/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 51 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 107 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(1(2(x1)))) -> 3(1(3(x1))) 4(2(5(5(x1)))) -> 1(3(0(5(x1)))) 1(5(3(5(2(x1))))) -> 1(2(3(2(4(x1))))) 2(2(0(2(0(4(x1)))))) -> 2(2(1(1(4(x1))))) 3(4(0(2(2(0(x1)))))) -> 3(4(4(5(2(x1))))) 0(3(0(1(2(1(0(x1))))))) -> 3(5(0(2(0(2(x1)))))) 1(2(3(1(2(0(2(x1))))))) -> 1(4(5(0(3(5(x1)))))) 5(4(1(4(5(1(5(x1))))))) -> 5(2(0(2(1(4(5(x1))))))) 2(0(4(5(5(1(2(2(x1)))))))) -> 2(5(5(4(4(2(2(x1))))))) 5(3(1(5(2(0(5(0(x1)))))))) -> 5(0(0(0(4(3(1(0(x1)))))))) 0(2(3(2(0(5(1(0(2(x1))))))))) -> 5(1(2(5(5(0(3(4(x1)))))))) 2(1(3(1(1(0(0(2(2(x1))))))))) -> 1(1(3(1(2(4(5(4(x1)))))))) 3(3(2(0(5(0(1(5(2(x1))))))))) -> 3(0(2(1(4(2(5(2(x1)))))))) 4(3(2(2(5(3(1(5(0(4(x1)))))))))) -> 4(4(3(0(2(0(4(4(0(3(0(x1))))))))))) 2(4(3(3(2(4(1(1(2(0(0(x1))))))))))) -> 4(3(4(4(4(5(0(3(5(1(x1)))))))))) 0(0(0(2(3(5(5(2(4(3(2(0(x1)))))))))))) -> 2(5(1(0(3(0(5(0(2(3(2(0(x1)))))))))))) 1(4(1(3(3(4(5(1(0(1(2(0(x1)))))))))))) -> 1(3(1(0(5(3(2(4(0(2(1(x1))))))))))) 0(1(5(1(1(4(3(0(0(0(2(1(2(x1))))))))))))) -> 5(5(1(1(4(0(0(4(5(5(0(3(x1)))))))))))) 5(4(2(2(4(5(0(4(3(1(2(5(4(x1))))))))))))) -> 5(3(5(5(1(5(0(0(2(1(1(3(4(x1))))))))))))) 3(0(3(3(1(0(1(0(1(4(3(3(3(4(x1)))))))))))))) -> 3(5(4(0(3(2(4(0(1(4(0(5(4(x1))))))))))))) 0(2(4(1(5(1(5(0(1(1(2(4(1(5(4(x1))))))))))))))) -> 0(3(1(5(3(4(0(2(5(5(4(5(1(4(4(x1))))))))))))))) 3(0(2(3(3(4(1(1(4(2(1(1(1(5(4(x1))))))))))))))) -> 3(3(1(5(1(3(0(1(4(5(1(0(3(0(4(x1))))))))))))))) 4(4(4(3(4(1(1(1(0(1(0(1(0(3(1(0(4(x1))))))))))))))))) -> 3(4(2(1(5(0(0(3(1(3(0(3(3(1(3(5(x1)))))))))))))))) 5(2(1(3(0(3(5(4(3(2(2(4(5(2(5(2(3(x1))))))))))))))))) -> 0(5(2(5(4(5(0(0(5(5(4(2(3(0(0(1(2(3(x1)))))))))))))))))) 3(0(4(4(5(5(5(5(2(4(4(1(0(0(2(3(2(4(x1)))))))))))))))))) -> 3(0(5(2(0(2(3(1(2(0(4(4(5(4(0(4(2(3(4(x1))))))))))))))))))) 3(2(0(3(2(1(2(4(3(5(1(3(3(2(5(5(0(4(x1)))))))))))))))))) -> 4(2(5(2(4(3(4(2(0(2(2(4(1(0(2(1(0(4(x1)))))))))))))))))) 4(0(3(3(2(5(0(3(0(1(3(3(4(4(0(5(0(2(4(x1))))))))))))))))))) -> 4(0(1(1(2(3(5(5(2(4(2(3(1(5(5(2(5(x1))))))))))))))))) 2(3(2(3(4(0(4(2(3(1(4(4(2(0(0(3(3(2(0(0(x1)))))))))))))))))))) -> 2(4(2(0(2(5(2(4(4(2(5(4(3(1(4(1(5(1(x1)))))))))))))))))) 0(4(1(5(0(2(3(3(0(3(0(5(2(0(4(3(4(2(1(2(2(x1))))))))))))))))))))) -> 5(4(3(1(4(5(0(5(5(2(3(1(0(4(0(4(4(2(3(0(x1)))))))))))))))))))) 1(5(5(3(2(4(0(4(1(5(5(0(4(2(4(3(4(2(2(1(4(x1))))))))))))))))))))) -> 1(2(2(1(4(5(1(4(4(1(5(3(4(0(2(0(4(4(2(5(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(1(2(x1)))) -> 3(1(3(x1))) 4(2(5(5(x1)))) -> 1(3(0(5(x1)))) 1(5(3(5(2(x1))))) -> 1(2(3(2(4(x1))))) 2(2(0(2(0(4(x1)))))) -> 2(2(1(1(4(x1))))) 3(4(0(2(2(0(x1)))))) -> 3(4(4(5(2(x1))))) 0(3(0(1(2(1(0(x1))))))) -> 3(5(0(2(0(2(x1)))))) 1(2(3(1(2(0(2(x1))))))) -> 1(4(5(0(3(5(x1)))))) 5(4(1(4(5(1(5(x1))))))) -> 5(2(0(2(1(4(5(x1))))))) 2(0(4(5(5(1(2(2(x1)))))))) -> 2(5(5(4(4(2(2(x1))))))) 5(3(1(5(2(0(5(0(x1)))))))) -> 5(0(0(0(4(3(1(0(x1)))))))) 0(2(3(2(0(5(1(0(2(x1))))))))) -> 5(1(2(5(5(0(3(4(x1)))))))) 2(1(3(1(1(0(0(2(2(x1))))))))) -> 1(1(3(1(2(4(5(4(x1)))))))) 3(3(2(0(5(0(1(5(2(x1))))))))) -> 3(0(2(1(4(2(5(2(x1)))))))) 4(3(2(2(5(3(1(5(0(4(x1)))))))))) -> 4(4(3(0(2(0(4(4(0(3(0(x1))))))))))) 2(4(3(3(2(4(1(1(2(0(0(x1))))))))))) -> 4(3(4(4(4(5(0(3(5(1(x1)))))))))) 0(0(0(2(3(5(5(2(4(3(2(0(x1)))))))))))) -> 2(5(1(0(3(0(5(0(2(3(2(0(x1)))))))))))) 1(4(1(3(3(4(5(1(0(1(2(0(x1)))))))))))) -> 1(3(1(0(5(3(2(4(0(2(1(x1))))))))))) 0(1(5(1(1(4(3(0(0(0(2(1(2(x1))))))))))))) -> 5(5(1(1(4(0(0(4(5(5(0(3(x1)))))))))))) 5(4(2(2(4(5(0(4(3(1(2(5(4(x1))))))))))))) -> 5(3(5(5(1(5(0(0(2(1(1(3(4(x1))))))))))))) 3(0(3(3(1(0(1(0(1(4(3(3(3(4(x1)))))))))))))) -> 3(5(4(0(3(2(4(0(1(4(0(5(4(x1))))))))))))) 0(2(4(1(5(1(5(0(1(1(2(4(1(5(4(x1))))))))))))))) -> 0(3(1(5(3(4(0(2(5(5(4(5(1(4(4(x1))))))))))))))) 3(0(2(3(3(4(1(1(4(2(1(1(1(5(4(x1))))))))))))))) -> 3(3(1(5(1(3(0(1(4(5(1(0(3(0(4(x1))))))))))))))) 4(4(4(3(4(1(1(1(0(1(0(1(0(3(1(0(4(x1))))))))))))))))) -> 3(4(2(1(5(0(0(3(1(3(0(3(3(1(3(5(x1)))))))))))))))) 5(2(1(3(0(3(5(4(3(2(2(4(5(2(5(2(3(x1))))))))))))))))) -> 0(5(2(5(4(5(0(0(5(5(4(2(3(0(0(1(2(3(x1)))))))))))))))))) 3(0(4(4(5(5(5(5(2(4(4(1(0(0(2(3(2(4(x1)))))))))))))))))) -> 3(0(5(2(0(2(3(1(2(0(4(4(5(4(0(4(2(3(4(x1))))))))))))))))))) 3(2(0(3(2(1(2(4(3(5(1(3(3(2(5(5(0(4(x1)))))))))))))))))) -> 4(2(5(2(4(3(4(2(0(2(2(4(1(0(2(1(0(4(x1)))))))))))))))))) 4(0(3(3(2(5(0(3(0(1(3(3(4(4(0(5(0(2(4(x1))))))))))))))))))) -> 4(0(1(1(2(3(5(5(2(4(2(3(1(5(5(2(5(x1))))))))))))))))) 2(3(2(3(4(0(4(2(3(1(4(4(2(0(0(3(3(2(0(0(x1)))))))))))))))))))) -> 2(4(2(0(2(5(2(4(4(2(5(4(3(1(4(1(5(1(x1)))))))))))))))))) 0(4(1(5(0(2(3(3(0(3(0(5(2(0(4(3(4(2(1(2(2(x1))))))))))))))))))))) -> 5(4(3(1(4(5(0(5(5(2(3(1(0(4(0(4(4(2(3(0(x1)))))))))))))))))))) 1(5(5(3(2(4(0(4(1(5(5(0(4(2(4(3(4(2(2(1(4(x1))))))))))))))))))))) -> 1(2(2(1(4(5(1(4(4(1(5(3(4(0(2(0(4(4(2(5(0(x1))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(1(2(x1)))) -> 3(1(3(x1))) 4(2(5(5(x1)))) -> 1(3(0(5(x1)))) 1(5(3(5(2(x1))))) -> 1(2(3(2(4(x1))))) 2(2(0(2(0(4(x1)))))) -> 2(2(1(1(4(x1))))) 3(4(0(2(2(0(x1)))))) -> 3(4(4(5(2(x1))))) 0(3(0(1(2(1(0(x1))))))) -> 3(5(0(2(0(2(x1)))))) 1(2(3(1(2(0(2(x1))))))) -> 1(4(5(0(3(5(x1)))))) 5(4(1(4(5(1(5(x1))))))) -> 5(2(0(2(1(4(5(x1))))))) 2(0(4(5(5(1(2(2(x1)))))))) -> 2(5(5(4(4(2(2(x1))))))) 5(3(1(5(2(0(5(0(x1)))))))) -> 5(0(0(0(4(3(1(0(x1)))))))) 0(2(3(2(0(5(1(0(2(x1))))))))) -> 5(1(2(5(5(0(3(4(x1)))))))) 2(1(3(1(1(0(0(2(2(x1))))))))) -> 1(1(3(1(2(4(5(4(x1)))))))) 3(3(2(0(5(0(1(5(2(x1))))))))) -> 3(0(2(1(4(2(5(2(x1)))))))) 4(3(2(2(5(3(1(5(0(4(x1)))))))))) -> 4(4(3(0(2(0(4(4(0(3(0(x1))))))))))) 2(4(3(3(2(4(1(1(2(0(0(x1))))))))))) -> 4(3(4(4(4(5(0(3(5(1(x1)))))))))) 0(0(0(2(3(5(5(2(4(3(2(0(x1)))))))))))) -> 2(5(1(0(3(0(5(0(2(3(2(0(x1)))))))))))) 1(4(1(3(3(4(5(1(0(1(2(0(x1)))))))))))) -> 1(3(1(0(5(3(2(4(0(2(1(x1))))))))))) 0(1(5(1(1(4(3(0(0(0(2(1(2(x1))))))))))))) -> 5(5(1(1(4(0(0(4(5(5(0(3(x1)))))))))))) 5(4(2(2(4(5(0(4(3(1(2(5(4(x1))))))))))))) -> 5(3(5(5(1(5(0(0(2(1(1(3(4(x1))))))))))))) 3(0(3(3(1(0(1(0(1(4(3(3(3(4(x1)))))))))))))) -> 3(5(4(0(3(2(4(0(1(4(0(5(4(x1))))))))))))) 0(2(4(1(5(1(5(0(1(1(2(4(1(5(4(x1))))))))))))))) -> 0(3(1(5(3(4(0(2(5(5(4(5(1(4(4(x1))))))))))))))) 3(0(2(3(3(4(1(1(4(2(1(1(1(5(4(x1))))))))))))))) -> 3(3(1(5(1(3(0(1(4(5(1(0(3(0(4(x1))))))))))))))) 4(4(4(3(4(1(1(1(0(1(0(1(0(3(1(0(4(x1))))))))))))))))) -> 3(4(2(1(5(0(0(3(1(3(0(3(3(1(3(5(x1)))))))))))))))) 5(2(1(3(0(3(5(4(3(2(2(4(5(2(5(2(3(x1))))))))))))))))) -> 0(5(2(5(4(5(0(0(5(5(4(2(3(0(0(1(2(3(x1)))))))))))))))))) 3(0(4(4(5(5(5(5(2(4(4(1(0(0(2(3(2(4(x1)))))))))))))))))) -> 3(0(5(2(0(2(3(1(2(0(4(4(5(4(0(4(2(3(4(x1))))))))))))))))))) 3(2(0(3(2(1(2(4(3(5(1(3(3(2(5(5(0(4(x1)))))))))))))))))) -> 4(2(5(2(4(3(4(2(0(2(2(4(1(0(2(1(0(4(x1)))))))))))))))))) 4(0(3(3(2(5(0(3(0(1(3(3(4(4(0(5(0(2(4(x1))))))))))))))))))) -> 4(0(1(1(2(3(5(5(2(4(2(3(1(5(5(2(5(x1))))))))))))))))) 2(3(2(3(4(0(4(2(3(1(4(4(2(0(0(3(3(2(0(0(x1)))))))))))))))))))) -> 2(4(2(0(2(5(2(4(4(2(5(4(3(1(4(1(5(1(x1)))))))))))))))))) 0(4(1(5(0(2(3(3(0(3(0(5(2(0(4(3(4(2(1(2(2(x1))))))))))))))))))))) -> 5(4(3(1(4(5(0(5(5(2(3(1(0(4(0(4(4(2(3(0(x1)))))))))))))))))))) 1(5(5(3(2(4(0(4(1(5(5(0(4(2(4(3(4(2(2(1(4(x1))))))))))))))))))))) -> 1(2(2(1(4(5(1(4(4(1(5(3(4(0(2(0(4(4(2(5(0(x1))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(1(2(x1)))) -> 3(1(3(x1))) 4(2(5(5(x1)))) -> 1(3(0(5(x1)))) 1(5(3(5(2(x1))))) -> 1(2(3(2(4(x1))))) 2(2(0(2(0(4(x1)))))) -> 2(2(1(1(4(x1))))) 3(4(0(2(2(0(x1)))))) -> 3(4(4(5(2(x1))))) 0(3(0(1(2(1(0(x1))))))) -> 3(5(0(2(0(2(x1)))))) 1(2(3(1(2(0(2(x1))))))) -> 1(4(5(0(3(5(x1)))))) 5(4(1(4(5(1(5(x1))))))) -> 5(2(0(2(1(4(5(x1))))))) 2(0(4(5(5(1(2(2(x1)))))))) -> 2(5(5(4(4(2(2(x1))))))) 5(3(1(5(2(0(5(0(x1)))))))) -> 5(0(0(0(4(3(1(0(x1)))))))) 0(2(3(2(0(5(1(0(2(x1))))))))) -> 5(1(2(5(5(0(3(4(x1)))))))) 2(1(3(1(1(0(0(2(2(x1))))))))) -> 1(1(3(1(2(4(5(4(x1)))))))) 3(3(2(0(5(0(1(5(2(x1))))))))) -> 3(0(2(1(4(2(5(2(x1)))))))) 4(3(2(2(5(3(1(5(0(4(x1)))))))))) -> 4(4(3(0(2(0(4(4(0(3(0(x1))))))))))) 2(4(3(3(2(4(1(1(2(0(0(x1))))))))))) -> 4(3(4(4(4(5(0(3(5(1(x1)))))))))) 0(0(0(2(3(5(5(2(4(3(2(0(x1)))))))))))) -> 2(5(1(0(3(0(5(0(2(3(2(0(x1)))))))))))) 1(4(1(3(3(4(5(1(0(1(2(0(x1)))))))))))) -> 1(3(1(0(5(3(2(4(0(2(1(x1))))))))))) 0(1(5(1(1(4(3(0(0(0(2(1(2(x1))))))))))))) -> 5(5(1(1(4(0(0(4(5(5(0(3(x1)))))))))))) 5(4(2(2(4(5(0(4(3(1(2(5(4(x1))))))))))))) -> 5(3(5(5(1(5(0(0(2(1(1(3(4(x1))))))))))))) 3(0(3(3(1(0(1(0(1(4(3(3(3(4(x1)))))))))))))) -> 3(5(4(0(3(2(4(0(1(4(0(5(4(x1))))))))))))) 0(2(4(1(5(1(5(0(1(1(2(4(1(5(4(x1))))))))))))))) -> 0(3(1(5(3(4(0(2(5(5(4(5(1(4(4(x1))))))))))))))) 3(0(2(3(3(4(1(1(4(2(1(1(1(5(4(x1))))))))))))))) -> 3(3(1(5(1(3(0(1(4(5(1(0(3(0(4(x1))))))))))))))) 4(4(4(3(4(1(1(1(0(1(0(1(0(3(1(0(4(x1))))))))))))))))) -> 3(4(2(1(5(0(0(3(1(3(0(3(3(1(3(5(x1)))))))))))))))) 5(2(1(3(0(3(5(4(3(2(2(4(5(2(5(2(3(x1))))))))))))))))) -> 0(5(2(5(4(5(0(0(5(5(4(2(3(0(0(1(2(3(x1)))))))))))))))))) 3(0(4(4(5(5(5(5(2(4(4(1(0(0(2(3(2(4(x1)))))))))))))))))) -> 3(0(5(2(0(2(3(1(2(0(4(4(5(4(0(4(2(3(4(x1))))))))))))))))))) 3(2(0(3(2(1(2(4(3(5(1(3(3(2(5(5(0(4(x1)))))))))))))))))) -> 4(2(5(2(4(3(4(2(0(2(2(4(1(0(2(1(0(4(x1)))))))))))))))))) 4(0(3(3(2(5(0(3(0(1(3(3(4(4(0(5(0(2(4(x1))))))))))))))))))) -> 4(0(1(1(2(3(5(5(2(4(2(3(1(5(5(2(5(x1))))))))))))))))) 2(3(2(3(4(0(4(2(3(1(4(4(2(0(0(3(3(2(0(0(x1)))))))))))))))))))) -> 2(4(2(0(2(5(2(4(4(2(5(4(3(1(4(1(5(1(x1)))))))))))))))))) 0(4(1(5(0(2(3(3(0(3(0(5(2(0(4(3(4(2(1(2(2(x1))))))))))))))))))))) -> 5(4(3(1(4(5(0(5(5(2(3(1(0(4(0(4(4(2(3(0(x1)))))))))))))))))))) 1(5(5(3(2(4(0(4(1(5(5(0(4(2(4(3(4(2(2(1(4(x1))))))))))))))))))))) -> 1(2(2(1(4(5(1(4(4(1(5(3(4(0(2(0(4(4(2(5(0(x1))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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. "[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] {(151,152,[0_1|0, 4_1|0, 1_1|0, 2_1|0, 3_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]), (151,153,[0_1|1, 4_1|1, 1_1|1, 2_1|1, 3_1|1, 5_1|1]), (151,154,[3_1|2]), (151,156,[2_1|2]), (151,167,[3_1|2]), (151,172,[5_1|2]), (151,179,[0_1|2]), (151,193,[5_1|2]), (151,204,[5_1|2]), (151,223,[1_1|2]), (151,226,[4_1|2]), (151,236,[3_1|2]), (151,251,[4_1|2]), (151,267,[1_1|2]), (151,271,[1_1|2]), (151,291,[1_1|2]), (151,296,[1_1|2]), (151,306,[2_1|2]), (151,310,[2_1|2]), (151,316,[1_1|2]), (151,323,[4_1|2]), (151,332,[2_1|2]), (151,349,[3_1|2]), (151,353,[3_1|2]), (151,360,[3_1|2]), (151,372,[3_1|2]), (151,386,[3_1|2]), (151,404,[4_1|2]), (151,421,[5_1|2]), (151,427,[5_1|2]), (151,439,[5_1|2]), (151,446,[0_1|2]), (152,152,[cons_0_1|0, cons_4_1|0, cons_1_1|0, cons_2_1|0, cons_3_1|0, cons_5_1|0]), (153,152,[encArg_1|1]), (153,153,[0_1|1, 4_1|1, 1_1|1, 2_1|1, 3_1|1, 5_1|1]), (153,154,[3_1|2]), (153,156,[2_1|2]), (153,167,[3_1|2]), (153,172,[5_1|2]), (153,179,[0_1|2]), (153,193,[5_1|2]), (153,204,[5_1|2]), (153,223,[1_1|2]), (153,226,[4_1|2]), (153,236,[3_1|2]), (153,251,[4_1|2]), (153,267,[1_1|2]), (153,271,[1_1|2]), (153,291,[1_1|2]), (153,296,[1_1|2]), (153,306,[2_1|2]), (153,310,[2_1|2]), (153,316,[1_1|2]), (153,323,[4_1|2]), (153,332,[2_1|2]), (153,349,[3_1|2]), (153,353,[3_1|2]), (153,360,[3_1|2]), (153,372,[3_1|2]), (153,386,[3_1|2]), (153,404,[4_1|2]), (153,421,[5_1|2]), (153,427,[5_1|2]), (153,439,[5_1|2]), (153,446,[0_1|2]), (154,155,[1_1|2]), (155,153,[3_1|2]), (155,156,[3_1|2]), (155,306,[3_1|2]), (155,310,[3_1|2]), (155,332,[3_1|2]), (155,268,[3_1|2]), (155,272,[3_1|2]), (155,349,[3_1|2]), (155,353,[3_1|2]), (155,360,[3_1|2]), (155,372,[3_1|2]), (155,386,[3_1|2]), (155,404,[4_1|2]), (156,157,[5_1|2]), (157,158,[1_1|2]), (158,159,[0_1|2]), (159,160,[3_1|2]), (160,161,[0_1|2]), (161,162,[5_1|2]), (162,163,[0_1|2]), (162,172,[5_1|2]), (163,164,[2_1|2]), (164,165,[3_1|2]), (164,404,[4_1|2]), (165,166,[2_1|2]), (165,310,[2_1|2]), (166,153,[0_1|2]), (166,179,[0_1|2]), (166,446,[0_1|2]), (166,154,[3_1|2]), (166,156,[2_1|2]), (166,167,[3_1|2]), (166,172,[5_1|2]), (166,193,[5_1|2]), (166,204,[5_1|2]), (167,168,[5_1|2]), (168,169,[0_1|2]), (169,170,[2_1|2]), (170,171,[0_1|2]), (170,172,[5_1|2]), (170,179,[0_1|2]), (171,153,[2_1|2]), (171,179,[2_1|2]), (171,446,[2_1|2]), (171,306,[2_1|2]), (171,310,[2_1|2]), (171,316,[1_1|2]), (171,323,[4_1|2]), (171,332,[2_1|2]), (172,173,[1_1|2]), (173,174,[2_1|2]), (174,175,[5_1|2]), (175,176,[5_1|2]), (176,177,[0_1|2]), (177,178,[3_1|2]), (177,349,[3_1|2]), (178,153,[4_1|2]), (178,156,[4_1|2]), (178,306,[4_1|2]), (178,310,[4_1|2]), (178,332,[4_1|2]), (178,223,[1_1|2]), (178,226,[4_1|2]), (178,236,[3_1|2]), (178,251,[4_1|2]), (178,463,[1_1|3]), (179,180,[3_1|2]), (180,181,[1_1|2]), (181,182,[5_1|2]), (182,183,[3_1|2]), (183,184,[4_1|2]), (184,185,[0_1|2]), (185,186,[2_1|2]), (186,187,[5_1|2]), (187,188,[5_1|2]), (188,189,[4_1|2]), (189,190,[5_1|2]), (190,191,[1_1|2]), (191,192,[4_1|2]), (191,236,[3_1|2]), (192,153,[4_1|2]), (192,226,[4_1|2]), (192,251,[4_1|2]), (192,323,[4_1|2]), (192,404,[4_1|2]), (192,205,[4_1|2]), (192,223,[1_1|2]), (192,236,[3_1|2]), (192,463,[1_1|3]), (193,194,[5_1|2]), (194,195,[1_1|2]), (195,196,[1_1|2]), (196,197,[4_1|2]), (197,198,[0_1|2]), (198,199,[0_1|2]), (199,200,[4_1|2]), (200,201,[5_1|2]), (201,202,[5_1|2]), (202,203,[0_1|2]), (202,167,[3_1|2]), (203,153,[3_1|2]), (203,156,[3_1|2]), (203,306,[3_1|2]), (203,310,[3_1|2]), (203,332,[3_1|2]), (203,268,[3_1|2]), (203,272,[3_1|2]), (203,349,[3_1|2]), (203,353,[3_1|2]), (203,360,[3_1|2]), (203,372,[3_1|2]), (203,386,[3_1|2]), (203,404,[4_1|2]), (204,205,[4_1|2]), (205,206,[3_1|2]), (206,207,[1_1|2]), (207,208,[4_1|2]), (208,209,[5_1|2]), (209,210,[0_1|2]), (210,211,[5_1|2]), (211,212,[5_1|2]), (212,213,[2_1|2]), (213,214,[3_1|2]), (214,215,[1_1|2]), (215,216,[0_1|2]), (216,217,[4_1|2]), (217,218,[0_1|2]), (218,219,[4_1|2]), (219,220,[4_1|2]), (220,221,[2_1|2]), (221,222,[3_1|2]), (221,360,[3_1|2]), (221,372,[3_1|2]), (221,386,[3_1|2]), (222,153,[0_1|2]), (222,156,[0_1|2, 2_1|2]), (222,306,[0_1|2]), (222,310,[0_1|2]), (222,332,[0_1|2]), (222,307,[0_1|2]), (222,273,[0_1|2]), (222,154,[3_1|2]), (222,167,[3_1|2]), (222,172,[5_1|2]), (222,179,[0_1|2]), (222,193,[5_1|2]), (222,204,[5_1|2]), (223,224,[3_1|2]), (224,225,[0_1|2]), (225,153,[5_1|2]), (225,172,[5_1|2]), (225,193,[5_1|2]), (225,204,[5_1|2]), (225,421,[5_1|2]), (225,427,[5_1|2]), (225,439,[5_1|2]), (225,194,[5_1|2]), (225,312,[5_1|2]), (225,446,[0_1|2]), (226,227,[4_1|2]), (227,228,[3_1|2]), (228,229,[0_1|2]), (229,230,[2_1|2]), (230,231,[0_1|2]), (231,232,[4_1|2]), (232,233,[4_1|2]), (233,234,[0_1|2]), (233,167,[3_1|2]), (234,235,[3_1|2]), (234,360,[3_1|2]), (234,372,[3_1|2]), (234,386,[3_1|2]), (235,153,[0_1|2]), (235,226,[0_1|2]), (235,251,[0_1|2]), (235,323,[0_1|2]), (235,404,[0_1|2]), (235,154,[3_1|2]), (235,156,[2_1|2]), (235,167,[3_1|2]), (235,172,[5_1|2]), (235,179,[0_1|2]), (235,193,[5_1|2]), (235,204,[5_1|2]), (236,237,[4_1|2]), (237,238,[2_1|2]), (238,239,[1_1|2]), (239,240,[5_1|2]), (240,241,[0_1|2]), (241,242,[0_1|2]), (242,243,[3_1|2]), (243,244,[1_1|2]), (244,245,[3_1|2]), (245,246,[0_1|2]), (246,247,[3_1|2]), (247,248,[3_1|2]), (248,249,[1_1|2]), (249,250,[3_1|2]), (250,153,[5_1|2]), (250,226,[5_1|2]), (250,251,[5_1|2]), (250,323,[5_1|2]), (250,404,[5_1|2]), (250,421,[5_1|2]), (250,427,[5_1|2]), (250,439,[5_1|2]), (250,446,[0_1|2]), (251,252,[0_1|2]), (252,253,[1_1|2]), (253,254,[1_1|2]), (254,255,[2_1|2]), (255,256,[3_1|2]), (256,257,[5_1|2]), (257,258,[5_1|2]), (258,259,[2_1|2]), (259,260,[4_1|2]), (260,261,[2_1|2]), (261,262,[3_1|2]), (262,263,[1_1|2]), (263,264,[5_1|2]), (264,265,[5_1|2]), (265,266,[2_1|2]), (266,153,[5_1|2]), (266,226,[5_1|2]), (266,251,[5_1|2]), (266,323,[5_1|2]), (266,404,[5_1|2]), (266,333,[5_1|2]), (266,421,[5_1|2]), (266,427,[5_1|2]), (266,439,[5_1|2]), (266,446,[0_1|2]), (267,268,[2_1|2]), (268,269,[3_1|2]), (269,270,[2_1|2]), (269,323,[4_1|2]), (270,153,[4_1|2]), (270,156,[4_1|2]), (270,306,[4_1|2]), (270,310,[4_1|2]), (270,332,[4_1|2]), (270,422,[4_1|2]), (270,223,[1_1|2]), (270,226,[4_1|2]), (270,236,[3_1|2]), (270,251,[4_1|2]), (270,463,[1_1|3]), (271,272,[2_1|2]), (272,273,[2_1|2]), (273,274,[1_1|2]), (274,275,[4_1|2]), (275,276,[5_1|2]), (276,277,[1_1|2]), (277,278,[4_1|2]), (278,279,[4_1|2]), (279,280,[1_1|2]), (280,281,[5_1|2]), (281,282,[3_1|2]), (282,283,[4_1|2]), (283,284,[0_1|2]), (284,285,[2_1|2]), (285,286,[0_1|2]), (286,287,[4_1|2]), (287,288,[4_1|2]), (287,468,[1_1|3]), (288,289,[2_1|2]), (289,290,[5_1|2]), (290,153,[0_1|2]), (290,226,[0_1|2]), (290,251,[0_1|2]), (290,323,[0_1|2]), (290,404,[0_1|2]), (290,292,[0_1|2]), (290,154,[3_1|2]), (290,156,[2_1|2]), (290,167,[3_1|2]), (290,172,[5_1|2]), (290,179,[0_1|2]), (290,193,[5_1|2]), (290,204,[5_1|2]), (291,292,[4_1|2]), (292,293,[5_1|2]), (293,294,[0_1|2]), (294,295,[3_1|2]), (295,153,[5_1|2]), (295,156,[5_1|2]), (295,306,[5_1|2]), (295,310,[5_1|2]), (295,332,[5_1|2]), (295,421,[5_1|2]), (295,427,[5_1|2]), (295,439,[5_1|2]), (295,446,[0_1|2]), (296,297,[3_1|2]), (297,298,[1_1|2]), (298,299,[0_1|2]), (299,300,[5_1|2]), (300,301,[3_1|2]), (301,302,[2_1|2]), (302,303,[4_1|2]), (303,304,[0_1|2]), (304,305,[2_1|2]), (304,316,[1_1|2]), (305,153,[1_1|2]), (305,179,[1_1|2]), (305,446,[1_1|2]), (305,267,[1_1|2]), (305,271,[1_1|2]), (305,291,[1_1|2]), (305,296,[1_1|2]), (306,307,[2_1|2]), (307,308,[1_1|2]), (308,309,[1_1|2]), (308,296,[1_1|2]), (309,153,[4_1|2]), (309,226,[4_1|2]), (309,251,[4_1|2]), (309,323,[4_1|2]), (309,404,[4_1|2]), (309,223,[1_1|2]), (309,236,[3_1|2]), (309,463,[1_1|3]), (310,311,[5_1|2]), (311,312,[5_1|2]), (312,313,[4_1|2]), (313,314,[4_1|2]), (314,315,[2_1|2]), (314,306,[2_1|2]), (315,153,[2_1|2]), (315,156,[2_1|2]), (315,306,[2_1|2]), (315,310,[2_1|2]), (315,332,[2_1|2]), (315,307,[2_1|2]), (315,273,[2_1|2]), (315,316,[1_1|2]), (315,323,[4_1|2]), (316,317,[1_1|2]), (317,318,[3_1|2]), (318,319,[1_1|2]), (319,320,[2_1|2]), (320,321,[4_1|2]), (321,322,[5_1|2]), (321,421,[5_1|2]), (321,427,[5_1|2]), (322,153,[4_1|2]), (322,156,[4_1|2]), (322,306,[4_1|2]), (322,310,[4_1|2]), (322,332,[4_1|2]), (322,307,[4_1|2]), (322,223,[1_1|2]), (322,226,[4_1|2]), (322,236,[3_1|2]), (322,251,[4_1|2]), (322,463,[1_1|3]), (323,324,[3_1|2]), (324,325,[4_1|2]), (325,326,[4_1|2]), (326,327,[4_1|2]), (327,328,[5_1|2]), (328,329,[0_1|2]), (329,330,[3_1|2]), (330,331,[5_1|2]), (331,153,[1_1|2]), (331,179,[1_1|2]), (331,446,[1_1|2]), (331,267,[1_1|2]), (331,271,[1_1|2]), (331,291,[1_1|2]), (331,296,[1_1|2]), (332,333,[4_1|2]), (333,334,[2_1|2]), (334,335,[0_1|2]), (335,336,[2_1|2]), (336,337,[5_1|2]), (337,338,[2_1|2]), (338,339,[4_1|2]), (339,340,[4_1|2]), (340,341,[2_1|2]), (341,342,[5_1|2]), (342,343,[4_1|2]), (343,344,[3_1|2]), (344,345,[1_1|2]), (345,346,[4_1|2]), (346,347,[1_1|2]), (347,348,[5_1|2]), (348,153,[1_1|2]), (348,179,[1_1|2]), (348,446,[1_1|2]), (348,267,[1_1|2]), (348,271,[1_1|2]), (348,291,[1_1|2]), (348,296,[1_1|2]), (349,350,[4_1|2]), (350,351,[4_1|2]), (351,352,[5_1|2]), (351,446,[0_1|2]), (352,153,[2_1|2]), (352,179,[2_1|2]), (352,446,[2_1|2]), (352,306,[2_1|2]), (352,310,[2_1|2]), (352,316,[1_1|2]), (352,323,[4_1|2]), (352,332,[2_1|2]), (353,354,[0_1|2]), (354,355,[2_1|2]), (355,356,[1_1|2]), (356,357,[4_1|2]), (357,358,[2_1|2]), (358,359,[5_1|2]), (358,446,[0_1|2]), (359,153,[2_1|2]), (359,156,[2_1|2]), (359,306,[2_1|2]), (359,310,[2_1|2]), (359,332,[2_1|2]), (359,422,[2_1|2]), (359,316,[1_1|2]), (359,323,[4_1|2]), (360,361,[5_1|2]), (361,362,[4_1|2]), (362,363,[0_1|2]), (363,364,[3_1|2]), (364,365,[2_1|2]), (365,366,[4_1|2]), (366,367,[0_1|2]), (367,368,[1_1|2]), (368,369,[4_1|2]), (369,370,[0_1|2]), (370,371,[5_1|2]), (370,421,[5_1|2]), (370,427,[5_1|2]), (371,153,[4_1|2]), (371,226,[4_1|2]), (371,251,[4_1|2]), (371,323,[4_1|2]), (371,404,[4_1|2]), (371,237,[4_1|2]), (371,350,[4_1|2]), (371,223,[1_1|2]), (371,236,[3_1|2]), (371,463,[1_1|3]), (372,373,[3_1|2]), (373,374,[1_1|2]), (374,375,[5_1|2]), (375,376,[1_1|2]), (376,377,[3_1|2]), (377,378,[0_1|2]), (378,379,[1_1|2]), (379,380,[4_1|2]), (380,381,[5_1|2]), (381,382,[1_1|2]), (382,383,[0_1|2]), (383,384,[3_1|2]), (383,386,[3_1|2]), (384,385,[0_1|2]), (384,204,[5_1|2]), (385,153,[4_1|2]), (385,226,[4_1|2]), (385,251,[4_1|2]), (385,323,[4_1|2]), (385,404,[4_1|2]), (385,205,[4_1|2]), (385,223,[1_1|2]), (385,236,[3_1|2]), (385,463,[1_1|3]), (386,387,[0_1|2]), (387,388,[5_1|2]), (388,389,[2_1|2]), (389,390,[0_1|2]), (390,391,[2_1|2]), (391,392,[3_1|2]), (392,393,[1_1|2]), (393,394,[2_1|2]), (394,395,[0_1|2]), (395,396,[4_1|2]), (396,397,[4_1|2]), (397,398,[5_1|2]), (398,399,[4_1|2]), (399,400,[0_1|2]), (400,401,[4_1|2]), (401,402,[2_1|2]), (402,403,[3_1|2]), (402,349,[3_1|2]), (403,153,[4_1|2]), (403,226,[4_1|2]), (403,251,[4_1|2]), (403,323,[4_1|2]), (403,404,[4_1|2]), (403,333,[4_1|2]), (403,223,[1_1|2]), (403,236,[3_1|2]), (403,463,[1_1|3]), (404,405,[2_1|2]), (405,406,[5_1|2]), (406,407,[2_1|2]), (407,408,[4_1|2]), (408,409,[3_1|2]), (409,410,[4_1|2]), (410,411,[2_1|2]), (411,412,[0_1|2]), (412,413,[2_1|2]), (413,414,[2_1|2]), (414,415,[4_1|2]), (415,416,[1_1|2]), (416,417,[0_1|2]), (417,418,[2_1|2]), (418,419,[1_1|2]), (419,420,[0_1|2]), (419,204,[5_1|2]), (420,153,[4_1|2]), (420,226,[4_1|2]), (420,251,[4_1|2]), (420,323,[4_1|2]), (420,404,[4_1|2]), (420,223,[1_1|2]), (420,236,[3_1|2]), (420,463,[1_1|3]), (421,422,[2_1|2]), (422,423,[0_1|2]), (423,424,[2_1|2]), (424,425,[1_1|2]), (425,426,[4_1|2]), (426,153,[5_1|2]), (426,172,[5_1|2]), (426,193,[5_1|2]), (426,204,[5_1|2]), (426,421,[5_1|2]), (426,427,[5_1|2]), (426,439,[5_1|2]), (426,446,[0_1|2]), (427,428,[3_1|2]), (428,429,[5_1|2]), (429,430,[5_1|2]), (430,431,[1_1|2]), (431,432,[5_1|2]), (432,433,[0_1|2]), (433,434,[0_1|2]), (434,435,[2_1|2]), (435,436,[1_1|2]), (436,437,[1_1|2]), (437,438,[3_1|2]), (437,349,[3_1|2]), (438,153,[4_1|2]), (438,226,[4_1|2]), (438,251,[4_1|2]), (438,323,[4_1|2]), (438,404,[4_1|2]), (438,205,[4_1|2]), (438,223,[1_1|2]), (438,236,[3_1|2]), (438,463,[1_1|3]), (439,440,[0_1|2]), (440,441,[0_1|2]), (441,442,[0_1|2]), (442,443,[4_1|2]), (443,444,[3_1|2]), (444,445,[1_1|2]), (445,153,[0_1|2]), (445,179,[0_1|2]), (445,446,[0_1|2]), (445,440,[0_1|2]), (445,154,[3_1|2]), (445,156,[2_1|2]), (445,167,[3_1|2]), (445,172,[5_1|2]), (445,193,[5_1|2]), (445,204,[5_1|2]), (446,447,[5_1|2]), (447,448,[2_1|2]), (448,449,[5_1|2]), (449,450,[4_1|2]), (450,451,[5_1|2]), (451,452,[0_1|2]), (452,453,[0_1|2]), (453,454,[5_1|2]), (454,455,[5_1|2]), (455,456,[4_1|2]), (456,457,[2_1|2]), (457,458,[3_1|2]), (458,459,[0_1|2]), (458,466,[3_1|3]), (459,460,[0_1|2]), (460,461,[1_1|2]), (460,291,[1_1|2]), (461,462,[2_1|2]), (461,332,[2_1|2]), (462,153,[3_1|2]), (462,154,[3_1|2]), (462,167,[3_1|2]), (462,236,[3_1|2]), (462,349,[3_1|2]), (462,353,[3_1|2]), (462,360,[3_1|2]), (462,372,[3_1|2]), (462,386,[3_1|2]), (462,404,[4_1|2]), (463,464,[3_1|3]), (464,465,[0_1|3]), (465,312,[5_1|3]), (466,467,[1_1|3]), (467,462,[3_1|3]), (467,332,[3_1|3]), (467,353,[3_1|2]), (468,469,[3_1|3]), (469,470,[0_1|3]), (470,172,[5_1|3]), (470,193,[5_1|3]), (470,204,[5_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)