/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 43 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 97 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(2(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: 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(2(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: 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(2(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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: 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. "[98, 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, 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, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503] {(98,99,[0_1|0, 2_1|0, 3_1|0, 4_1|0, 1_1|0, 5_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]), (98,100,[0_1|1, 2_1|1, 3_1|1, 4_1|1, 1_1|1, 5_1|1]), (98,101,[3_1|2]), (98,103,[3_1|2]), (98,108,[1_1|2]), (98,127,[3_1|2]), (98,134,[5_1|2]), (98,145,[5_1|2]), (98,151,[3_1|2]), (98,158,[5_1|2]), (98,169,[1_1|2]), (98,184,[5_1|2]), (98,201,[2_1|2]), (98,221,[2_1|2]), (98,225,[2_1|2]), (98,232,[2_1|2]), (98,249,[2_1|2]), (98,261,[2_1|2]), (98,275,[2_1|2]), (98,290,[2_1|2]), (98,309,[1_1|2]), (98,312,[3_1|2]), (98,332,[1_1|2]), (98,338,[1_1|2]), (98,346,[4_1|2]), (98,354,[4_1|2]), (98,367,[4_1|2]), (98,384,[1_1|2]), (98,393,[1_1|2]), (98,418,[3_1|2]), (98,429,[1_1|2]), (98,439,[3_1|2]), (99,99,[cons_0_1|0, cons_2_1|0, cons_3_1|0, cons_4_1|0, cons_1_1|0, cons_5_1|0]), (100,99,[encArg_1|1]), (100,100,[0_1|1, 2_1|1, 3_1|1, 4_1|1, 1_1|1, 5_1|1]), (100,101,[3_1|2]), (100,103,[3_1|2]), (100,108,[1_1|2]), (100,127,[3_1|2]), (100,134,[5_1|2]), (100,145,[5_1|2]), (100,151,[3_1|2]), (100,158,[5_1|2]), (100,169,[1_1|2]), (100,184,[5_1|2]), (100,201,[2_1|2]), (100,221,[2_1|2]), (100,225,[2_1|2]), (100,232,[2_1|2]), (100,249,[2_1|2]), (100,261,[2_1|2]), 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(158,159,[5_1|2]), (159,160,[1_1|2]), (160,161,[3_1|2]), (161,162,[3_1|2]), (162,163,[3_1|2]), (163,164,[3_1|2]), (164,165,[1_1|2]), (165,166,[5_1|2]), (166,167,[5_1|2]), (166,418,[3_1|2]), (167,168,[1_1|2]), (168,100,[3_1|2]), (168,201,[3_1|2]), (168,221,[3_1|2]), (168,225,[3_1|2]), (168,232,[3_1|2]), (168,249,[3_1|2]), (168,261,[3_1|2]), (168,275,[3_1|2]), (168,290,[3_1|2]), (168,102,[3_1|2]), (168,309,[1_1|2]), (168,312,[3_1|2]), (168,332,[1_1|2]), (168,338,[1_1|2]), (169,170,[4_1|2]), (170,171,[2_1|2]), (171,172,[2_1|2]), (172,173,[4_1|2]), (173,174,[4_1|2]), (174,175,[0_1|2]), (175,176,[0_1|2]), (176,177,[0_1|2]), (177,178,[4_1|2]), (178,179,[0_1|2]), (179,180,[3_1|2]), (180,181,[1_1|2]), (181,182,[5_1|2]), (182,183,[5_1|2]), (182,418,[3_1|2]), (183,100,[1_1|2]), (183,201,[1_1|2]), (183,221,[1_1|2]), (183,225,[1_1|2]), (183,232,[1_1|2]), (183,249,[1_1|2]), (183,261,[1_1|2]), (183,275,[1_1|2]), (183,290,[1_1|2]), (183,333,[1_1|2]), (183,339,[1_1|2]), (183,430,[1_1|2]), (183,384,[1_1|2]), (183,393,[1_1|2]), (184,185,[2_1|2]), (185,186,[1_1|2]), (186,187,[0_1|2]), (187,188,[4_1|2]), (188,189,[4_1|2]), (189,190,[0_1|2]), (190,191,[1_1|2]), (191,192,[0_1|2]), (191,491,[3_1|3]), (192,193,[1_1|2]), (193,194,[2_1|2]), (194,195,[2_1|2]), (195,196,[5_1|2]), (196,197,[1_1|2]), (197,198,[2_1|2]), (198,199,[5_1|2]), (199,200,[4_1|2]), (200,100,[5_1|2]), (200,201,[5_1|2]), (200,221,[5_1|2]), (200,225,[5_1|2]), (200,232,[5_1|2]), (200,249,[5_1|2]), (200,261,[5_1|2]), (200,275,[5_1|2]), (200,290,[5_1|2]), (200,136,[5_1|2]), (200,418,[3_1|2]), (200,429,[1_1|2]), (200,439,[3_1|2]), (201,202,[1_1|2]), (202,203,[4_1|2]), (203,204,[0_1|2]), (204,205,[1_1|2]), (205,206,[0_1|2]), (206,207,[2_1|2]), (207,208,[0_1|2]), (208,209,[3_1|2]), (209,210,[3_1|2]), (210,211,[4_1|2]), (211,212,[0_1|2]), (212,213,[3_1|2]), (213,214,[5_1|2]), (214,215,[4_1|2]), (215,216,[3_1|2]), (216,217,[2_1|2]), (217,218,[5_1|2]), (218,219,[4_1|2]), (219,220,[1_1|2]), (220,100,[3_1|2]), (220,101,[3_1|2]), (220,103,[3_1|2]), (220,127,[3_1|2]), (220,151,[3_1|2]), (220,312,[3_1|2]), (220,418,[3_1|2]), (220,439,[3_1|2]), (220,309,[1_1|2]), (220,332,[1_1|2]), (220,338,[1_1|2]), (221,222,[2_1|2]), (222,223,[2_1|2]), (223,224,[4_1|2]), (224,100,[4_1|2]), (224,201,[4_1|2]), (224,221,[4_1|2]), (224,225,[4_1|2]), (224,232,[4_1|2]), (224,249,[4_1|2]), (224,261,[4_1|2]), (224,275,[4_1|2]), (224,290,[4_1|2]), (224,185,[4_1|2]), (224,346,[4_1|2]), (224,354,[4_1|2]), (224,367,[4_1|2]), (225,226,[1_1|2]), (226,227,[2_1|2]), (227,228,[0_1|2]), (228,229,[4_1|2]), (229,230,[5_1|2]), (230,231,[3_1|2]), (230,309,[1_1|2]), (230,312,[3_1|2]), (231,100,[0_1|2]), (231,101,[3_1|2]), (231,103,[3_1|2]), (231,108,[1_1|2]), (231,127,[3_1|2]), (231,134,[5_1|2]), (231,145,[5_1|2]), (231,151,[3_1|2]), (231,158,[5_1|2]), (231,169,[1_1|2]), (231,184,[5_1|2]), (231,201,[2_1|2]), (232,233,[5_1|2]), (233,234,[4_1|2]), (234,235,[0_1|2]), (235,236,[3_1|2]), (236,237,[5_1|2]), (237,238,[2_1|2]), (238,239,[4_1|2]), (239,240,[2_1|2]), (240,241,[3_1|2]), (241,242,[0_1|2]), (242,243,[2_1|2]), (243,244,[2_1|2]), (244,245,[0_1|2]), (245,246,[1_1|2]), (246,247,[1_1|2]), (247,248,[4_1|2]), (248,100,[1_1|2]), (248,201,[1_1|2]), (248,221,[1_1|2]), (248,225,[1_1|2]), (248,232,[1_1|2]), (248,249,[1_1|2]), (248,261,[1_1|2]), (248,275,[1_1|2]), (248,290,[1_1|2]), (248,384,[1_1|2]), (248,393,[1_1|2]), (249,250,[4_1|2]), (250,251,[0_1|2]), (251,252,[3_1|2]), (252,253,[5_1|2]), (253,254,[0_1|2]), (254,255,[3_1|2]), (255,256,[1_1|2]), (256,257,[4_1|2]), (257,258,[4_1|2]), (258,259,[2_1|2]), (259,260,[0_1|2]), (259,101,[3_1|2]), (259,103,[3_1|2]), (259,108,[1_1|2]), (259,127,[3_1|2]), (259,134,[5_1|2]), (259,493,[3_1|3]), (260,100,[1_1|2]), (260,108,[1_1|2]), (260,169,[1_1|2]), (260,309,[1_1|2]), (260,332,[1_1|2]), (260,338,[1_1|2]), (260,384,[1_1|2]), (260,393,[1_1|2]), (260,429,[1_1|2]), (260,314,[1_1|2]), (261,262,[1_1|2]), (262,263,[3_1|2]), (263,264,[4_1|2]), (264,265,[1_1|2]), 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(302,303,[1_1|2]), (303,304,[4_1|2]), (304,305,[5_1|2]), (305,306,[3_1|2]), (306,307,[1_1|2]), (307,308,[3_1|2]), (308,100,[1_1|2]), (308,108,[1_1|2]), (308,169,[1_1|2]), (308,309,[1_1|2]), (308,332,[1_1|2]), (308,338,[1_1|2]), (308,384,[1_1|2]), (308,393,[1_1|2]), (308,429,[1_1|2]), (308,348,[1_1|2]), (308,369,[1_1|2]), (309,310,[1_1|2]), (310,311,[3_1|2]), (310,309,[1_1|2]), (310,312,[3_1|2]), (311,100,[0_1|2]), (311,101,[3_1|2]), (311,103,[3_1|2]), (311,108,[1_1|2]), (311,127,[3_1|2]), (311,134,[5_1|2]), (311,145,[5_1|2]), (311,151,[3_1|2]), (311,158,[5_1|2]), (311,169,[1_1|2]), (311,184,[5_1|2]), (311,201,[2_1|2]), (312,313,[0_1|2]), (313,314,[1_1|2]), (314,315,[5_1|2]), (315,316,[5_1|2]), (316,317,[0_1|2]), (317,318,[3_1|2]), (318,319,[2_1|2]), (319,320,[4_1|2]), (320,321,[4_1|2]), (321,322,[0_1|2]), (322,323,[2_1|2]), (323,324,[4_1|2]), (324,325,[5_1|2]), (325,326,[3_1|2]), (326,327,[3_1|2]), (327,328,[2_1|2]), (328,329,[5_1|2]), (329,330,[1_1|2]), (330,331,[1_1|2]), 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(349,350,[1_1|2]), (350,351,[3_1|2]), (351,352,[2_1|2]), (352,353,[2_1|2]), (352,249,[2_1|2]), (353,100,[3_1|2]), (353,346,[3_1|2]), (353,354,[3_1|2]), (353,367,[3_1|2]), (353,135,[3_1|2]), (353,309,[1_1|2]), (353,312,[3_1|2]), (353,332,[1_1|2]), (353,338,[1_1|2]), (354,355,[1_1|2]), (355,356,[3_1|2]), (356,357,[1_1|2]), (357,358,[3_1|2]), (358,359,[3_1|2]), (359,360,[2_1|2]), (360,361,[5_1|2]), (361,362,[0_1|2]), (362,363,[4_1|2]), (363,364,[5_1|2]), (364,365,[5_1|2]), (365,366,[4_1|2]), (365,346,[4_1|2]), (366,100,[0_1|2]), (366,421,[0_1|2]), (366,101,[3_1|2]), (366,103,[3_1|2]), (366,108,[1_1|2]), (366,127,[3_1|2]), (366,134,[5_1|2]), (366,145,[5_1|2]), (366,151,[3_1|2]), (366,158,[5_1|2]), (366,169,[1_1|2]), (366,184,[5_1|2]), (366,201,[2_1|2]), (367,368,[0_1|2]), (368,369,[1_1|2]), (369,370,[4_1|2]), (370,371,[5_1|2]), (371,372,[0_1|2]), (372,373,[1_1|2]), (373,374,[5_1|2]), (374,375,[1_1|2]), (375,376,[2_1|2]), (376,377,[2_1|2]), (377,378,[5_1|2]), (378,379,[2_1|2]), (379,380,[2_1|2]), (380,381,[5_1|2]), (381,382,[4_1|2]), (382,383,[1_1|2]), (383,100,[5_1|2]), (383,134,[5_1|2]), (383,145,[5_1|2]), (383,158,[5_1|2]), (383,184,[5_1|2]), (383,233,[5_1|2]), (383,291,[5_1|2]), (383,418,[3_1|2]), (383,429,[1_1|2]), (383,439,[3_1|2]), (384,385,[1_1|2]), (385,386,[1_1|2]), (386,387,[0_1|2]), (387,388,[1_1|2]), (388,389,[3_1|2]), (389,390,[1_1|2]), (390,391,[2_1|2]), (391,392,[1_1|2]), (392,100,[0_1|2]), (392,101,[3_1|2]), (392,103,[3_1|2]), (392,108,[1_1|2]), (392,127,[3_1|2]), (392,134,[5_1|2]), (392,145,[5_1|2]), (392,151,[3_1|2]), (392,158,[5_1|2]), (392,169,[1_1|2]), (392,184,[5_1|2]), (392,201,[2_1|2]), (393,394,[3_1|2]), (394,395,[2_1|2]), (395,396,[2_1|2]), (396,397,[0_1|2]), (397,398,[0_1|2]), (398,399,[5_1|2]), (399,400,[1_1|2]), (400,401,[2_1|2]), (401,402,[0_1|2]), (401,151,[3_1|2]), (401,495,[3_1|3]), (402,403,[4_1|2]), (403,404,[0_1|2]), (403,158,[5_1|2]), (403,169,[1_1|2]), (403,184,[5_1|2]), (403,127,[3_1|2]), (404,100,[3_1|2]), (404,101,[3_1|2]), (404,103,[3_1|2]), (404,127,[3_1|2]), (404,151,[3_1|2]), (404,312,[3_1|2]), (404,418,[3_1|2]), (404,439,[3_1|2]), (404,146,[3_1|2]), (404,309,[1_1|2]), (404,332,[1_1|2]), (404,338,[1_1|2]), (418,419,[5_1|2]), (419,420,[1_1|2]), (420,421,[0_1|2]), (421,422,[3_1|2]), (422,423,[4_1|2]), (423,424,[3_1|2]), (424,425,[5_1|2]), (425,426,[2_1|2]), (426,427,[0_1|2]), (427,428,[4_1|2]), (428,100,[3_1|2]), (428,201,[3_1|2]), (428,221,[3_1|2]), (428,225,[3_1|2]), (428,232,[3_1|2]), (428,249,[3_1|2]), (428,261,[3_1|2]), (428,275,[3_1|2]), (428,290,[3_1|2]), (428,222,[3_1|2]), (428,276,[3_1|2]), (428,309,[1_1|2]), (428,312,[3_1|2]), (428,332,[1_1|2]), (428,338,[1_1|2]), (429,430,[2_1|2]), (430,431,[0_1|2]), (431,432,[5_1|2]), (432,433,[5_1|2]), (433,434,[0_1|2]), (434,435,[3_1|2]), (435,436,[2_1|2]), (436,437,[2_1|2]), (437,438,[1_1|2]), (438,100,[3_1|2]), (438,101,[3_1|2]), (438,103,[3_1|2]), (438,127,[3_1|2]), (438,151,[3_1|2]), (438,312,[3_1|2]), (438,418,[3_1|2]), (438,439,[3_1|2]), (438,309,[1_1|2]), (438,332,[1_1|2]), (438,338,[1_1|2]), (439,440,[0_1|2]), (440,441,[0_1|2]), (441,442,[0_1|2]), (442,443,[2_1|2]), (443,444,[5_1|2]), (444,445,[5_1|2]), (445,446,[1_1|2]), (446,447,[0_1|2]), (446,502,[3_1|3]), (447,448,[1_1|2]), (448,449,[2_1|2]), (449,450,[2_1|2]), (450,451,[0_1|2]), (450,151,[3_1|2]), (451,100,[4_1|2]), (451,346,[4_1|2]), (451,354,[4_1|2]), (451,367,[4_1|2]), (491,492,[2_1|3]), (492,195,[1_1|3]), (493,494,[2_1|3]), (494,222,[1_1|3]), (494,276,[1_1|3]), (495,496,[4_1|3]), (496,497,[2_1|3]), (497,498,[3_1|3]), (498,499,[1_1|3]), (499,500,[3_1|3]), (500,501,[1_1|3]), (501,394,[1_1|3]), (501,263,[1_1|3]), (502,503,[2_1|3]), (503,450,[1_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)