/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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, 157 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(0(0(x1)))) -> 1(1(2(x1))) 2(3(3(4(x1)))) -> 4(1(4(x1))) 4(2(4(0(x1)))) -> 0(0(1(1(x1)))) 2(1(0(1(0(x1))))) -> 1(3(4(0(x1)))) 5(3(3(4(5(x1))))) -> 5(4(1(5(x1)))) 0(3(4(5(3(0(x1)))))) -> 0(3(5(1(0(2(x1)))))) 2(4(1(1(3(5(x1)))))) -> 2(0(4(4(5(x1))))) 0(3(1(2(1(1(0(x1))))))) -> 1(1(3(4(2(0(x1)))))) 0(3(5(4(5(3(0(x1))))))) -> 0(3(4(3(5(0(4(x1))))))) 2(1(1(3(5(5(3(x1))))))) -> 1(3(5(4(0(5(x1)))))) 1(2(3(3(3(5(1(1(x1)))))))) -> 1(5(0(1(5(0(1(x1))))))) 2(0(2(2(4(3(1(0(x1)))))))) -> 2(1(1(5(4(2(2(x1))))))) 3(0(5(0(5(0(3(3(x1)))))))) -> 5(5(1(0(2(4(1(x1))))))) 3(3(2(4(2(3(4(3(3(x1))))))))) -> 5(4(4(5(0(4(3(x1))))))) 4(2(0(2(5(2(2(3(1(0(x1)))))))))) -> 5(0(5(1(2(5(0(5(x1)))))))) 4(3(5(5(2(5(3(3(0(1(x1)))))))))) -> 4(2(1(4(0(2(3(3(4(2(x1)))))))))) 1(3(1(1(0(0(2(0(3(4(0(x1))))))))))) -> 1(2(0(0(2(3(4(3(2(1(x1)))))))))) 2(1(2(1(3(2(4(4(0(0(3(x1))))))))))) -> 5(1(0(5(3(1(4(3(0(3(x1)))))))))) 4(0(0(1(0(4(3(3(0(2(2(x1))))))))))) -> 5(3(5(3(4(2(4(4(2(x1))))))))) 1(3(1(1(1(3(0(2(0(0(3(0(x1)))))))))))) -> 1(3(1(4(5(5(4(3(0(5(x1)))))))))) 3(3(5(5(5(5(2(4(1(0(1(5(x1)))))))))))) -> 5(0(5(1(1(1(3(0(1(2(0(5(x1)))))))))))) 5(2(1(4(2(2(1(2(4(1(2(0(x1)))))))))))) -> 5(2(5(2(2(4(1(3(4(0(4(x1))))))))))) 3(5(1(4(4(1(2(2(5(2(3(4(4(x1))))))))))))) -> 0(0(3(2(3(5(2(4(0(1(0(0(4(5(x1)))))))))))))) 4(0(3(4(0(4(3(0(2(3(0(5(2(x1))))))))))))) -> 2(3(4(5(1(3(5(0(5(5(2(2(x1)))))))))))) 0(5(3(3(0(1(4(5(4(4(1(1(3(0(x1)))))))))))))) -> 0(1(5(0(3(5(2(4(0(5(3(1(3(2(x1)))))))))))))) 2(3(0(4(3(3(1(1(0(2(1(4(5(5(x1)))))))))))))) -> 2(5(0(2(2(2(0(3(5(3(0(4(5(x1))))))))))))) 3(5(0(2(5(3(2(2(1(0(3(0(0(2(x1)))))))))))))) -> 3(5(3(2(1(2(3(4(3(1(1(0(5(x1))))))))))))) 4(1(0(5(1(2(5(5(3(4(2(1(3(0(0(x1))))))))))))))) -> 1(5(4(5(0(3(5(1(2(4(3(3(0(2(x1)))))))))))))) 3(0(0(4(1(4(0(1(4(4(0(4(4(5(5(0(x1)))))))))))))))) -> 2(3(0(4(4(1(1(1(0(2(0(0(1(4(0(5(1(x1))))))))))))))))) 3(1(0(2(5(0(0(4(1(3(0(1(3(4(2(3(x1)))))))))))))))) -> 5(3(0(3(1(3(4(5(2(1(2(4(3(4(3(x1))))))))))))))) 2(0(2(3(5(1(1(5(4(4(3(4(2(2(2(4(5(x1))))))))))))))))) -> 2(2(5(3(0(3(5(4(3(4(0(1(5(4(5(5(x1)))))))))))))))) 2(5(4(3(0(1(2(2(5(2(3(4(3(3(5(0(2(x1))))))))))))))))) -> 4(4(3(3(4(0(2(1(3(3(5(5(0(3(3(1(4(x1))))))))))))))))) 5(3(3(3(5(0(2(1(0(0(4(5(1(3(2(0(0(x1))))))))))))))))) -> 5(4(3(3(0(3(1(2(5(2(4(3(1(1(4(1(x1)))))))))))))))) 5(4(3(4(3(4(5(2(2(0(5(0(4(3(4(0(2(2(x1)))))))))))))))))) -> 5(1(2(5(1(1(3(3(0(4(3(2(5(2(1(2(4(2(x1)))))))))))))))))) 1(0(3(1(1(2(2(0(3(3(2(0(0(1(1(5(3(5(2(x1))))))))))))))))))) -> 0(1(3(1(1(4(5(0(1(2(1(4(2(2(2(4(4(x1))))))))))))))))) 2(4(5(2(3(2(5(1(4(1(3(5(0(0(1(1(3(1(4(x1))))))))))))))))))) -> 5(2(4(4(4(4(5(4(5(3(4(1(4(5(5(0(x1)))))))))))))))) 4(0(0(1(2(4(0(2(4(1(2(1(0(4(3(1(1(0(2(x1))))))))))))))))))) -> 4(0(3(1(4(1(0(3(1(0(4(5(3(0(0(1(4(4(x1)))))))))))))))))) 1(2(4(1(0(2(1(5(0(0(5(1(4(0(4(5(3(1(2(5(0(x1))))))))))))))))))))) -> 1(0(2(3(5(4(1(1(1(1(1(5(5(4(1(4(4(4(1(2(0(x1))))))))))))))))))))) 2(4(4(3(3(3(5(2(3(0(2(5(1(3(3(0(4(3(4(0(1(x1))))))))))))))))))))) -> 5(1(1(4(1(4(2(5(2(3(0(1(3(1(1(5(5(1(3(3(x1)))))))))))))))))))) 5(3(4(2(5(2(5(1(1(5(3(5(1(5(0(4(1(0(2(2(0(x1))))))))))))))))))))) -> 5(0(3(5(4(5(5(1(2(2(1(4(0(0(1(2(5(1(5(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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(0(x1)))) -> 1(1(2(x1))) 2(3(3(4(x1)))) -> 4(1(4(x1))) 4(2(4(0(x1)))) -> 0(0(1(1(x1)))) 2(1(0(1(0(x1))))) -> 1(3(4(0(x1)))) 5(3(3(4(5(x1))))) -> 5(4(1(5(x1)))) 0(3(4(5(3(0(x1)))))) -> 0(3(5(1(0(2(x1)))))) 2(4(1(1(3(5(x1)))))) -> 2(0(4(4(5(x1))))) 0(3(1(2(1(1(0(x1))))))) -> 1(1(3(4(2(0(x1)))))) 0(3(5(4(5(3(0(x1))))))) -> 0(3(4(3(5(0(4(x1))))))) 2(1(1(3(5(5(3(x1))))))) -> 1(3(5(4(0(5(x1)))))) 1(2(3(3(3(5(1(1(x1)))))))) -> 1(5(0(1(5(0(1(x1))))))) 2(0(2(2(4(3(1(0(x1)))))))) -> 2(1(1(5(4(2(2(x1))))))) 3(0(5(0(5(0(3(3(x1)))))))) -> 5(5(1(0(2(4(1(x1))))))) 3(3(2(4(2(3(4(3(3(x1))))))))) -> 5(4(4(5(0(4(3(x1))))))) 4(2(0(2(5(2(2(3(1(0(x1)))))))))) -> 5(0(5(1(2(5(0(5(x1)))))))) 4(3(5(5(2(5(3(3(0(1(x1)))))))))) -> 4(2(1(4(0(2(3(3(4(2(x1)))))))))) 1(3(1(1(0(0(2(0(3(4(0(x1))))))))))) -> 1(2(0(0(2(3(4(3(2(1(x1)))))))))) 2(1(2(1(3(2(4(4(0(0(3(x1))))))))))) -> 5(1(0(5(3(1(4(3(0(3(x1)))))))))) 4(0(0(1(0(4(3(3(0(2(2(x1))))))))))) -> 5(3(5(3(4(2(4(4(2(x1))))))))) 1(3(1(1(1(3(0(2(0(0(3(0(x1)))))))))))) -> 1(3(1(4(5(5(4(3(0(5(x1)))))))))) 3(3(5(5(5(5(2(4(1(0(1(5(x1)))))))))))) -> 5(0(5(1(1(1(3(0(1(2(0(5(x1)))))))))))) 5(2(1(4(2(2(1(2(4(1(2(0(x1)))))))))))) -> 5(2(5(2(2(4(1(3(4(0(4(x1))))))))))) 3(5(1(4(4(1(2(2(5(2(3(4(4(x1))))))))))))) -> 0(0(3(2(3(5(2(4(0(1(0(0(4(5(x1)))))))))))))) 4(0(3(4(0(4(3(0(2(3(0(5(2(x1))))))))))))) -> 2(3(4(5(1(3(5(0(5(5(2(2(x1)))))))))))) 0(5(3(3(0(1(4(5(4(4(1(1(3(0(x1)))))))))))))) -> 0(1(5(0(3(5(2(4(0(5(3(1(3(2(x1)))))))))))))) 2(3(0(4(3(3(1(1(0(2(1(4(5(5(x1)))))))))))))) -> 2(5(0(2(2(2(0(3(5(3(0(4(5(x1))))))))))))) 3(5(0(2(5(3(2(2(1(0(3(0(0(2(x1)))))))))))))) -> 3(5(3(2(1(2(3(4(3(1(1(0(5(x1))))))))))))) 4(1(0(5(1(2(5(5(3(4(2(1(3(0(0(x1))))))))))))))) -> 1(5(4(5(0(3(5(1(2(4(3(3(0(2(x1)))))))))))))) 3(0(0(4(1(4(0(1(4(4(0(4(4(5(5(0(x1)))))))))))))))) -> 2(3(0(4(4(1(1(1(0(2(0(0(1(4(0(5(1(x1))))))))))))))))) 3(1(0(2(5(0(0(4(1(3(0(1(3(4(2(3(x1)))))))))))))))) -> 5(3(0(3(1(3(4(5(2(1(2(4(3(4(3(x1))))))))))))))) 2(0(2(3(5(1(1(5(4(4(3(4(2(2(2(4(5(x1))))))))))))))))) -> 2(2(5(3(0(3(5(4(3(4(0(1(5(4(5(5(x1)))))))))))))))) 2(5(4(3(0(1(2(2(5(2(3(4(3(3(5(0(2(x1))))))))))))))))) -> 4(4(3(3(4(0(2(1(3(3(5(5(0(3(3(1(4(x1))))))))))))))))) 5(3(3(3(5(0(2(1(0(0(4(5(1(3(2(0(0(x1))))))))))))))))) -> 5(4(3(3(0(3(1(2(5(2(4(3(1(1(4(1(x1)))))))))))))))) 5(4(3(4(3(4(5(2(2(0(5(0(4(3(4(0(2(2(x1)))))))))))))))))) -> 5(1(2(5(1(1(3(3(0(4(3(2(5(2(1(2(4(2(x1)))))))))))))))))) 1(0(3(1(1(2(2(0(3(3(2(0(0(1(1(5(3(5(2(x1))))))))))))))))))) -> 0(1(3(1(1(4(5(0(1(2(1(4(2(2(2(4(4(x1))))))))))))))))) 2(4(5(2(3(2(5(1(4(1(3(5(0(0(1(1(3(1(4(x1))))))))))))))))))) -> 5(2(4(4(4(4(5(4(5(3(4(1(4(5(5(0(x1)))))))))))))))) 4(0(0(1(2(4(0(2(4(1(2(1(0(4(3(1(1(0(2(x1))))))))))))))))))) -> 4(0(3(1(4(1(0(3(1(0(4(5(3(0(0(1(4(4(x1)))))))))))))))))) 1(2(4(1(0(2(1(5(0(0(5(1(4(0(4(5(3(1(2(5(0(x1))))))))))))))))))))) -> 1(0(2(3(5(4(1(1(1(1(1(5(5(4(1(4(4(4(1(2(0(x1))))))))))))))))))))) 2(4(4(3(3(3(5(2(3(0(2(5(1(3(3(0(4(3(4(0(1(x1))))))))))))))))))))) -> 5(1(1(4(1(4(2(5(2(3(0(1(3(1(1(5(5(1(3(3(x1)))))))))))))))))))) 5(3(4(2(5(2(5(1(1(5(3(5(1(5(0(4(1(0(2(2(0(x1))))))))))))))))))))) -> 5(0(3(5(4(5(5(1(2(2(1(4(0(0(1(2(5(1(5(4(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(0(x1)))) -> 1(1(2(x1))) 2(3(3(4(x1)))) -> 4(1(4(x1))) 4(2(4(0(x1)))) -> 0(0(1(1(x1)))) 2(1(0(1(0(x1))))) -> 1(3(4(0(x1)))) 5(3(3(4(5(x1))))) -> 5(4(1(5(x1)))) 0(3(4(5(3(0(x1)))))) -> 0(3(5(1(0(2(x1)))))) 2(4(1(1(3(5(x1)))))) -> 2(0(4(4(5(x1))))) 0(3(1(2(1(1(0(x1))))))) -> 1(1(3(4(2(0(x1)))))) 0(3(5(4(5(3(0(x1))))))) -> 0(3(4(3(5(0(4(x1))))))) 2(1(1(3(5(5(3(x1))))))) -> 1(3(5(4(0(5(x1)))))) 1(2(3(3(3(5(1(1(x1)))))))) -> 1(5(0(1(5(0(1(x1))))))) 2(0(2(2(4(3(1(0(x1)))))))) -> 2(1(1(5(4(2(2(x1))))))) 3(0(5(0(5(0(3(3(x1)))))))) -> 5(5(1(0(2(4(1(x1))))))) 3(3(2(4(2(3(4(3(3(x1))))))))) -> 5(4(4(5(0(4(3(x1))))))) 4(2(0(2(5(2(2(3(1(0(x1)))))))))) -> 5(0(5(1(2(5(0(5(x1)))))))) 4(3(5(5(2(5(3(3(0(1(x1)))))))))) -> 4(2(1(4(0(2(3(3(4(2(x1)))))))))) 1(3(1(1(0(0(2(0(3(4(0(x1))))))))))) -> 1(2(0(0(2(3(4(3(2(1(x1)))))))))) 2(1(2(1(3(2(4(4(0(0(3(x1))))))))))) -> 5(1(0(5(3(1(4(3(0(3(x1)))))))))) 4(0(0(1(0(4(3(3(0(2(2(x1))))))))))) -> 5(3(5(3(4(2(4(4(2(x1))))))))) 1(3(1(1(1(3(0(2(0(0(3(0(x1)))))))))))) -> 1(3(1(4(5(5(4(3(0(5(x1)))))))))) 3(3(5(5(5(5(2(4(1(0(1(5(x1)))))))))))) -> 5(0(5(1(1(1(3(0(1(2(0(5(x1)))))))))))) 5(2(1(4(2(2(1(2(4(1(2(0(x1)))))))))))) -> 5(2(5(2(2(4(1(3(4(0(4(x1))))))))))) 3(5(1(4(4(1(2(2(5(2(3(4(4(x1))))))))))))) -> 0(0(3(2(3(5(2(4(0(1(0(0(4(5(x1)))))))))))))) 4(0(3(4(0(4(3(0(2(3(0(5(2(x1))))))))))))) -> 2(3(4(5(1(3(5(0(5(5(2(2(x1)))))))))))) 0(5(3(3(0(1(4(5(4(4(1(1(3(0(x1)))))))))))))) -> 0(1(5(0(3(5(2(4(0(5(3(1(3(2(x1)))))))))))))) 2(3(0(4(3(3(1(1(0(2(1(4(5(5(x1)))))))))))))) -> 2(5(0(2(2(2(0(3(5(3(0(4(5(x1))))))))))))) 3(5(0(2(5(3(2(2(1(0(3(0(0(2(x1)))))))))))))) -> 3(5(3(2(1(2(3(4(3(1(1(0(5(x1))))))))))))) 4(1(0(5(1(2(5(5(3(4(2(1(3(0(0(x1))))))))))))))) -> 1(5(4(5(0(3(5(1(2(4(3(3(0(2(x1)))))))))))))) 3(0(0(4(1(4(0(1(4(4(0(4(4(5(5(0(x1)))))))))))))))) -> 2(3(0(4(4(1(1(1(0(2(0(0(1(4(0(5(1(x1))))))))))))))))) 3(1(0(2(5(0(0(4(1(3(0(1(3(4(2(3(x1)))))))))))))))) -> 5(3(0(3(1(3(4(5(2(1(2(4(3(4(3(x1))))))))))))))) 2(0(2(3(5(1(1(5(4(4(3(4(2(2(2(4(5(x1))))))))))))))))) -> 2(2(5(3(0(3(5(4(3(4(0(1(5(4(5(5(x1)))))))))))))))) 2(5(4(3(0(1(2(2(5(2(3(4(3(3(5(0(2(x1))))))))))))))))) -> 4(4(3(3(4(0(2(1(3(3(5(5(0(3(3(1(4(x1))))))))))))))))) 5(3(3(3(5(0(2(1(0(0(4(5(1(3(2(0(0(x1))))))))))))))))) -> 5(4(3(3(0(3(1(2(5(2(4(3(1(1(4(1(x1)))))))))))))))) 5(4(3(4(3(4(5(2(2(0(5(0(4(3(4(0(2(2(x1)))))))))))))))))) -> 5(1(2(5(1(1(3(3(0(4(3(2(5(2(1(2(4(2(x1)))))))))))))))))) 1(0(3(1(1(2(2(0(3(3(2(0(0(1(1(5(3(5(2(x1))))))))))))))))))) -> 0(1(3(1(1(4(5(0(1(2(1(4(2(2(2(4(4(x1))))))))))))))))) 2(4(5(2(3(2(5(1(4(1(3(5(0(0(1(1(3(1(4(x1))))))))))))))))))) -> 5(2(4(4(4(4(5(4(5(3(4(1(4(5(5(0(x1)))))))))))))))) 4(0(0(1(2(4(0(2(4(1(2(1(0(4(3(1(1(0(2(x1))))))))))))))))))) -> 4(0(3(1(4(1(0(3(1(0(4(5(3(0(0(1(4(4(x1)))))))))))))))))) 1(2(4(1(0(2(1(5(0(0(5(1(4(0(4(5(3(1(2(5(0(x1))))))))))))))))))))) -> 1(0(2(3(5(4(1(1(1(1(1(5(5(4(1(4(4(4(1(2(0(x1))))))))))))))))))))) 2(4(4(3(3(3(5(2(3(0(2(5(1(3(3(0(4(3(4(0(1(x1))))))))))))))))))))) -> 5(1(1(4(1(4(2(5(2(3(0(1(3(1(1(5(5(1(3(3(x1)))))))))))))))))))) 5(3(4(2(5(2(5(1(1(5(3(5(1(5(0(4(1(0(2(2(0(x1))))))))))))))))))))) -> 5(0(3(5(4(5(5(1(2(2(1(4(0(0(1(2(5(1(5(4(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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(0(0(x1)))) -> 1(1(2(x1))) 2(3(3(4(x1)))) -> 4(1(4(x1))) 4(2(4(0(x1)))) -> 0(0(1(1(x1)))) 2(1(0(1(0(x1))))) -> 1(3(4(0(x1)))) 5(3(3(4(5(x1))))) -> 5(4(1(5(x1)))) 0(3(4(5(3(0(x1)))))) -> 0(3(5(1(0(2(x1)))))) 2(4(1(1(3(5(x1)))))) -> 2(0(4(4(5(x1))))) 0(3(1(2(1(1(0(x1))))))) -> 1(1(3(4(2(0(x1)))))) 0(3(5(4(5(3(0(x1))))))) -> 0(3(4(3(5(0(4(x1))))))) 2(1(1(3(5(5(3(x1))))))) -> 1(3(5(4(0(5(x1)))))) 1(2(3(3(3(5(1(1(x1)))))))) -> 1(5(0(1(5(0(1(x1))))))) 2(0(2(2(4(3(1(0(x1)))))))) -> 2(1(1(5(4(2(2(x1))))))) 3(0(5(0(5(0(3(3(x1)))))))) -> 5(5(1(0(2(4(1(x1))))))) 3(3(2(4(2(3(4(3(3(x1))))))))) -> 5(4(4(5(0(4(3(x1))))))) 4(2(0(2(5(2(2(3(1(0(x1)))))))))) -> 5(0(5(1(2(5(0(5(x1)))))))) 4(3(5(5(2(5(3(3(0(1(x1)))))))))) -> 4(2(1(4(0(2(3(3(4(2(x1)))))))))) 1(3(1(1(0(0(2(0(3(4(0(x1))))))))))) -> 1(2(0(0(2(3(4(3(2(1(x1)))))))))) 2(1(2(1(3(2(4(4(0(0(3(x1))))))))))) -> 5(1(0(5(3(1(4(3(0(3(x1)))))))))) 4(0(0(1(0(4(3(3(0(2(2(x1))))))))))) -> 5(3(5(3(4(2(4(4(2(x1))))))))) 1(3(1(1(1(3(0(2(0(0(3(0(x1)))))))))))) -> 1(3(1(4(5(5(4(3(0(5(x1)))))))))) 3(3(5(5(5(5(2(4(1(0(1(5(x1)))))))))))) -> 5(0(5(1(1(1(3(0(1(2(0(5(x1)))))))))))) 5(2(1(4(2(2(1(2(4(1(2(0(x1)))))))))))) -> 5(2(5(2(2(4(1(3(4(0(4(x1))))))))))) 3(5(1(4(4(1(2(2(5(2(3(4(4(x1))))))))))))) -> 0(0(3(2(3(5(2(4(0(1(0(0(4(5(x1)))))))))))))) 4(0(3(4(0(4(3(0(2(3(0(5(2(x1))))))))))))) -> 2(3(4(5(1(3(5(0(5(5(2(2(x1)))))))))))) 0(5(3(3(0(1(4(5(4(4(1(1(3(0(x1)))))))))))))) -> 0(1(5(0(3(5(2(4(0(5(3(1(3(2(x1)))))))))))))) 2(3(0(4(3(3(1(1(0(2(1(4(5(5(x1)))))))))))))) -> 2(5(0(2(2(2(0(3(5(3(0(4(5(x1))))))))))))) 3(5(0(2(5(3(2(2(1(0(3(0(0(2(x1)))))))))))))) -> 3(5(3(2(1(2(3(4(3(1(1(0(5(x1))))))))))))) 4(1(0(5(1(2(5(5(3(4(2(1(3(0(0(x1))))))))))))))) -> 1(5(4(5(0(3(5(1(2(4(3(3(0(2(x1)))))))))))))) 3(0(0(4(1(4(0(1(4(4(0(4(4(5(5(0(x1)))))))))))))))) -> 2(3(0(4(4(1(1(1(0(2(0(0(1(4(0(5(1(x1))))))))))))))))) 3(1(0(2(5(0(0(4(1(3(0(1(3(4(2(3(x1)))))))))))))))) -> 5(3(0(3(1(3(4(5(2(1(2(4(3(4(3(x1))))))))))))))) 2(0(2(3(5(1(1(5(4(4(3(4(2(2(2(4(5(x1))))))))))))))))) -> 2(2(5(3(0(3(5(4(3(4(0(1(5(4(5(5(x1)))))))))))))))) 2(5(4(3(0(1(2(2(5(2(3(4(3(3(5(0(2(x1))))))))))))))))) -> 4(4(3(3(4(0(2(1(3(3(5(5(0(3(3(1(4(x1))))))))))))))))) 5(3(3(3(5(0(2(1(0(0(4(5(1(3(2(0(0(x1))))))))))))))))) -> 5(4(3(3(0(3(1(2(5(2(4(3(1(1(4(1(x1)))))))))))))))) 5(4(3(4(3(4(5(2(2(0(5(0(4(3(4(0(2(2(x1)))))))))))))))))) -> 5(1(2(5(1(1(3(3(0(4(3(2(5(2(1(2(4(2(x1)))))))))))))))))) 1(0(3(1(1(2(2(0(3(3(2(0(0(1(1(5(3(5(2(x1))))))))))))))))))) -> 0(1(3(1(1(4(5(0(1(2(1(4(2(2(2(4(4(x1))))))))))))))))) 2(4(5(2(3(2(5(1(4(1(3(5(0(0(1(1(3(1(4(x1))))))))))))))))))) -> 5(2(4(4(4(4(5(4(5(3(4(1(4(5(5(0(x1)))))))))))))))) 4(0(0(1(2(4(0(2(4(1(2(1(0(4(3(1(1(0(2(x1))))))))))))))))))) -> 4(0(3(1(4(1(0(3(1(0(4(5(3(0(0(1(4(4(x1)))))))))))))))))) 1(2(4(1(0(2(1(5(0(0(5(1(4(0(4(5(3(1(2(5(0(x1))))))))))))))))))))) -> 1(0(2(3(5(4(1(1(1(1(1(5(5(4(1(4(4(4(1(2(0(x1))))))))))))))))))))) 2(4(4(3(3(3(5(2(3(0(2(5(1(3(3(0(4(3(4(0(1(x1))))))))))))))))))))) -> 5(1(1(4(1(4(2(5(2(3(0(1(3(1(1(5(5(1(3(3(x1)))))))))))))))))))) 5(3(4(2(5(2(5(1(1(5(3(5(1(5(0(4(1(0(2(2(0(x1))))))))))))))))))))) -> 5(0(3(5(4(5(5(1(2(2(1(4(0(0(1(2(5(1(5(4(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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. "[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, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576] {(148,149,[0_1|0, 2_1|0, 4_1|0, 5_1|0, 1_1|0, 3_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]), (148,150,[0_1|1, 2_1|1, 4_1|1, 5_1|1, 1_1|1, 3_1|1]), (148,151,[1_1|2]), (148,153,[0_1|2]), (148,158,[1_1|2]), (148,163,[0_1|2]), (148,169,[0_1|2]), (148,182,[4_1|2]), (148,184,[2_1|2]), (148,196,[1_1|2]), (148,199,[1_1|2]), (148,204,[5_1|2]), (148,213,[2_1|2]), (148,217,[5_1|2]), (148,232,[5_1|2]), (148,251,[2_1|2]), (148,257,[2_1|2]), (148,272,[4_1|2]), (148,288,[0_1|2]), (148,291,[5_1|2]), (148,298,[4_1|2]), (148,307,[5_1|2]), (148,315,[4_1|2]), (148,332,[2_1|2]), (148,343,[1_1|2]), (148,356,[5_1|2]), (148,359,[5_1|2]), (148,374,[5_1|2]), (148,393,[5_1|2]), (148,403,[5_1|2]), (148,420,[1_1|2]), (148,426,[1_1|2]), (148,446,[1_1|2]), (148,455,[1_1|2]), (148,464,[0_1|2]), (148,480,[5_1|2]), (148,486,[2_1|2]), (148,502,[5_1|2]), (148,508,[5_1|2]), (148,519,[0_1|2]), (148,532,[3_1|2]), (148,544,[5_1|2]), (149,149,[cons_0_1|0, cons_2_1|0, cons_4_1|0, cons_5_1|0, cons_1_1|0, cons_3_1|0]), (150,149,[encArg_1|1]), (150,150,[0_1|1, 2_1|1, 4_1|1, 5_1|1, 1_1|1, 3_1|1]), (150,151,[1_1|2]), (150,153,[0_1|2]), (150,158,[1_1|2]), (150,163,[0_1|2]), (150,169,[0_1|2]), (150,182,[4_1|2]), (150,184,[2_1|2]), (150,196,[1_1|2]), (150,199,[1_1|2]), (150,204,[5_1|2]), (150,213,[2_1|2]), (150,217,[5_1|2]), (150,232,[5_1|2]), (150,251,[2_1|2]), (150,257,[2_1|2]), (150,272,[4_1|2]), (150,288,[0_1|2]), (150,291,[5_1|2]), (150,298,[4_1|2]), (150,307,[5_1|2]), (150,315,[4_1|2]), (150,332,[2_1|2]), (150,343,[1_1|2]), (150,356,[5_1|2]), (150,359,[5_1|2]), (150,374,[5_1|2]), (150,393,[5_1|2]), (150,403,[5_1|2]), (150,420,[1_1|2]), (150,426,[1_1|2]), (150,446,[1_1|2]), (150,455,[1_1|2]), (150,464,[0_1|2]), (150,480,[5_1|2]), (150,486,[2_1|2]), (150,502,[5_1|2]), (150,508,[5_1|2]), (150,519,[0_1|2]), (150,532,[3_1|2]), (150,544,[5_1|2]), (151,152,[1_1|2]), (151,420,[1_1|2]), (151,426,[1_1|2]), (152,150,[2_1|2]), (152,153,[2_1|2]), (152,163,[2_1|2]), (152,169,[2_1|2]), (152,288,[2_1|2]), (152,464,[2_1|2]), (152,519,[2_1|2]), (152,289,[2_1|2]), (152,520,[2_1|2]), (152,182,[4_1|2]), (152,184,[2_1|2]), (152,196,[1_1|2]), (152,199,[1_1|2]), (152,204,[5_1|2]), (152,213,[2_1|2]), (152,217,[5_1|2]), (152,232,[5_1|2]), (152,251,[2_1|2]), (152,257,[2_1|2]), (152,272,[4_1|2]), (153,154,[3_1|2]), (154,155,[5_1|2]), (155,156,[1_1|2]), (156,157,[0_1|2]), (157,150,[2_1|2]), (157,153,[2_1|2]), (157,163,[2_1|2]), (157,169,[2_1|2]), (157,288,[2_1|2]), (157,464,[2_1|2]), (157,519,[2_1|2]), (157,546,[2_1|2]), (157,182,[4_1|2]), (157,184,[2_1|2]), (157,196,[1_1|2]), (157,199,[1_1|2]), (157,204,[5_1|2]), (157,213,[2_1|2]), (157,217,[5_1|2]), (157,232,[5_1|2]), (157,251,[2_1|2]), (157,257,[2_1|2]), (157,272,[4_1|2]), (158,159,[1_1|2]), (159,160,[3_1|2]), (160,161,[4_1|2]), (160,291,[5_1|2]), (161,162,[2_1|2]), (161,251,[2_1|2]), (161,257,[2_1|2]), (162,150,[0_1|2]), (162,153,[0_1|2]), (162,163,[0_1|2]), (162,169,[0_1|2]), (162,288,[0_1|2]), (162,464,[0_1|2]), (162,519,[0_1|2]), (162,427,[0_1|2]), (162,151,[1_1|2]), (162,158,[1_1|2]), (163,164,[3_1|2]), (164,165,[4_1|2]), (165,166,[3_1|2]), (166,167,[5_1|2]), (167,168,[0_1|2]), (168,150,[4_1|2]), (168,153,[4_1|2]), (168,163,[4_1|2]), (168,169,[4_1|2]), (168,288,[4_1|2, 0_1|2]), (168,464,[4_1|2]), (168,519,[4_1|2]), (168,546,[4_1|2]), (168,291,[5_1|2]), (168,298,[4_1|2]), (168,307,[5_1|2]), (168,315,[4_1|2]), (168,332,[2_1|2]), (168,343,[1_1|2]), (169,170,[1_1|2]), (170,171,[5_1|2]), (171,172,[0_1|2]), (172,173,[3_1|2]), (173,174,[5_1|2]), (174,175,[2_1|2]), (175,176,[4_1|2]), (176,177,[0_1|2]), (177,178,[5_1|2]), (178,179,[3_1|2]), (179,180,[1_1|2]), (180,181,[3_1|2]), (181,150,[2_1|2]), (181,153,[2_1|2]), (181,163,[2_1|2]), (181,169,[2_1|2]), (181,288,[2_1|2]), (181,464,[2_1|2]), (181,519,[2_1|2]), (181,182,[4_1|2]), (181,184,[2_1|2]), (181,196,[1_1|2]), (181,199,[1_1|2]), (181,204,[5_1|2]), (181,213,[2_1|2]), (181,217,[5_1|2]), (181,232,[5_1|2]), (181,251,[2_1|2]), (181,257,[2_1|2]), (181,272,[4_1|2]), (182,183,[1_1|2]), (183,150,[4_1|2]), (183,182,[4_1|2]), (183,272,[4_1|2]), (183,298,[4_1|2]), (183,315,[4_1|2]), (183,288,[0_1|2]), (183,291,[5_1|2]), (183,307,[5_1|2]), (183,332,[2_1|2]), (183,343,[1_1|2]), (184,185,[5_1|2]), (185,186,[0_1|2]), (186,187,[2_1|2]), (187,188,[2_1|2]), (188,189,[2_1|2]), (189,190,[0_1|2]), (190,191,[3_1|2]), (191,192,[5_1|2]), (192,193,[3_1|2]), (193,194,[0_1|2]), (194,195,[4_1|2]), (195,150,[5_1|2]), (195,204,[5_1|2]), (195,217,[5_1|2]), (195,232,[5_1|2]), (195,291,[5_1|2]), (195,307,[5_1|2]), (195,356,[5_1|2]), (195,359,[5_1|2]), (195,374,[5_1|2]), (195,393,[5_1|2]), (195,403,[5_1|2]), (195,480,[5_1|2]), (195,502,[5_1|2]), (195,508,[5_1|2]), (195,544,[5_1|2]), (195,481,[5_1|2]), (196,197,[3_1|2]), (197,198,[4_1|2]), (197,307,[5_1|2]), (197,315,[4_1|2]), (197,332,[2_1|2]), (198,150,[0_1|2]), (198,153,[0_1|2]), (198,163,[0_1|2]), (198,169,[0_1|2]), (198,288,[0_1|2]), (198,464,[0_1|2]), (198,519,[0_1|2]), (198,427,[0_1|2]), (198,151,[1_1|2]), (198,158,[1_1|2]), (199,200,[3_1|2]), (200,201,[5_1|2]), (201,202,[4_1|2]), (202,203,[0_1|2]), (202,169,[0_1|2]), (203,150,[5_1|2]), (203,532,[5_1|2]), (203,308,[5_1|2]), (203,545,[5_1|2]), (203,356,[5_1|2]), (203,359,[5_1|2]), (203,374,[5_1|2]), (203,393,[5_1|2]), (203,403,[5_1|2]), (204,205,[1_1|2]), (205,206,[0_1|2]), (206,207,[5_1|2]), (207,208,[3_1|2]), (208,209,[1_1|2]), (209,210,[4_1|2]), (210,211,[3_1|2]), (211,212,[0_1|2]), (211,153,[0_1|2]), (211,158,[1_1|2]), (211,163,[0_1|2]), (212,150,[3_1|2]), (212,532,[3_1|2]), (212,154,[3_1|2]), (212,164,[3_1|2]), (212,521,[3_1|2]), (212,480,[5_1|2]), (212,486,[2_1|2]), (212,502,[5_1|2]), (212,508,[5_1|2]), (212,519,[0_1|2]), (212,544,[5_1|2]), (213,214,[0_1|2]), (214,215,[4_1|2]), (215,216,[4_1|2]), (216,150,[5_1|2]), (216,204,[5_1|2]), 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(536,537,[2_1|2]), (537,538,[3_1|2]), (538,539,[4_1|2]), (539,540,[3_1|2]), (540,541,[1_1|2]), (541,542,[1_1|2]), (542,543,[0_1|2]), (542,169,[0_1|2]), (543,150,[5_1|2]), (543,184,[5_1|2]), (543,213,[5_1|2]), (543,251,[5_1|2]), (543,257,[5_1|2]), (543,332,[5_1|2]), (543,486,[5_1|2]), (543,356,[5_1|2]), (543,359,[5_1|2]), (543,374,[5_1|2]), (543,393,[5_1|2]), (543,403,[5_1|2]), (544,545,[3_1|2]), (545,546,[0_1|2]), (546,547,[3_1|2]), (547,548,[1_1|2]), (548,549,[3_1|2]), (549,550,[4_1|2]), (550,551,[5_1|2]), (551,552,[2_1|2]), (552,553,[1_1|2]), (553,554,[2_1|2]), (554,555,[4_1|2]), (555,556,[3_1|2]), (556,557,[4_1|2]), (556,298,[4_1|2]), (557,150,[3_1|2]), (557,532,[3_1|2]), (557,333,[3_1|2]), (557,487,[3_1|2]), (557,480,[5_1|2]), (557,486,[2_1|2]), (557,502,[5_1|2]), (557,508,[5_1|2]), (557,519,[0_1|2]), (557,544,[5_1|2]), (558,559,[1_1|3]), (559,306,[4_1|3]), (559,288,[0_1|2]), (559,291,[5_1|2]), (559,560,[0_1|3]), (560,561,[0_1|3]), (561,562,[1_1|3]), (562,316,[1_1|3]), (563,564,[1_1|3]), (564,289,[2_1|3]), (564,520,[2_1|3]), (564,199,[1_1|2]), (565,566,[0_1|3]), (566,567,[4_1|3]), (567,568,[4_1|3]), (568,201,[5_1|3]), (568,307,[5_1|3]), (569,570,[1_1|3]), (570,530,[2_1|3]), (570,217,[5_1|2]), (571,572,[0_1|3]), (572,573,[1_1|3]), (573,288,[1_1|3]), (573,560,[1_1|3]), (574,575,[3_1|3]), (575,576,[4_1|3]), (576,464,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)