/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 76 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 69 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(0(1(0(x1)))) -> 2(3(4(x1))) 3(5(1(1(2(1(x1)))))) -> 3(3(3(1(3(1(x1)))))) 0(0(3(3(5(5(4(x1))))))) -> 2(3(3(0(3(4(x1)))))) 0(4(0(0(4(4(5(x1))))))) -> 3(5(4(1(2(4(x1)))))) 5(5(5(2(5(2(4(4(3(x1))))))))) -> 5(0(1(5(0(4(5(2(x1)))))))) 0(4(3(0(1(1(1(4(1(0(x1)))))))))) -> 2(5(0(2(2(1(0(5(1(3(x1)))))))))) 0(4(5(2(1(5(3(0(1(1(x1)))))))))) -> 5(3(0(2(3(0(5(0(0(0(x1)))))))))) 3(3(1(4(2(0(3(5(0(0(x1)))))))))) -> 0(1(1(2(5(3(1(2(2(2(x1)))))))))) 4(5(2(2(1(5(2(4(5(0(x1)))))))))) -> 4(2(5(4(1(4(5(5(5(0(x1)))))))))) 0(1(1(3(4(4(0(4(1(5(5(x1))))))))))) -> 0(2(4(4(5(5(5(3(2(4(0(x1))))))))))) 2(1(1(3(5(5(4(1(0(4(4(1(x1)))))))))))) -> 2(5(5(3(4(2(0(4(0(1(2(0(x1)))))))))))) 4(4(5(1(1(1(3(2(5(5(4(1(x1)))))))))))) -> 4(2(1(0(2(0(0(0(4(3(3(1(x1)))))))))))) 4(5(2(1(0(5(2(0(2(5(0(4(x1)))))))))))) -> 3(0(0(5(3(2(0(0(2(0(0(4(x1)))))))))))) 5(2(2(4(2(5(1(4(5(4(0(4(x1)))))))))))) -> 2(4(5(5(5(4(3(2(5(3(1(4(x1)))))))))))) 0(4(1(5(4(3(5(5(0(0(5(1(2(x1))))))))))))) -> 0(3(5(4(5(0(5(2(0(4(2(4(x1)))))))))))) 2(2(2(1(0(2(2(5(1(0(1(5(1(x1))))))))))))) -> 5(4(0(3(2(5(0(1(2(4(3(3(x1)))))))))))) 0(2(5(5(1(2(2(5(5(0(5(2(1(0(x1)))))))))))))) -> 2(0(5(5(5(0(1(2(3(4(0(1(1(x1))))))))))))) 2(4(1(3(4(1(0(0(3(2(4(4(1(2(x1)))))))))))))) -> 3(4(5(1(4(4(4(5(4(5(1(4(3(1(x1)))))))))))))) 0(3(3(2(0(2(2(3(4(0(5(1(2(5(1(2(x1)))))))))))))))) -> 0(1(0(1(3(0(1(1(2(2(4(0(3(4(4(x1))))))))))))))) 2(2(0(0(2(0(5(2(4(0(4(3(2(5(2(0(x1)))))))))))))))) -> 4(5(4(1(3(5(1(3(0(2(5(0(2(5(3(x1))))))))))))))) 1(4(4(3(4(5(3(5(1(2(5(0(4(4(4(0(4(x1))))))))))))))))) -> 1(2(3(3(1(0(1(1(1(0(3(3(3(3(5(3(4(x1))))))))))))))))) 3(0(2(0(3(2(3(3(0(4(3(0(0(2(0(2(0(x1))))))))))))))))) -> 3(5(5(3(2(1(2(2(2(0(3(0(4(5(1(1(x1)))))))))))))))) 4(0(4(3(3(5(2(1(3(5(5(3(2(4(5(1(0(x1))))))))))))))))) -> 4(4(5(5(4(2(2(0(2(1(5(2(1(5(4(5(0(x1))))))))))))))))) 1(2(4(3(4(4(2(2(1(4(2(1(3(1(5(1(5(3(x1)))))))))))))))))) -> 1(0(3(2(0(0(1(3(0(3(4(0(0(0(3(5(3(x1))))))))))))))))) 4(5(5(5(4(0(0(3(3(4(4(3(3(5(5(3(1(0(x1)))))))))))))))))) -> 3(3(2(5(1(1(5(4(1(5(1(2(5(4(1(2(1(x1))))))))))))))))) 1(1(1(3(0(3(5(0(5(1(1(4(2(2(0(3(2(3(3(x1))))))))))))))))))) -> 3(5(2(2(4(4(1(2(4(3(0(5(5(0(0(1(2(2(3(x1))))))))))))))))))) 0(2(1(3(1(2(0(3(4(2(2(5(3(0(1(3(3(5(3(2(x1)))))))))))))))))))) -> 1(3(1(0(2(5(3(1(1(3(1(5(0(1(1(0(3(0(3(x1))))))))))))))))))) 1(0(0(3(2(0(4(5(1(1(2(5(0(1(0(2(1(4(3(2(x1)))))))))))))))))))) -> 3(1(1(4(4(5(4(3(1(0(3(0(4(0(1(0(2(2(2(1(x1)))))))))))))))))))) 0(0(3(3(3(1(1(0(0(2(5(0(4(5(1(4(2(4(1(0(5(x1))))))))))))))))))))) -> 1(2(3(0(5(2(3(0(3(5(4(1(4(4(5(0(1(1(1(5(x1)))))))))))))))))))) 0(5(3(1(0(2(1(4(5(1(1(0(1(1(3(2(5(4(3(0(4(x1))))))))))))))))))))) -> 1(1(5(5(4(2(3(5(3(1(0(0(5(3(5(5(4(1(3(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(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(1, n^1). The TRS R consists of the following rules: 0(0(1(0(x1)))) -> 2(3(4(x1))) 3(5(1(1(2(1(x1)))))) -> 3(3(3(1(3(1(x1)))))) 0(0(3(3(5(5(4(x1))))))) -> 2(3(3(0(3(4(x1)))))) 0(4(0(0(4(4(5(x1))))))) -> 3(5(4(1(2(4(x1)))))) 5(5(5(2(5(2(4(4(3(x1))))))))) -> 5(0(1(5(0(4(5(2(x1)))))))) 0(4(3(0(1(1(1(4(1(0(x1)))))))))) -> 2(5(0(2(2(1(0(5(1(3(x1)))))))))) 0(4(5(2(1(5(3(0(1(1(x1)))))))))) -> 5(3(0(2(3(0(5(0(0(0(x1)))))))))) 3(3(1(4(2(0(3(5(0(0(x1)))))))))) -> 0(1(1(2(5(3(1(2(2(2(x1)))))))))) 4(5(2(2(1(5(2(4(5(0(x1)))))))))) -> 4(2(5(4(1(4(5(5(5(0(x1)))))))))) 0(1(1(3(4(4(0(4(1(5(5(x1))))))))))) -> 0(2(4(4(5(5(5(3(2(4(0(x1))))))))))) 2(1(1(3(5(5(4(1(0(4(4(1(x1)))))))))))) -> 2(5(5(3(4(2(0(4(0(1(2(0(x1)))))))))))) 4(4(5(1(1(1(3(2(5(5(4(1(x1)))))))))))) -> 4(2(1(0(2(0(0(0(4(3(3(1(x1)))))))))))) 4(5(2(1(0(5(2(0(2(5(0(4(x1)))))))))))) -> 3(0(0(5(3(2(0(0(2(0(0(4(x1)))))))))))) 5(2(2(4(2(5(1(4(5(4(0(4(x1)))))))))))) -> 2(4(5(5(5(4(3(2(5(3(1(4(x1)))))))))))) 0(4(1(5(4(3(5(5(0(0(5(1(2(x1))))))))))))) -> 0(3(5(4(5(0(5(2(0(4(2(4(x1)))))))))))) 2(2(2(1(0(2(2(5(1(0(1(5(1(x1))))))))))))) -> 5(4(0(3(2(5(0(1(2(4(3(3(x1)))))))))))) 0(2(5(5(1(2(2(5(5(0(5(2(1(0(x1)))))))))))))) -> 2(0(5(5(5(0(1(2(3(4(0(1(1(x1))))))))))))) 2(4(1(3(4(1(0(0(3(2(4(4(1(2(x1)))))))))))))) -> 3(4(5(1(4(4(4(5(4(5(1(4(3(1(x1)))))))))))))) 0(3(3(2(0(2(2(3(4(0(5(1(2(5(1(2(x1)))))))))))))))) -> 0(1(0(1(3(0(1(1(2(2(4(0(3(4(4(x1))))))))))))))) 2(2(0(0(2(0(5(2(4(0(4(3(2(5(2(0(x1)))))))))))))))) -> 4(5(4(1(3(5(1(3(0(2(5(0(2(5(3(x1))))))))))))))) 1(4(4(3(4(5(3(5(1(2(5(0(4(4(4(0(4(x1))))))))))))))))) -> 1(2(3(3(1(0(1(1(1(0(3(3(3(3(5(3(4(x1))))))))))))))))) 3(0(2(0(3(2(3(3(0(4(3(0(0(2(0(2(0(x1))))))))))))))))) -> 3(5(5(3(2(1(2(2(2(0(3(0(4(5(1(1(x1)))))))))))))))) 4(0(4(3(3(5(2(1(3(5(5(3(2(4(5(1(0(x1))))))))))))))))) -> 4(4(5(5(4(2(2(0(2(1(5(2(1(5(4(5(0(x1))))))))))))))))) 1(2(4(3(4(4(2(2(1(4(2(1(3(1(5(1(5(3(x1)))))))))))))))))) -> 1(0(3(2(0(0(1(3(0(3(4(0(0(0(3(5(3(x1))))))))))))))))) 4(5(5(5(4(0(0(3(3(4(4(3(3(5(5(3(1(0(x1)))))))))))))))))) -> 3(3(2(5(1(1(5(4(1(5(1(2(5(4(1(2(1(x1))))))))))))))))) 1(1(1(3(0(3(5(0(5(1(1(4(2(2(0(3(2(3(3(x1))))))))))))))))))) -> 3(5(2(2(4(4(1(2(4(3(0(5(5(0(0(1(2(2(3(x1))))))))))))))))))) 0(2(1(3(1(2(0(3(4(2(2(5(3(0(1(3(3(5(3(2(x1)))))))))))))))))))) -> 1(3(1(0(2(5(3(1(1(3(1(5(0(1(1(0(3(0(3(x1))))))))))))))))))) 1(0(0(3(2(0(4(5(1(1(2(5(0(1(0(2(1(4(3(2(x1)))))))))))))))))))) -> 3(1(1(4(4(5(4(3(1(0(3(0(4(0(1(0(2(2(2(1(x1)))))))))))))))))))) 0(0(3(3(3(1(1(0(0(2(5(0(4(5(1(4(2(4(1(0(5(x1))))))))))))))))))))) -> 1(2(3(0(5(2(3(0(3(5(4(1(4(4(5(0(1(1(1(5(x1)))))))))))))))))))) 0(5(3(1(0(2(1(4(5(1(1(0(1(1(3(2(5(4(3(0(4(x1))))))))))))))))))))) -> 1(1(5(5(4(2(3(5(3(1(0(0(5(3(5(5(4(1(3(4(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(1, n^1). The TRS R consists of the following rules: 0(0(1(0(x1)))) -> 2(3(4(x1))) 3(5(1(1(2(1(x1)))))) -> 3(3(3(1(3(1(x1)))))) 0(0(3(3(5(5(4(x1))))))) -> 2(3(3(0(3(4(x1)))))) 0(4(0(0(4(4(5(x1))))))) -> 3(5(4(1(2(4(x1)))))) 5(5(5(2(5(2(4(4(3(x1))))))))) -> 5(0(1(5(0(4(5(2(x1)))))))) 0(4(3(0(1(1(1(4(1(0(x1)))))))))) -> 2(5(0(2(2(1(0(5(1(3(x1)))))))))) 0(4(5(2(1(5(3(0(1(1(x1)))))))))) -> 5(3(0(2(3(0(5(0(0(0(x1)))))))))) 3(3(1(4(2(0(3(5(0(0(x1)))))))))) -> 0(1(1(2(5(3(1(2(2(2(x1)))))))))) 4(5(2(2(1(5(2(4(5(0(x1)))))))))) -> 4(2(5(4(1(4(5(5(5(0(x1)))))))))) 0(1(1(3(4(4(0(4(1(5(5(x1))))))))))) -> 0(2(4(4(5(5(5(3(2(4(0(x1))))))))))) 2(1(1(3(5(5(4(1(0(4(4(1(x1)))))))))))) -> 2(5(5(3(4(2(0(4(0(1(2(0(x1)))))))))))) 4(4(5(1(1(1(3(2(5(5(4(1(x1)))))))))))) -> 4(2(1(0(2(0(0(0(4(3(3(1(x1)))))))))))) 4(5(2(1(0(5(2(0(2(5(0(4(x1)))))))))))) -> 3(0(0(5(3(2(0(0(2(0(0(4(x1)))))))))))) 5(2(2(4(2(5(1(4(5(4(0(4(x1)))))))))))) -> 2(4(5(5(5(4(3(2(5(3(1(4(x1)))))))))))) 0(4(1(5(4(3(5(5(0(0(5(1(2(x1))))))))))))) -> 0(3(5(4(5(0(5(2(0(4(2(4(x1)))))))))))) 2(2(2(1(0(2(2(5(1(0(1(5(1(x1))))))))))))) -> 5(4(0(3(2(5(0(1(2(4(3(3(x1)))))))))))) 0(2(5(5(1(2(2(5(5(0(5(2(1(0(x1)))))))))))))) -> 2(0(5(5(5(0(1(2(3(4(0(1(1(x1))))))))))))) 2(4(1(3(4(1(0(0(3(2(4(4(1(2(x1)))))))))))))) -> 3(4(5(1(4(4(4(5(4(5(1(4(3(1(x1)))))))))))))) 0(3(3(2(0(2(2(3(4(0(5(1(2(5(1(2(x1)))))))))))))))) -> 0(1(0(1(3(0(1(1(2(2(4(0(3(4(4(x1))))))))))))))) 2(2(0(0(2(0(5(2(4(0(4(3(2(5(2(0(x1)))))))))))))))) -> 4(5(4(1(3(5(1(3(0(2(5(0(2(5(3(x1))))))))))))))) 1(4(4(3(4(5(3(5(1(2(5(0(4(4(4(0(4(x1))))))))))))))))) -> 1(2(3(3(1(0(1(1(1(0(3(3(3(3(5(3(4(x1))))))))))))))))) 3(0(2(0(3(2(3(3(0(4(3(0(0(2(0(2(0(x1))))))))))))))))) -> 3(5(5(3(2(1(2(2(2(0(3(0(4(5(1(1(x1)))))))))))))))) 4(0(4(3(3(5(2(1(3(5(5(3(2(4(5(1(0(x1))))))))))))))))) -> 4(4(5(5(4(2(2(0(2(1(5(2(1(5(4(5(0(x1))))))))))))))))) 1(2(4(3(4(4(2(2(1(4(2(1(3(1(5(1(5(3(x1)))))))))))))))))) -> 1(0(3(2(0(0(1(3(0(3(4(0(0(0(3(5(3(x1))))))))))))))))) 4(5(5(5(4(0(0(3(3(4(4(3(3(5(5(3(1(0(x1)))))))))))))))))) -> 3(3(2(5(1(1(5(4(1(5(1(2(5(4(1(2(1(x1))))))))))))))))) 1(1(1(3(0(3(5(0(5(1(1(4(2(2(0(3(2(3(3(x1))))))))))))))))))) -> 3(5(2(2(4(4(1(2(4(3(0(5(5(0(0(1(2(2(3(x1))))))))))))))))))) 0(2(1(3(1(2(0(3(4(2(2(5(3(0(1(3(3(5(3(2(x1)))))))))))))))))))) -> 1(3(1(0(2(5(3(1(1(3(1(5(0(1(1(0(3(0(3(x1))))))))))))))))))) 1(0(0(3(2(0(4(5(1(1(2(5(0(1(0(2(1(4(3(2(x1)))))))))))))))))))) -> 3(1(1(4(4(5(4(3(1(0(3(0(4(0(1(0(2(2(2(1(x1)))))))))))))))))))) 0(0(3(3(3(1(1(0(0(2(5(0(4(5(1(4(2(4(1(0(5(x1))))))))))))))))))))) -> 1(2(3(0(5(2(3(0(3(5(4(1(4(4(5(0(1(1(1(5(x1)))))))))))))))))))) 0(5(3(1(0(2(1(4(5(1(1(0(1(1(3(2(5(4(3(0(4(x1))))))))))))))))))))) -> 1(1(5(5(4(2(3(5(3(1(0(0(5(3(5(5(4(1(3(4(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(0(1(0(x1)))) -> 2(3(4(x1))) 3(5(1(1(2(1(x1)))))) -> 3(3(3(1(3(1(x1)))))) 0(0(3(3(5(5(4(x1))))))) -> 2(3(3(0(3(4(x1)))))) 0(4(0(0(4(4(5(x1))))))) -> 3(5(4(1(2(4(x1)))))) 5(5(5(2(5(2(4(4(3(x1))))))))) -> 5(0(1(5(0(4(5(2(x1)))))))) 0(4(3(0(1(1(1(4(1(0(x1)))))))))) -> 2(5(0(2(2(1(0(5(1(3(x1)))))))))) 0(4(5(2(1(5(3(0(1(1(x1)))))))))) -> 5(3(0(2(3(0(5(0(0(0(x1)))))))))) 3(3(1(4(2(0(3(5(0(0(x1)))))))))) -> 0(1(1(2(5(3(1(2(2(2(x1)))))))))) 4(5(2(2(1(5(2(4(5(0(x1)))))))))) -> 4(2(5(4(1(4(5(5(5(0(x1)))))))))) 0(1(1(3(4(4(0(4(1(5(5(x1))))))))))) -> 0(2(4(4(5(5(5(3(2(4(0(x1))))))))))) 2(1(1(3(5(5(4(1(0(4(4(1(x1)))))))))))) -> 2(5(5(3(4(2(0(4(0(1(2(0(x1)))))))))))) 4(4(5(1(1(1(3(2(5(5(4(1(x1)))))))))))) -> 4(2(1(0(2(0(0(0(4(3(3(1(x1)))))))))))) 4(5(2(1(0(5(2(0(2(5(0(4(x1)))))))))))) -> 3(0(0(5(3(2(0(0(2(0(0(4(x1)))))))))))) 5(2(2(4(2(5(1(4(5(4(0(4(x1)))))))))))) -> 2(4(5(5(5(4(3(2(5(3(1(4(x1)))))))))))) 0(4(1(5(4(3(5(5(0(0(5(1(2(x1))))))))))))) -> 0(3(5(4(5(0(5(2(0(4(2(4(x1)))))))))))) 2(2(2(1(0(2(2(5(1(0(1(5(1(x1))))))))))))) -> 5(4(0(3(2(5(0(1(2(4(3(3(x1)))))))))))) 0(2(5(5(1(2(2(5(5(0(5(2(1(0(x1)))))))))))))) -> 2(0(5(5(5(0(1(2(3(4(0(1(1(x1))))))))))))) 2(4(1(3(4(1(0(0(3(2(4(4(1(2(x1)))))))))))))) -> 3(4(5(1(4(4(4(5(4(5(1(4(3(1(x1)))))))))))))) 0(3(3(2(0(2(2(3(4(0(5(1(2(5(1(2(x1)))))))))))))))) -> 0(1(0(1(3(0(1(1(2(2(4(0(3(4(4(x1))))))))))))))) 2(2(0(0(2(0(5(2(4(0(4(3(2(5(2(0(x1)))))))))))))))) -> 4(5(4(1(3(5(1(3(0(2(5(0(2(5(3(x1))))))))))))))) 1(4(4(3(4(5(3(5(1(2(5(0(4(4(4(0(4(x1))))))))))))))))) -> 1(2(3(3(1(0(1(1(1(0(3(3(3(3(5(3(4(x1))))))))))))))))) 3(0(2(0(3(2(3(3(0(4(3(0(0(2(0(2(0(x1))))))))))))))))) -> 3(5(5(3(2(1(2(2(2(0(3(0(4(5(1(1(x1)))))))))))))))) 4(0(4(3(3(5(2(1(3(5(5(3(2(4(5(1(0(x1))))))))))))))))) -> 4(4(5(5(4(2(2(0(2(1(5(2(1(5(4(5(0(x1))))))))))))))))) 1(2(4(3(4(4(2(2(1(4(2(1(3(1(5(1(5(3(x1)))))))))))))))))) -> 1(0(3(2(0(0(1(3(0(3(4(0(0(0(3(5(3(x1))))))))))))))))) 4(5(5(5(4(0(0(3(3(4(4(3(3(5(5(3(1(0(x1)))))))))))))))))) -> 3(3(2(5(1(1(5(4(1(5(1(2(5(4(1(2(1(x1))))))))))))))))) 1(1(1(3(0(3(5(0(5(1(1(4(2(2(0(3(2(3(3(x1))))))))))))))))))) -> 3(5(2(2(4(4(1(2(4(3(0(5(5(0(0(1(2(2(3(x1))))))))))))))))))) 0(2(1(3(1(2(0(3(4(2(2(5(3(0(1(3(3(5(3(2(x1)))))))))))))))))))) -> 1(3(1(0(2(5(3(1(1(3(1(5(0(1(1(0(3(0(3(x1))))))))))))))))))) 1(0(0(3(2(0(4(5(1(1(2(5(0(1(0(2(1(4(3(2(x1)))))))))))))))))))) -> 3(1(1(4(4(5(4(3(1(0(3(0(4(0(1(0(2(2(2(1(x1)))))))))))))))))))) 0(0(3(3(3(1(1(0(0(2(5(0(4(5(1(4(2(4(1(0(5(x1))))))))))))))))))))) -> 1(2(3(0(5(2(3(0(3(5(4(1(4(4(5(0(1(1(1(5(x1)))))))))))))))))))) 0(5(3(1(0(2(1(4(5(1(1(0(1(1(3(2(5(4(3(0(4(x1))))))))))))))))))))) -> 1(1(5(5(4(2(3(5(3(1(0(0(5(3(5(5(4(1(3(4(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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. "[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] {(99,100,[0_1|0, 3_1|0, 5_1|0, 4_1|0, 2_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]), (99,101,[0_1|1, 3_1|1, 5_1|1, 4_1|1, 2_1|1, 1_1|1]), (99,102,[2_1|2]), (99,104,[2_1|2]), (99,109,[1_1|2]), (99,128,[3_1|2]), (99,133,[2_1|2]), (99,142,[5_1|2]), (99,151,[0_1|2]), (99,162,[0_1|2]), (99,172,[2_1|2]), (99,184,[1_1|2]), (99,202,[0_1|2]), (99,216,[1_1|2]), (99,235,[3_1|2]), (99,240,[0_1|2]), (99,249,[3_1|2]), (99,264,[5_1|2]), (99,271,[2_1|2]), (99,282,[4_1|2]), (99,291,[3_1|2]), (99,302,[3_1|2]), (99,318,[4_1|2]), (99,329,[4_1|2]), (99,345,[2_1|2]), (99,356,[5_1|2]), (99,367,[4_1|2]), (99,381,[3_1|2]), (99,394,[1_1|2]), (99,410,[1_1|2]), (99,426,[3_1|2]), (99,444,[3_1|2]), (100,100,[cons_0_1|0, cons_3_1|0, cons_5_1|0, cons_4_1|0, cons_2_1|0, cons_1_1|0]), (101,100,[encArg_1|1]), (101,101,[0_1|1, 3_1|1, 5_1|1, 4_1|1, 2_1|1, 1_1|1]), (101,102,[2_1|2]), (101,104,[2_1|2]), (101,109,[1_1|2]), (101,128,[3_1|2]), (101,133,[2_1|2]), (101,142,[5_1|2]), (101,151,[0_1|2]), (101,162,[0_1|2]), (101,172,[2_1|2]), (101,184,[1_1|2]), (101,202,[0_1|2]), (101,216,[1_1|2]), 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(188,189,[5_1|2]), (189,190,[3_1|2]), (190,191,[1_1|2]), (191,192,[1_1|2]), (192,193,[3_1|2]), (193,194,[1_1|2]), (194,195,[5_1|2]), (195,196,[0_1|2]), (196,197,[1_1|2]), (197,198,[1_1|2]), (198,199,[0_1|2]), (199,200,[3_1|2]), (200,201,[0_1|2]), (200,202,[0_1|2]), (201,101,[3_1|2]), (201,102,[3_1|2]), (201,104,[3_1|2]), (201,133,[3_1|2]), (201,172,[3_1|2]), (201,271,[3_1|2]), (201,345,[3_1|2]), (201,235,[3_1|2]), (201,240,[0_1|2]), (201,249,[3_1|2]), (202,203,[1_1|2]), (203,204,[0_1|2]), (204,205,[1_1|2]), (205,206,[3_1|2]), (206,207,[0_1|2]), (207,208,[1_1|2]), (208,209,[1_1|2]), (209,210,[2_1|2]), (210,211,[2_1|2]), (211,212,[4_1|2]), (212,213,[0_1|2]), (213,214,[3_1|2]), (214,215,[4_1|2]), (214,318,[4_1|2]), (215,101,[4_1|2]), (215,102,[4_1|2]), (215,104,[4_1|2]), (215,133,[4_1|2]), (215,172,[4_1|2]), (215,271,[4_1|2]), (215,345,[4_1|2]), (215,110,[4_1|2]), (215,395,[4_1|2]), (215,282,[4_1|2]), (215,291,[3_1|2]), (215,302,[3_1|2]), (215,318,[4_1|2]), (215,329,[4_1|2]), (216,217,[1_1|2]), (217,218,[5_1|2]), (218,219,[5_1|2]), (219,220,[4_1|2]), (220,221,[2_1|2]), (221,222,[3_1|2]), (222,223,[5_1|2]), (223,224,[3_1|2]), (224,225,[1_1|2]), (225,226,[0_1|2]), (226,227,[0_1|2]), (227,228,[5_1|2]), (228,229,[3_1|2]), (229,230,[5_1|2]), (230,231,[5_1|2]), (231,232,[4_1|2]), (232,233,[1_1|2]), (233,234,[3_1|2]), (234,101,[4_1|2]), (234,282,[4_1|2]), (234,318,[4_1|2]), (234,329,[4_1|2]), (234,367,[4_1|2]), (234,291,[3_1|2]), (234,302,[3_1|2]), (235,236,[3_1|2]), (236,237,[3_1|2]), (237,238,[1_1|2]), (238,239,[3_1|2]), (239,101,[1_1|2]), (239,109,[1_1|2]), (239,184,[1_1|2]), (239,216,[1_1|2]), (239,394,[1_1|2]), (239,410,[1_1|2]), (239,426,[3_1|2]), (239,444,[3_1|2]), (240,241,[1_1|2]), (241,242,[1_1|2]), (242,243,[2_1|2]), (243,244,[5_1|2]), (244,245,[3_1|2]), (245,246,[1_1|2]), (246,247,[2_1|2]), (246,356,[5_1|2]), (247,248,[2_1|2]), (247,356,[5_1|2]), (247,367,[4_1|2]), (248,101,[2_1|2]), (248,151,[2_1|2]), (248,162,[2_1|2]), (248,202,[2_1|2]), 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(328,394,[1_1|2]), (328,410,[1_1|2]), (328,426,[3_1|2]), (328,444,[3_1|2]), (329,330,[4_1|2]), (330,331,[5_1|2]), (331,332,[5_1|2]), (332,333,[4_1|2]), (333,334,[2_1|2]), (334,335,[2_1|2]), (335,336,[0_1|2]), (336,337,[2_1|2]), (337,338,[1_1|2]), (338,339,[5_1|2]), (339,340,[2_1|2]), (340,341,[1_1|2]), (341,342,[5_1|2]), (342,343,[4_1|2]), (343,344,[5_1|2]), (344,101,[0_1|2]), (344,151,[0_1|2]), (344,162,[0_1|2]), (344,202,[0_1|2]), (344,240,[0_1|2]), (344,411,[0_1|2]), (344,102,[2_1|2]), (344,104,[2_1|2]), (344,109,[1_1|2]), (344,128,[3_1|2]), (344,133,[2_1|2]), (344,142,[5_1|2]), (344,172,[2_1|2]), (344,184,[1_1|2]), (344,216,[1_1|2]), (344,465,[2_1|3]), (345,346,[5_1|2]), (346,347,[5_1|2]), (347,348,[3_1|2]), (348,349,[4_1|2]), (349,350,[2_1|2]), (350,351,[0_1|2]), (351,352,[4_1|2]), (352,353,[0_1|2]), (353,354,[1_1|2]), (354,355,[2_1|2]), (355,101,[0_1|2]), (355,109,[0_1|2, 1_1|2]), (355,184,[0_1|2, 1_1|2]), (355,216,[0_1|2, 1_1|2]), (355,394,[0_1|2]), (355,410,[0_1|2]), (355,102,[2_1|2]), (355,104,[2_1|2]), (355,128,[3_1|2]), (355,133,[2_1|2]), (355,142,[5_1|2]), (355,151,[0_1|2]), (355,162,[0_1|2]), (355,172,[2_1|2]), (355,202,[0_1|2]), (355,465,[2_1|3]), (356,357,[4_1|2]), (357,358,[0_1|2]), (358,359,[3_1|2]), (359,360,[2_1|2]), (360,361,[5_1|2]), (361,362,[0_1|2]), (362,363,[1_1|2]), (363,364,[2_1|2]), (364,365,[4_1|2]), (365,366,[3_1|2]), (365,240,[0_1|2]), (366,101,[3_1|2]), (366,109,[3_1|2]), (366,184,[3_1|2]), (366,216,[3_1|2]), (366,394,[3_1|2]), (366,410,[3_1|2]), (366,235,[3_1|2]), (366,240,[0_1|2]), (366,249,[3_1|2]), (367,368,[5_1|2]), (368,369,[4_1|2]), (369,370,[1_1|2]), (370,371,[3_1|2]), (371,372,[5_1|2]), (372,373,[1_1|2]), (373,374,[3_1|2]), (374,375,[0_1|2]), (375,376,[2_1|2]), (376,377,[5_1|2]), (377,378,[0_1|2]), (378,379,[2_1|2]), (379,380,[5_1|2]), (380,101,[3_1|2]), (380,151,[3_1|2]), (380,162,[3_1|2]), (380,202,[3_1|2]), (380,240,[3_1|2, 0_1|2]), (380,173,[3_1|2]), (380,235,[3_1|2]), (380,249,[3_1|2]), (381,382,[4_1|2]), (382,383,[5_1|2]), (383,384,[1_1|2]), (384,385,[4_1|2]), (385,386,[4_1|2]), (386,387,[4_1|2]), (387,388,[5_1|2]), (388,389,[4_1|2]), (389,390,[5_1|2]), (390,391,[1_1|2]), (391,392,[4_1|2]), (392,393,[3_1|2]), (393,101,[1_1|2]), (393,102,[1_1|2]), (393,104,[1_1|2]), (393,133,[1_1|2]), (393,172,[1_1|2]), (393,271,[1_1|2]), (393,345,[1_1|2]), (393,110,[1_1|2]), (393,395,[1_1|2]), (393,394,[1_1|2]), (393,410,[1_1|2]), (393,426,[3_1|2]), (393,444,[3_1|2]), (394,395,[2_1|2]), (395,396,[3_1|2]), (396,397,[3_1|2]), (397,398,[1_1|2]), (398,399,[0_1|2]), (399,400,[1_1|2]), (400,401,[1_1|2]), (401,402,[1_1|2]), (402,403,[0_1|2]), (403,404,[3_1|2]), (404,405,[3_1|2]), (405,406,[3_1|2]), (406,407,[3_1|2]), (407,408,[5_1|2]), (408,409,[3_1|2]), (409,101,[4_1|2]), (409,282,[4_1|2]), (409,318,[4_1|2]), (409,329,[4_1|2]), (409,367,[4_1|2]), (409,291,[3_1|2]), (409,302,[3_1|2]), (410,411,[0_1|2]), (411,412,[3_1|2]), (412,413,[2_1|2]), (413,414,[0_1|2]), (414,415,[0_1|2]), (415,416,[1_1|2]), (416,417,[3_1|2]), (417,418,[0_1|2]), (418,419,[3_1|2]), (419,420,[4_1|2]), (420,421,[0_1|2]), (421,422,[0_1|2]), (422,423,[0_1|2]), (423,424,[3_1|2]), (424,425,[5_1|2]), (425,101,[3_1|2]), (425,128,[3_1|2]), (425,235,[3_1|2]), (425,249,[3_1|2]), (425,291,[3_1|2]), (425,302,[3_1|2]), (425,381,[3_1|2]), (425,426,[3_1|2]), (425,444,[3_1|2]), (425,143,[3_1|2]), (425,240,[0_1|2]), (426,427,[5_1|2]), (427,428,[2_1|2]), (428,429,[2_1|2]), (429,430,[4_1|2]), (430,431,[4_1|2]), (431,432,[1_1|2]), (432,433,[2_1|2]), (433,434,[4_1|2]), (434,435,[3_1|2]), (435,436,[0_1|2]), (436,437,[5_1|2]), (437,438,[5_1|2]), (438,439,[0_1|2]), (439,440,[0_1|2]), (440,441,[1_1|2]), (441,442,[2_1|2]), (442,443,[2_1|2]), (443,101,[3_1|2]), (443,128,[3_1|2]), (443,235,[3_1|2]), (443,249,[3_1|2]), (443,291,[3_1|2]), (443,302,[3_1|2]), (443,381,[3_1|2]), (443,426,[3_1|2]), (443,444,[3_1|2]), (443,236,[3_1|2]), (443,303,[3_1|2]), (443,106,[3_1|2]), (443,240,[0_1|2]), (444,445,[1_1|2]), (445,446,[1_1|2]), (446,447,[4_1|2]), (447,448,[4_1|2]), (448,449,[5_1|2]), (449,450,[4_1|2]), (450,451,[3_1|2]), (451,452,[1_1|2]), (452,453,[0_1|2]), (453,454,[3_1|2]), (454,455,[0_1|2]), (455,456,[4_1|2]), (456,457,[0_1|2]), (457,458,[1_1|2]), (458,459,[0_1|2]), (459,460,[2_1|2]), (459,356,[5_1|2]), (460,461,[2_1|2]), (461,462,[2_1|2]), (461,345,[2_1|2]), (462,101,[1_1|2]), (462,102,[1_1|2]), (462,104,[1_1|2]), (462,133,[1_1|2]), (462,172,[1_1|2]), (462,271,[1_1|2]), (462,345,[1_1|2]), (462,394,[1_1|2]), (462,410,[1_1|2]), (462,426,[3_1|2]), (462,444,[3_1|2]), (463,464,[3_1|3]), (464,411,[4_1|3]), (464,204,[4_1|3]), (465,466,[3_1|3]), (466,204,[4_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)