4(1(0(0(x1)))))))))))) -> 0(1(2(5(2(5(5(0(2(3(0(x1))))))))))) 2(5(4(0(3(1(2(5(1(1(4(5(x1)))))))))))) -> 1(0(3(4(4(1(3(5(1(1(4(5(x1)))))))))))) 1(5(2(1(5(2(3(3(2(1(5(1(1(x1))))))))))))) -> 1(5(0(3(2(5(3(5(2(2(0(x1))))))))))) 5(4(4(1(1(4(2(0(5(1(4(4(4(x1))))))))))))) -> 5(4(0(1(3(4(0(4(2(1(2(5(x1)))))))))))) 4(5(5(2(1(3(4(2(0(5(1(5(1(1(x1)))))))))))))) -> 4(5(2(3(2(0(5(4(3(5(4(2(x1)))))))))))) 3(0(5(2(2(0(4(5(3(4(3(5(2(5(2(x1))))))))))))))) -> 2(0(4(2(0(2(2(1(2(4(5(0(4(3(2(x1))))))))))))))) 4(2(5(0(5(4(2(5(0(3(2(1(5(5(4(x1))))))))))))))) -> 4(4(0(5(2(1(2(1(1(5(1(4(3(5(1(3(x1)))))))))))))))) 2(0(5(2(3(1(3(3(2(1(0(4(2(0(3(0(x1)))))))))))))))) -> 3(2(2(3(2(0(1(2(0(1(2(0(5(5(4(0(x1)))))))))))))))) 4(4(5(1(0(1(5(0(1(0(4(5(3(5(4(4(x1)))))))))))))))) -> 4(1(2(1(2(2(0(1(1(1(0(3(3(2(3(3(4(x1))))))))))))))))) 2(2(0(4(3(1(5(1(1(4(2(3(0(4(1(1(3(x1))))))))))))))))) -> 0(3(5(0(3(1(2(2(5(2(2(2(5(4(4(x1))))))))))))))) 5(5(1(0(5(0(5(0(5(0(3(0(2(5(5(4(1(x1))))))))))))))))) -> 2(1(1(3(1(5(0(2(0(0(4(5(5(5(5(2(5(1(x1)))))))))))))))))) 1(0(0(0(2(5(4(1(0(1(2(2(5(0(4(3(4(4(x1)))))))))))))))))) -> 1(2(2(5(1(5(2(0(2(4(3(4(3(0(4(2(0(x1))))))))))))))))) 2(2(3(0(5(3(0(4(0(3(1(1(1(5(1(3(1(0(x1)))))))))))))))))) -> 2(4(1(5(5(0(1(2(0(2(5(1(4(1(0(0(x1)))))))))))))))) 2(1(1(4(4(2(1(0(3(0(0(3(3(4(3(1(0(4(4(x1))))))))))))))))))) -> 2(1(0(3(4(1(5(4(0(1(2(1(5(2(1(1(2(3(4(x1))))))))))))))))))) 4(1(4(0(2(0(5(5(2(0(2(2(0(1(2(1(5(5(2(4(x1)))))))))))))))))))) -> 4(3(1(4(5(0(0(5(2(5(3(0(5(0(1(1(0(3(5(x1))))))))))))))))))) 4(3(3(5(3(5(1(3(1(0(3(1(5(4(1(3(4(4(5(2(x1)))))))))))))))))))) -> 4(3(5(5(5(0(5(2(5(5(3(3(2(4(3(4(4(2(x1)))))))))))))))))) 5(4(4(0(5(0(4(0(3(4(4(4(0(5(3(3(3(5(2(4(x1)))))))))))))))))))) -> 3(2(4(1(5(2(4(5(5(3(5(0(3(4(1(4(3(1(2(2(2(x1))))))))))))))))))))) 0(0(2(4(0(2(0(0(1(5(4(5(5(1(3(0(1(5(1(5(4(x1))))))))))))))))))))) -> 0(2(3(2(4(0(4(4(1(5(2(2(2(0(0(1(3(4(2(5(x1)))))))))))))))))))) 1(1(1(2(5(4(2(3(5(1(3(3(2(2(1(4(0(2(5(4(1(x1))))))))))))))))))))) -> 4(3(2(0(1(1(5(1(3(2(4(3(3(4(3(4(4(3(0(2(x1)))))))))))))))))))) 2(2(2(2(2(1(2(0(0(5(3(0(1(0(2(0(1(3(1(1(2(x1))))))))))))))))))))) -> 0(3(3(4(2(2(3(0(0(3(1(2(2(4(4(4(1(5(0(x1))))))))))))))))))) 2(3(3(2(1(4(0(4(5(1(1(2(5(2(2(4(1(2(1(0(2(x1))))))))))))))))))))) -> 5(5(4(2(3(5(2(0(4(2(0(2(3(3(0(5(1(1(4(2(x1)))))))))))))))))))) 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_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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 3. The certificate found is represented by the following graph. 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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] {(148,149,[0_1|0, 2_1|0, 3_1|0, 1_1|0, 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(422,150,[5_1|2]), (422,359,[5_1|2]), (422,460,[5_1|2]), (422,471,[5_1|2]), (422,486,[5_1|2]), (422,502,[5_1|2]), (422,520,[5_1|2]), (422,472,[5_1|2]), (422,404,[5_1|2]), (422,412,[5_1|2]), (422,423,[3_1|2]), (422,443,[2_1|2]), (423,424,[2_1|2]), (424,425,[4_1|2]), (425,426,[1_1|2]), (426,427,[5_1|2]), (427,428,[2_1|2]), (428,429,[4_1|2]), (429,430,[5_1|2]), (430,431,[5_1|2]), (431,432,[3_1|2]), (432,433,[5_1|2]), (433,434,[0_1|2]), (434,435,[3_1|2]), (435,436,[4_1|2]), (436,437,[1_1|2]), (437,438,[4_1|2]), (438,439,[3_1|2]), (439,440,[1_1|2]), (440,441,[2_1|2]), (440,269,[0_1|2]), (441,442,[2_1|2]), (441,240,[0_1|2]), (441,254,[2_1|2]), (441,269,[0_1|2]), (442,150,[2_1|2]), (442,359,[2_1|2]), (442,460,[2_1|2]), (442,471,[2_1|2]), (442,486,[2_1|2]), (442,502,[2_1|2]), (442,520,[2_1|2]), (442,255,[2_1|2]), (442,321,[2_1|2]), (442,179,[2_1|2]), (442,186,[0_1|2]), (442,196,[2_1|2]), (442,214,[1_1|2]), (442,225,[3_1|2]), (442,240,[0_1|2]), (442,254,[2_1|2]), (442,269,[0_1|2]), (442,287,[5_1|2]), (443,444,[1_1|2]), (444,445,[1_1|2]), (445,446,[3_1|2]), (446,447,[1_1|2]), (447,448,[5_1|2]), (448,449,[0_1|2]), (449,450,[2_1|2]), (450,451,[0_1|2]), (451,452,[0_1|2]), (452,453,[4_1|2]), (453,454,[5_1|2]), (454,455,[5_1|2]), (455,456,[5_1|2]), (456,457,[5_1|2]), (457,458,[2_1|2]), (458,459,[5_1|2]), (459,150,[1_1|2]), (459,214,[1_1|2]), (459,351,[1_1|2]), (459,378,[1_1|2]), (459,388,[1_1|2]), (459,487,[1_1|2]), (459,359,[4_1|2]), (460,461,[5_1|2]), (461,462,[2_1|2]), (462,463,[3_1|2]), (463,464,[2_1|2]), (464,465,[0_1|2]), (465,466,[5_1|2]), (466,467,[4_1|2]), (467,468,[3_1|2]), (468,469,[5_1|2]), (469,470,[4_1|2]), (469,471,[4_1|2]), (470,150,[2_1|2]), (470,214,[2_1|2, 1_1|2]), (470,351,[2_1|2]), (470,378,[2_1|2]), (470,388,[2_1|2]), (470,352,[2_1|2]), (470,179,[2_1|2]), (470,186,[0_1|2]), (470,196,[2_1|2]), (470,225,[3_1|2]), (470,240,[0_1|2]), (470,254,[2_1|2]), (470,269,[0_1|2]), (470,287,[5_1|2]), (471,472,[4_1|2]), (472,473,[0_1|2]), (473,474,[5_1|2]), (474,475,[2_1|2]), (475,476,[1_1|2]), (476,477,[2_1|2]), (477,478,[1_1|2]), (478,479,[1_1|2]), (479,480,[5_1|2]), (480,481,[1_1|2]), (481,482,[4_1|2]), (482,483,[3_1|2]), (483,484,[5_1|2]), (484,485,[1_1|2]), (485,150,[3_1|2]), (485,359,[3_1|2]), (485,460,[3_1|2]), (485,471,[3_1|2]), (485,486,[3_1|2]), (485,502,[3_1|2]), (485,520,[3_1|2]), (485,413,[3_1|2]), (485,289,[3_1|2]), (485,306,[3_1|2]), (485,314,[5_1|2]), (485,320,[2_1|2]), (485,329,[3_1|2]), (485,337,[2_1|2]), (486,487,[1_1|2]), (487,488,[2_1|2]), (488,489,[1_1|2]), (489,490,[2_1|2]), (490,491,[2_1|2]), (491,492,[0_1|2]), (492,493,[1_1|2]), (493,494,[1_1|2]), (494,495,[1_1|2]), (495,496,[0_1|2]), (496,497,[3_1|2]), (497,498,[3_1|2]), (498,499,[2_1|2]), (499,500,[3_1|2]), (500,501,[3_1|2]), (500,314,[5_1|2]), (500,320,[2_1|2]), (501,150,[4_1|2]), (501,359,[4_1|2]), (501,460,[4_1|2]), (501,471,[4_1|2]), (501,486,[4_1|2]), (501,502,[4_1|2]), (501,520,[4_1|2]), (501,472,[4_1|2]), (502,503,[3_1|2]), (503,504,[1_1|2]), (504,505,[4_1|2]), (505,506,[5_1|2]), (506,507,[0_1|2]), (507,508,[0_1|2]), (508,509,[5_1|2]), (509,510,[2_1|2]), (510,511,[5_1|2]), (511,512,[3_1|2]), (512,513,[0_1|2]), (513,514,[5_1|2]), (514,515,[0_1|2]), (515,516,[1_1|2]), (516,517,[1_1|2]), (517,518,[0_1|2]), (518,519,[3_1|2]), (519,150,[5_1|2]), (519,359,[5_1|2]), (519,460,[5_1|2]), (519,471,[5_1|2]), (519,486,[5_1|2]), (519,502,[5_1|2]), (519,520,[5_1|2]), (519,255,[5_1|2]), (519,321,[5_1|2]), (519,404,[5_1|2]), (519,412,[5_1|2]), (519,423,[3_1|2]), (519,443,[2_1|2]), (520,521,[3_1|2]), (521,522,[5_1|2]), (522,523,[5_1|2]), (523,524,[5_1|2]), (524,525,[0_1|2]), (525,526,[5_1|2]), (526,527,[2_1|2]), (527,528,[5_1|2]), (528,529,[5_1|2]), (529,530,[3_1|2]), (530,531,[3_1|2]), (531,532,[2_1|2]), (532,533,[4_1|2]), (533,534,[3_1|2]), (534,535,[4_1|2]), (535,536,[4_1|2]), (535,471,[4_1|2]), (536,150,[2_1|2]), (536,179,[2_1|2]), (536,196,[2_1|2]), (536,254,[2_1|2]), (536,320,[2_1|2]), (536,337,[2_1|2]), (536,443,[2_1|2]), (536,462,[2_1|2]), (536,186,[0_1|2]), (536,214,[1_1|2]), (536,225,[3_1|2]), (536,240,[0_1|2]), (536,269,[0_1|2]), (536,287,[5_1|2]), (537,538,[1_1|3]), (538,152,[2_1|3]), (538,187,[2_1|3]), (538,240,[0_1|2]), (538,254,[2_1|2]), (538,269,[0_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)