/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 45 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 70 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(1(1(2(4(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(1(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(1(1(2(4(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(1(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(1(1(2(4(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(1(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(1(1(2(4(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 2. The certificate found is represented by the following graph. "[151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482] {(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,[2_1|2]), (151,157,[1_1|2]), (151,161,[4_1|2]), (151,179,[0_1|2]), (151,190,[4_1|2]), (151,202,[4_1|2]), (151,207,[4_1|2]), (151,214,[1_1|2]), (151,230,[4_1|2]), (151,241,[2_1|2]), (151,260,[5_1|2]), (151,268,[4_1|2]), (151,285,[4_1|2]), (151,302,[1_1|2]), (151,307,[4_1|2]), (151,325,[1_1|2]), (151,344,[2_1|2]), (151,351,[2_1|2]), (151,370,[3_1|2]), (151,382,[3_1|2]), (151,400,[0_1|2]), (151,415,[2_1|2]), (151,430,[5_1|2]), (151,446,[1_1|2]), (151,463,[5_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,[2_1|2]), (153,157,[1_1|2]), (153,161,[4_1|2]), (153,179,[0_1|2]), (153,190,[4_1|2]), (153,202,[4_1|2]), (153,207,[4_1|2]), (153,214,[1_1|2]), (153,230,[4_1|2]), (153,241,[2_1|2]), (153,260,[5_1|2]), (153,268,[4_1|2]), (153,285,[4_1|2]), (153,302,[1_1|2]), (153,307,[4_1|2]), (153,325,[1_1|2]), (153,344,[2_1|2]), (153,351,[2_1|2]), (153,370,[3_1|2]), (153,382,[3_1|2]), (153,400,[0_1|2]), (153,415,[2_1|2]), (153,430,[5_1|2]), (153,446,[1_1|2]), (153,463,[5_1|2]), (154,155,[1_1|2]), (155,156,[3_1|2]), (155,370,[3_1|2]), (155,382,[3_1|2]), (156,153,[2_1|2]), (156,154,[2_1|2]), (156,241,[2_1|2]), (156,344,[2_1|2]), (156,351,[2_1|2]), (156,415,[2_1|2]), (157,158,[3_1|2]), (158,159,[2_1|2]), (159,160,[1_1|2]), (160,153,[2_1|2]), (160,179,[2_1|2]), (160,400,[2_1|2]), (160,344,[2_1|2]), (160,351,[2_1|2]), (161,162,[2_1|2]), (162,163,[1_1|2]), (163,164,[4_1|2]), (164,165,[1_1|2]), (165,166,[1_1|2]), (166,167,[3_1|2]), (167,168,[2_1|2]), (168,169,[0_1|2]), (169,170,[4_1|2]), (170,171,[3_1|2]), (171,172,[3_1|2]), (172,173,[0_1|2]), (173,174,[4_1|2]), (174,175,[3_1|2]), (175,176,[3_1|2]), (176,177,[1_1|2]), (177,178,[3_1|2]), (178,153,[1_1|2]), (178,179,[1_1|2]), (178,400,[1_1|2]), (178,464,[1_1|2]), (178,302,[1_1|2]), (178,307,[4_1|2]), (178,325,[1_1|2]), (179,180,[2_1|2]), (180,181,[0_1|2]), (181,182,[0_1|2]), (182,183,[3_1|2]), (183,184,[5_1|2]), (184,185,[1_1|2]), (185,186,[1_1|2]), (186,187,[3_1|2]), (187,188,[2_1|2]), (188,189,[0_1|2]), (189,153,[5_1|2]), (189,157,[5_1|2]), (189,214,[5_1|2]), (189,302,[5_1|2]), (189,325,[5_1|2]), (189,446,[5_1|2, 1_1|2]), (189,400,[0_1|2]), (189,415,[2_1|2]), (189,430,[5_1|2]), (189,463,[5_1|2]), (190,191,[0_1|2]), (191,192,[2_1|2]), (192,193,[0_1|2]), (193,194,[4_1|2]), (194,195,[5_1|2]), (195,196,[1_1|2]), (196,197,[1_1|2]), (197,198,[3_1|2]), (198,199,[0_1|2]), (199,200,[1_1|2]), (200,201,[1_1|2]), (200,302,[1_1|2]), (201,153,[1_1|2]), (201,157,[1_1|2]), (201,214,[1_1|2]), (201,302,[1_1|2]), (201,325,[1_1|2]), (201,446,[1_1|2]), (201,307,[4_1|2]), (202,203,[3_1|2]), (203,204,[5_1|2]), (204,205,[1_1|2]), (205,206,[0_1|2]), (206,153,[0_1|2]), (206,179,[0_1|2]), (206,400,[0_1|2]), (206,215,[0_1|2]), (206,303,[0_1|2]), (206,154,[2_1|2]), (206,157,[1_1|2]), (206,161,[4_1|2]), (206,190,[4_1|2]), (207,208,[5_1|2]), (208,209,[5_1|2]), (209,210,[3_1|2]), (210,211,[5_1|2]), (211,212,[1_1|2]), (212,213,[0_1|2]), (212,154,[2_1|2]), (212,157,[1_1|2]), (212,161,[4_1|2]), (212,179,[0_1|2]), (213,153,[1_1|2]), (213,157,[1_1|2]), (213,214,[1_1|2]), (213,302,[1_1|2]), (213,325,[1_1|2]), (213,446,[1_1|2]), (213,307,[4_1|2]), (214,215,[0_1|2]), (215,216,[0_1|2]), (216,217,[4_1|2]), (217,218,[2_1|2]), (218,219,[2_1|2]), (219,220,[4_1|2]), (220,221,[4_1|2]), (221,222,[5_1|2]), (222,223,[5_1|2]), (223,224,[1_1|2]), (224,225,[4_1|2]), (225,226,[3_1|2]), (226,227,[5_1|2]), (227,228,[4_1|2]), (228,229,[1_1|2]), (229,153,[3_1|2]), (229,154,[3_1|2]), (229,241,[3_1|2]), (229,344,[3_1|2]), (229,351,[3_1|2]), (229,415,[3_1|2]), (229,180,[3_1|2]), (229,353,[3_1|2]), (229,370,[3_1|2]), (229,382,[3_1|2]), (230,231,[1_1|2]), (231,232,[0_1|2]), (232,233,[3_1|2]), (233,234,[1_1|2]), (234,235,[4_1|2]), (235,236,[0_1|2]), (236,237,[2_1|2]), (237,238,[2_1|2]), (238,239,[0_1|2]), (239,240,[3_1|2]), (240,153,[3_1|2]), (240,154,[3_1|2]), (240,241,[3_1|2]), (240,344,[3_1|2]), (240,351,[3_1|2]), (240,415,[3_1|2]), (240,370,[3_1|2]), (240,382,[3_1|2]), (241,242,[3_1|2]), (242,243,[4_1|2]), (243,244,[0_1|2]), (244,245,[5_1|2]), (245,246,[5_1|2]), (246,247,[4_1|2]), (247,248,[3_1|2]), (248,249,[4_1|2]), (249,250,[3_1|2]), (250,251,[5_1|2]), (251,252,[4_1|2]), (252,253,[1_1|2]), (253,254,[4_1|2]), (254,255,[2_1|2]), (255,256,[5_1|2]), (256,257,[4_1|2]), (257,258,[1_1|2]), (257,307,[4_1|2]), (258,259,[4_1|2]), (258,260,[5_1|2]), (258,268,[4_1|2]), (259,153,[1_1|2]), (259,157,[1_1|2]), (259,214,[1_1|2]), (259,302,[1_1|2]), (259,325,[1_1|2]), (259,446,[1_1|2]), (259,307,[4_1|2]), (260,261,[3_1|2]), (261,262,[0_1|2]), (262,263,[1_1|2]), (263,264,[4_1|2]), (264,265,[3_1|2]), (265,266,[4_1|2]), (266,267,[4_1|2]), (267,153,[2_1|2]), (267,179,[2_1|2]), (267,400,[2_1|2]), (267,191,[2_1|2]), (267,344,[2_1|2]), (267,351,[2_1|2]), (268,269,[2_1|2]), (269,270,[5_1|2]), (270,271,[4_1|2]), (271,272,[2_1|2]), (272,273,[3_1|2]), (273,274,[0_1|2]), (274,275,[0_1|2]), (275,276,[5_1|2]), (276,277,[3_1|2]), (277,278,[5_1|2]), (278,279,[3_1|2]), (279,280,[5_1|2]), (280,281,[5_1|2]), (281,282,[2_1|2]), (282,283,[2_1|2]), (283,284,[5_1|2]), (284,153,[2_1|2]), (284,154,[2_1|2]), (284,241,[2_1|2]), (284,344,[2_1|2]), (284,351,[2_1|2]), (284,415,[2_1|2]), (285,286,[5_1|2]), (286,287,[3_1|2]), (287,288,[2_1|2]), (288,289,[1_1|2]), (289,290,[5_1|2]), (290,291,[5_1|2]), (291,292,[4_1|2]), (292,293,[4_1|2]), (293,294,[3_1|2]), (294,295,[2_1|2]), (295,296,[3_1|2]), (296,297,[5_1|2]), (297,298,[5_1|2]), (298,299,[0_1|2]), (299,300,[0_1|2]), (300,301,[5_1|2]), (300,463,[5_1|2]), (301,153,[0_1|2]), (301,179,[0_1|2]), (301,400,[0_1|2]), (301,154,[2_1|2]), (301,157,[1_1|2]), (301,161,[4_1|2]), (301,190,[4_1|2]), (302,303,[0_1|2]), (303,304,[3_1|2]), (304,305,[4_1|2]), (304,260,[5_1|2]), (305,306,[1_1|2]), (306,153,[0_1|2]), (306,179,[0_1|2]), (306,400,[0_1|2]), (306,154,[2_1|2]), (306,157,[1_1|2]), (306,161,[4_1|2]), (306,190,[4_1|2]), (307,308,[4_1|2]), (308,309,[3_1|2]), (309,310,[5_1|2]), (310,311,[4_1|2]), (311,312,[2_1|2]), (312,313,[4_1|2]), (313,314,[0_1|2]), (314,315,[0_1|2]), (315,316,[4_1|2]), (316,317,[4_1|2]), (317,318,[4_1|2]), (318,319,[0_1|2]), (319,320,[3_1|2]), (320,321,[0_1|2]), (321,322,[1_1|2]), (322,323,[0_1|2]), (323,324,[0_1|2]), (324,153,[0_1|2]), (324,260,[0_1|2]), (324,430,[0_1|2]), (324,463,[0_1|2]), (324,417,[0_1|2]), (324,154,[2_1|2]), (324,157,[1_1|2]), (324,161,[4_1|2]), (324,179,[0_1|2]), (324,190,[4_1|2]), (325,326,[5_1|2]), (326,327,[0_1|2]), (327,328,[3_1|2]), (328,329,[4_1|2]), (329,330,[1_1|2]), (330,331,[5_1|2]), (331,332,[3_1|2]), (332,333,[3_1|2]), (333,334,[5_1|2]), (334,335,[1_1|2]), (335,336,[0_1|2]), (336,337,[0_1|2]), (337,338,[4_1|2]), (338,339,[3_1|2]), (339,340,[4_1|2]), (340,341,[2_1|2]), (341,342,[3_1|2]), (342,343,[0_1|2]), (342,190,[4_1|2]), (343,153,[4_1|2]), (343,161,[4_1|2]), (343,190,[4_1|2]), (343,202,[4_1|2]), (343,207,[4_1|2]), (343,230,[4_1|2]), (343,268,[4_1|2]), (343,285,[4_1|2]), (343,307,[4_1|2]), (343,448,[4_1|2]), (343,214,[1_1|2]), (343,241,[2_1|2]), (343,260,[5_1|2]), (344,345,[5_1|2]), (345,346,[5_1|2]), (346,347,[3_1|2]), (347,348,[1_1|2]), (348,349,[3_1|2]), (349,350,[1_1|2]), (350,153,[0_1|2]), (350,260,[0_1|2]), (350,430,[0_1|2]), (350,463,[0_1|2]), (350,326,[0_1|2]), (350,447,[0_1|2]), (350,154,[2_1|2]), (350,157,[1_1|2]), (350,161,[4_1|2]), (350,179,[0_1|2]), (350,190,[4_1|2]), (351,352,[0_1|2]), (352,353,[2_1|2]), (353,354,[4_1|2]), (354,355,[5_1|2]), (355,356,[0_1|2]), (356,357,[0_1|2]), (357,358,[5_1|2]), (358,359,[1_1|2]), (359,360,[3_1|2]), (360,361,[0_1|2]), (361,362,[5_1|2]), (362,363,[0_1|2]), (363,364,[2_1|2]), (364,365,[1_1|2]), (365,366,[4_1|2]), (366,367,[3_1|2]), (367,368,[3_1|2]), (368,369,[3_1|2]), (369,153,[0_1|2]), (369,157,[0_1|2, 1_1|2]), (369,214,[0_1|2]), (369,302,[0_1|2]), (369,325,[0_1|2]), (369,446,[0_1|2]), (369,155,[0_1|2]), (369,154,[2_1|2]), (369,161,[4_1|2]), (369,179,[0_1|2]), (369,190,[4_1|2]), (370,371,[2_1|2]), (371,372,[4_1|2]), (372,373,[1_1|2]), (373,374,[4_1|2]), (374,375,[5_1|2]), (375,376,[3_1|2]), (376,377,[2_1|2]), (377,378,[0_1|2]), (378,379,[1_1|2]), (379,380,[0_1|2]), (380,381,[4_1|2]), (381,153,[2_1|2]), (381,154,[2_1|2]), (381,241,[2_1|2]), (381,344,[2_1|2]), (381,351,[2_1|2]), (381,415,[2_1|2]), (382,383,[2_1|2]), (383,384,[0_1|2]), (384,385,[1_1|2]), (385,386,[2_1|2]), (386,387,[3_1|2]), (387,388,[0_1|2]), (388,389,[4_1|2]), (389,390,[4_1|2]), (390,391,[2_1|2]), (391,392,[4_1|2]), (392,393,[4_1|2]), (393,394,[3_1|2]), (394,395,[0_1|2]), (395,396,[0_1|2]), (396,397,[5_1|2]), (397,398,[4_1|2]), (398,399,[4_1|2]), (398,202,[4_1|2]), (398,207,[4_1|2]), (398,214,[1_1|2]), (398,230,[4_1|2]), (398,241,[2_1|2]), (399,153,[3_1|2]), (399,157,[3_1|2]), (399,214,[3_1|2]), (399,302,[3_1|2]), (399,325,[3_1|2]), (399,446,[3_1|2]), (399,431,[3_1|2]), (399,370,[3_1|2]), (399,382,[3_1|2]), (400,401,[1_1|2]), (401,402,[5_1|2]), (402,403,[0_1|2]), (403,404,[4_1|2]), (404,405,[0_1|2]), (405,406,[2_1|2]), (406,407,[1_1|2]), (407,408,[4_1|2]), (408,409,[4_1|2]), (409,410,[2_1|2]), (410,411,[5_1|2]), (411,412,[5_1|2]), (412,413,[3_1|2]), (413,414,[0_1|2]), (413,190,[4_1|2]), (414,153,[4_1|2]), (414,260,[4_1|2, 5_1|2]), (414,430,[4_1|2]), (414,463,[4_1|2]), (414,326,[4_1|2]), (414,447,[4_1|2]), (414,202,[4_1|2]), (414,207,[4_1|2]), (414,214,[1_1|2]), (414,230,[4_1|2]), (414,241,[2_1|2]), (414,268,[4_1|2]), (414,285,[4_1|2]), (415,416,[3_1|2]), (416,417,[5_1|2]), (417,418,[0_1|2]), (418,419,[0_1|2]), (419,420,[0_1|2]), (420,421,[3_1|2]), (421,422,[2_1|2]), (422,423,[1_1|2]), (423,424,[0_1|2]), (424,425,[3_1|2]), (425,426,[0_1|2]), (426,427,[4_1|2]), (427,428,[3_1|2]), (428,429,[3_1|2]), (428,370,[3_1|2]), (428,382,[3_1|2]), (429,153,[2_1|2]), (429,154,[2_1|2]), (429,241,[2_1|2]), (429,344,[2_1|2]), (429,351,[2_1|2]), (429,415,[2_1|2]), (429,371,[2_1|2]), (429,383,[2_1|2]), (430,431,[1_1|2]), (431,432,[5_1|2]), (432,433,[3_1|2]), (433,434,[0_1|2]), (434,435,[1_1|2]), (435,436,[2_1|2]), (436,437,[3_1|2]), (437,438,[2_1|2]), (438,439,[4_1|2]), (439,440,[5_1|2]), (440,441,[2_1|2]), (441,442,[0_1|2]), (442,443,[5_1|2]), (443,444,[3_1|2]), (444,445,[3_1|2]), (445,153,[0_1|2]), (445,179,[0_1|2]), (445,400,[0_1|2]), (445,154,[2_1|2]), (445,157,[1_1|2]), (445,161,[4_1|2]), (445,190,[4_1|2]), (446,447,[5_1|2]), (447,448,[4_1|2]), (448,449,[2_1|2]), (449,450,[2_1|2]), (450,451,[2_1|2]), (451,452,[2_1|2]), (452,453,[4_1|2]), (453,454,[2_1|2]), (454,455,[4_1|2]), (455,456,[5_1|2]), (456,457,[1_1|2]), (457,458,[2_1|2]), (458,459,[4_1|2]), (459,460,[3_1|2]), (460,461,[1_1|2]), (461,462,[3_1|2]), (461,370,[3_1|2]), (461,382,[3_1|2]), (462,153,[2_1|2]), (462,154,[2_1|2]), (462,241,[2_1|2]), (462,344,[2_1|2]), (462,351,[2_1|2]), (462,415,[2_1|2]), (463,464,[0_1|2]), (464,465,[1_1|2]), (465,466,[5_1|2]), (466,467,[0_1|2]), (467,468,[5_1|2]), (468,469,[5_1|2]), (469,470,[5_1|2]), (470,471,[3_1|2]), (471,472,[0_1|2]), (472,473,[0_1|2]), (473,474,[0_1|2]), (474,475,[1_1|2]), (475,476,[2_1|2]), (476,477,[0_1|2]), (477,478,[3_1|2]), (478,479,[1_1|2]), (479,480,[1_1|2]), (480,481,[1_1|2]), (481,482,[2_1|2]), (481,344,[2_1|2]), (482,153,[4_1|2]), (482,161,[4_1|2]), (482,190,[4_1|2]), (482,202,[4_1|2]), (482,207,[4_1|2]), (482,230,[4_1|2]), (482,268,[4_1|2]), (482,285,[4_1|2]), (482,307,[4_1|2]), (482,214,[1_1|2]), (482,241,[2_1|2]), (482,260,[5_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)