/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 150 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 144 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(x1)))) -> 0(0(1(x1))) 3(0(2(2(x1)))) -> 3(4(3(2(x1)))) 5(2(1(5(x1)))) -> 4(2(5(x1))) 5(2(3(0(x1)))) -> 2(0(0(x1))) 4(3(5(3(2(5(x1)))))) -> 2(3(1(2(5(x1))))) 3(1(0(0(0(1(1(x1))))))) -> 5(3(2(0(2(0(1(x1))))))) 3(2(5(3(1(0(3(x1))))))) -> 3(5(3(4(0(1(x1)))))) 5(2(2(2(4(0(3(x1))))))) -> 4(5(4(0(1(5(x1)))))) 0(1(2(5(1(5(3(0(x1)))))))) -> 0(0(5(2(5(4(4(x1))))))) 3(0(2(2(2(1(0(3(x1)))))))) -> 3(5(2(3(0(0(4(x1))))))) 3(3(1(5(0(3(2(4(x1)))))))) -> 4(3(3(1(0(4(4(x1))))))) 1(2(4(5(4(0(5(2(1(x1))))))))) -> 1(5(0(5(4(4(0(4(1(x1))))))))) 2(3(1(3(1(3(1(5(3(x1))))))))) -> 5(4(1(0(4(1(3(x1))))))) 3(5(5(4(4(1(5(1(4(x1))))))))) -> 3(2(5(0(4(0(0(5(4(x1))))))))) 1(0(1(5(3(3(4(5(5(1(x1)))))))))) -> 1(1(5(4(3(5(1(1(5(1(x1)))))))))) 2(3(0(3(0(5(3(2(1(3(x1)))))))))) -> 2(5(0(2(2(5(0(1(x1)))))))) 5(5(1(1(4(5(0(5(1(3(x1)))))))))) -> 5(1(1(2(5(3(3(4(3(3(x1)))))))))) 5(0(5(3(0(5(0(4(0(5(5(x1))))))))))) -> 3(4(4(3(4(0(3(5(3(5(x1)))))))))) 5(3(2(5(5(5(4(3(3(4(5(x1))))))))))) -> 3(0(2(3(2(5(3(0(2(4(x1)))))))))) 5(4(2(1(4(4(1(3(2(1(2(x1))))))))))) -> 5(4(1(2(3(1(3(0(0(0(1(x1))))))))))) 0(0(2(1(0(1(2(1(0(3(2(2(x1)))))))))))) -> 0(1(2(5(1(4(5(2(1(4(1(x1))))))))))) 5(3(2(0(5(4(5(2(2(3(4(5(x1)))))))))))) -> 2(4(3(5(0(2(0(2(0(4(4(x1))))))))))) 0(1(3(3(0(4(1(0(2(2(4(3(2(x1))))))))))))) -> 1(3(4(2(4(1(3(3(4(2(1(x1))))))))))) 2(4(4(5(1(5(1(0(2(5(4(1(2(x1))))))))))))) -> 5(1(5(2(1(2(3(2(0(3(1(2(3(3(x1)))))))))))))) 0(0(0(4(0(0(3(2(0(5(4(4(2(1(2(x1))))))))))))))) -> 0(4(4(2(1(0(2(1(5(0(2(3(4(2(x1)))))))))))))) 3(0(2(5(1(1(4(1(3(4(2(5(3(1(0(x1))))))))))))))) -> 3(5(5(4(0(5(2(1(5(0(4(0(4(4(3(0(x1)))))))))))))))) 5(3(4(3(2(1(4(2(2(1(3(3(1(4(4(x1))))))))))))))) -> 1(1(5(3(4(5(4(3(3(3(3(2(0(2(4(x1))))))))))))))) 4(3(3(4(2(0(0(1(3(3(0(1(3(1(2(1(x1)))))))))))))))) -> 4(3(3(1(1(4(4(4(5(2(0(5(0(5(4(1(x1)))))))))))))))) 0(3(3(4(2(4(3(1(4(1(0(4(1(4(0(4(1(1(x1)))))))))))))))))) -> 0(2(2(1(1(2(3(1(3(2(2(2(5(5(5(0(3(4(2(x1))))))))))))))))))) 3(1(3(2(4(2(3(3(1(5(4(0(3(1(0(4(1(0(x1)))))))))))))))))) -> 5(1(3(5(0(2(3(2(0(4(0(1(2(2(2(2(0(x1))))))))))))))))) 5(5(5(3(2(1(0(4(1(3(4(4(3(1(1(4(3(2(x1)))))))))))))))))) -> 5(5(4(3(2(2(3(0(3(4(4(5(5(4(4(3(0(3(x1)))))))))))))))))) 1(1(1(0(3(0(1(4(3(1(1(2(5(2(1(2(1(1(0(x1))))))))))))))))))) -> 1(5(5(2(3(1(3(3(2(0(1(5(5(4(1(2(4(x1))))))))))))))))) 1(1(5(3(2(0(4(1(3(4(5(3(0(2(5(2(3(3(0(x1))))))))))))))))))) -> 0(1(3(4(3(5(0(2(2(0(2(4(5(0(3(3(2(0(x1)))))))))))))))))) 5(2(2(2(1(1(4(0(4(3(5(3(4(4(4(4(0(5(0(x1))))))))))))))))))) -> 3(4(5(0(2(4(3(5(1(1(5(5(4(1(4(5(5(0(0(x1))))))))))))))))))) 5(5(4(3(2(1(3(3(1(4(1(1(3(3(3(4(0(1(2(x1))))))))))))))))))) -> 4(0(1(3(1(2(4(5(3(5(0(0(3(2(4(5(5(3(x1)))))))))))))))))) 2(0(4(4(2(4(3(3(2(0(0(3(3(0(1(0(2(2(2(5(x1)))))))))))))))))))) -> 2(3(2(3(5(1(1(0(1(3(0(4(3(2(4(2(5(2(5(x1))))))))))))))))))) 2(2(2(3(0(3(1(4(4(1(1(0(5(1(0(0(5(4(1(2(x1)))))))))))))))))))) -> 2(3(4(3(5(5(2(1(1(2(5(3(5(0(4(1(2(2(1(4(x1)))))))))))))))))))) 4(4(0(2(4(0(0(2(3(0(1(4(1(0(0(0(3(0(2(3(x1)))))))))))))))))))) -> 5(2(4(4(0(4(1(5(0(5(3(3(1(4(1(0(2(4(2(x1))))))))))))))))))) 0(3(4(1(5(3(4(5(1(5(0(2(3(1(1(0(1(0(5(4(2(x1))))))))))))))))))))) -> 0(5(5(0(2(4(1(0(1(4(1(5(3(0(1(5(2(5(2(x1))))))))))))))))))) 0(4(5(3(5(2(4(4(1(4(2(0(5(5(1(0(1(5(0(4(3(x1))))))))))))))))))))) -> 4(4(2(2(3(2(0(5(5(0(2(4(0(0(0(1(3(5(2(3(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(x1)))) -> 0(0(1(x1))) 3(0(2(2(x1)))) -> 3(4(3(2(x1)))) 5(2(1(5(x1)))) -> 4(2(5(x1))) 5(2(3(0(x1)))) -> 2(0(0(x1))) 4(3(5(3(2(5(x1)))))) -> 2(3(1(2(5(x1))))) 3(1(0(0(0(1(1(x1))))))) -> 5(3(2(0(2(0(1(x1))))))) 3(2(5(3(1(0(3(x1))))))) -> 3(5(3(4(0(1(x1)))))) 5(2(2(2(4(0(3(x1))))))) -> 4(5(4(0(1(5(x1)))))) 0(1(2(5(1(5(3(0(x1)))))))) -> 0(0(5(2(5(4(4(x1))))))) 3(0(2(2(2(1(0(3(x1)))))))) -> 3(5(2(3(0(0(4(x1))))))) 3(3(1(5(0(3(2(4(x1)))))))) -> 4(3(3(1(0(4(4(x1))))))) 1(2(4(5(4(0(5(2(1(x1))))))))) -> 1(5(0(5(4(4(0(4(1(x1))))))))) 2(3(1(3(1(3(1(5(3(x1))))))))) -> 5(4(1(0(4(1(3(x1))))))) 3(5(5(4(4(1(5(1(4(x1))))))))) -> 3(2(5(0(4(0(0(5(4(x1))))))))) 1(0(1(5(3(3(4(5(5(1(x1)))))))))) -> 1(1(5(4(3(5(1(1(5(1(x1)))))))))) 2(3(0(3(0(5(3(2(1(3(x1)))))))))) -> 2(5(0(2(2(5(0(1(x1)))))))) 5(5(1(1(4(5(0(5(1(3(x1)))))))))) -> 5(1(1(2(5(3(3(4(3(3(x1)))))))))) 5(0(5(3(0(5(0(4(0(5(5(x1))))))))))) -> 3(4(4(3(4(0(3(5(3(5(x1)))))))))) 5(3(2(5(5(5(4(3(3(4(5(x1))))))))))) -> 3(0(2(3(2(5(3(0(2(4(x1)))))))))) 5(4(2(1(4(4(1(3(2(1(2(x1))))))))))) -> 5(4(1(2(3(1(3(0(0(0(1(x1))))))))))) 0(0(2(1(0(1(2(1(0(3(2(2(x1)))))))))))) -> 0(1(2(5(1(4(5(2(1(4(1(x1))))))))))) 5(3(2(0(5(4(5(2(2(3(4(5(x1)))))))))))) -> 2(4(3(5(0(2(0(2(0(4(4(x1))))))))))) 0(1(3(3(0(4(1(0(2(2(4(3(2(x1))))))))))))) -> 1(3(4(2(4(1(3(3(4(2(1(x1))))))))))) 2(4(4(5(1(5(1(0(2(5(4(1(2(x1))))))))))))) -> 5(1(5(2(1(2(3(2(0(3(1(2(3(3(x1)))))))))))))) 0(0(0(4(0(0(3(2(0(5(4(4(2(1(2(x1))))))))))))))) -> 0(4(4(2(1(0(2(1(5(0(2(3(4(2(x1)))))))))))))) 3(0(2(5(1(1(4(1(3(4(2(5(3(1(0(x1))))))))))))))) -> 3(5(5(4(0(5(2(1(5(0(4(0(4(4(3(0(x1)))))))))))))))) 5(3(4(3(2(1(4(2(2(1(3(3(1(4(4(x1))))))))))))))) -> 1(1(5(3(4(5(4(3(3(3(3(2(0(2(4(x1))))))))))))))) 4(3(3(4(2(0(0(1(3(3(0(1(3(1(2(1(x1)))))))))))))))) -> 4(3(3(1(1(4(4(4(5(2(0(5(0(5(4(1(x1)))))))))))))))) 0(3(3(4(2(4(3(1(4(1(0(4(1(4(0(4(1(1(x1)))))))))))))))))) -> 0(2(2(1(1(2(3(1(3(2(2(2(5(5(5(0(3(4(2(x1))))))))))))))))))) 3(1(3(2(4(2(3(3(1(5(4(0(3(1(0(4(1(0(x1)))))))))))))))))) -> 5(1(3(5(0(2(3(2(0(4(0(1(2(2(2(2(0(x1))))))))))))))))) 5(5(5(3(2(1(0(4(1(3(4(4(3(1(1(4(3(2(x1)))))))))))))))))) -> 5(5(4(3(2(2(3(0(3(4(4(5(5(4(4(3(0(3(x1)))))))))))))))))) 1(1(1(0(3(0(1(4(3(1(1(2(5(2(1(2(1(1(0(x1))))))))))))))))))) -> 1(5(5(2(3(1(3(3(2(0(1(5(5(4(1(2(4(x1))))))))))))))))) 1(1(5(3(2(0(4(1(3(4(5(3(0(2(5(2(3(3(0(x1))))))))))))))))))) -> 0(1(3(4(3(5(0(2(2(0(2(4(5(0(3(3(2(0(x1)))))))))))))))))) 5(2(2(2(1(1(4(0(4(3(5(3(4(4(4(4(0(5(0(x1))))))))))))))))))) -> 3(4(5(0(2(4(3(5(1(1(5(5(4(1(4(5(5(0(0(x1))))))))))))))))))) 5(5(4(3(2(1(3(3(1(4(1(1(3(3(3(4(0(1(2(x1))))))))))))))))))) -> 4(0(1(3(1(2(4(5(3(5(0(0(3(2(4(5(5(3(x1)))))))))))))))))) 2(0(4(4(2(4(3(3(2(0(0(3(3(0(1(0(2(2(2(5(x1)))))))))))))))))))) -> 2(3(2(3(5(1(1(0(1(3(0(4(3(2(4(2(5(2(5(x1))))))))))))))))))) 2(2(2(3(0(3(1(4(4(1(1(0(5(1(0(0(5(4(1(2(x1)))))))))))))))))))) -> 2(3(4(3(5(5(2(1(1(2(5(3(5(0(4(1(2(2(1(4(x1)))))))))))))))))))) 4(4(0(2(4(0(0(2(3(0(1(4(1(0(0(0(3(0(2(3(x1)))))))))))))))))))) -> 5(2(4(4(0(4(1(5(0(5(3(3(1(4(1(0(2(4(2(x1))))))))))))))))))) 0(3(4(1(5(3(4(5(1(5(0(2(3(1(1(0(1(0(5(4(2(x1))))))))))))))))))))) -> 0(5(5(0(2(4(1(0(1(4(1(5(3(0(1(5(2(5(2(x1))))))))))))))))))) 0(4(5(3(5(2(4(4(1(4(2(0(5(5(1(0(1(5(0(4(3(x1))))))))))))))))))))) -> 4(4(2(2(3(2(0(5(5(0(2(4(0(0(0(1(3(5(2(3(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(x1)))) -> 0(0(1(x1))) 3(0(2(2(x1)))) -> 3(4(3(2(x1)))) 5(2(1(5(x1)))) -> 4(2(5(x1))) 5(2(3(0(x1)))) -> 2(0(0(x1))) 4(3(5(3(2(5(x1)))))) -> 2(3(1(2(5(x1))))) 3(1(0(0(0(1(1(x1))))))) -> 5(3(2(0(2(0(1(x1))))))) 3(2(5(3(1(0(3(x1))))))) -> 3(5(3(4(0(1(x1)))))) 5(2(2(2(4(0(3(x1))))))) -> 4(5(4(0(1(5(x1)))))) 0(1(2(5(1(5(3(0(x1)))))))) -> 0(0(5(2(5(4(4(x1))))))) 3(0(2(2(2(1(0(3(x1)))))))) -> 3(5(2(3(0(0(4(x1))))))) 3(3(1(5(0(3(2(4(x1)))))))) -> 4(3(3(1(0(4(4(x1))))))) 1(2(4(5(4(0(5(2(1(x1))))))))) -> 1(5(0(5(4(4(0(4(1(x1))))))))) 2(3(1(3(1(3(1(5(3(x1))))))))) -> 5(4(1(0(4(1(3(x1))))))) 3(5(5(4(4(1(5(1(4(x1))))))))) -> 3(2(5(0(4(0(0(5(4(x1))))))))) 1(0(1(5(3(3(4(5(5(1(x1)))))))))) -> 1(1(5(4(3(5(1(1(5(1(x1)))))))))) 2(3(0(3(0(5(3(2(1(3(x1)))))))))) -> 2(5(0(2(2(5(0(1(x1)))))))) 5(5(1(1(4(5(0(5(1(3(x1)))))))))) -> 5(1(1(2(5(3(3(4(3(3(x1)))))))))) 5(0(5(3(0(5(0(4(0(5(5(x1))))))))))) -> 3(4(4(3(4(0(3(5(3(5(x1)))))))))) 5(3(2(5(5(5(4(3(3(4(5(x1))))))))))) -> 3(0(2(3(2(5(3(0(2(4(x1)))))))))) 5(4(2(1(4(4(1(3(2(1(2(x1))))))))))) -> 5(4(1(2(3(1(3(0(0(0(1(x1))))))))))) 0(0(2(1(0(1(2(1(0(3(2(2(x1)))))))))))) -> 0(1(2(5(1(4(5(2(1(4(1(x1))))))))))) 5(3(2(0(5(4(5(2(2(3(4(5(x1)))))))))))) -> 2(4(3(5(0(2(0(2(0(4(4(x1))))))))))) 0(1(3(3(0(4(1(0(2(2(4(3(2(x1))))))))))))) -> 1(3(4(2(4(1(3(3(4(2(1(x1))))))))))) 2(4(4(5(1(5(1(0(2(5(4(1(2(x1))))))))))))) -> 5(1(5(2(1(2(3(2(0(3(1(2(3(3(x1)))))))))))))) 0(0(0(4(0(0(3(2(0(5(4(4(2(1(2(x1))))))))))))))) -> 0(4(4(2(1(0(2(1(5(0(2(3(4(2(x1)))))))))))))) 3(0(2(5(1(1(4(1(3(4(2(5(3(1(0(x1))))))))))))))) -> 3(5(5(4(0(5(2(1(5(0(4(0(4(4(3(0(x1)))))))))))))))) 5(3(4(3(2(1(4(2(2(1(3(3(1(4(4(x1))))))))))))))) -> 1(1(5(3(4(5(4(3(3(3(3(2(0(2(4(x1))))))))))))))) 4(3(3(4(2(0(0(1(3(3(0(1(3(1(2(1(x1)))))))))))))))) -> 4(3(3(1(1(4(4(4(5(2(0(5(0(5(4(1(x1)))))))))))))))) 0(3(3(4(2(4(3(1(4(1(0(4(1(4(0(4(1(1(x1)))))))))))))))))) -> 0(2(2(1(1(2(3(1(3(2(2(2(5(5(5(0(3(4(2(x1))))))))))))))))))) 3(1(3(2(4(2(3(3(1(5(4(0(3(1(0(4(1(0(x1)))))))))))))))))) -> 5(1(3(5(0(2(3(2(0(4(0(1(2(2(2(2(0(x1))))))))))))))))) 5(5(5(3(2(1(0(4(1(3(4(4(3(1(1(4(3(2(x1)))))))))))))))))) -> 5(5(4(3(2(2(3(0(3(4(4(5(5(4(4(3(0(3(x1)))))))))))))))))) 1(1(1(0(3(0(1(4(3(1(1(2(5(2(1(2(1(1(0(x1))))))))))))))))))) -> 1(5(5(2(3(1(3(3(2(0(1(5(5(4(1(2(4(x1))))))))))))))))) 1(1(5(3(2(0(4(1(3(4(5(3(0(2(5(2(3(3(0(x1))))))))))))))))))) -> 0(1(3(4(3(5(0(2(2(0(2(4(5(0(3(3(2(0(x1)))))))))))))))))) 5(2(2(2(1(1(4(0(4(3(5(3(4(4(4(4(0(5(0(x1))))))))))))))))))) -> 3(4(5(0(2(4(3(5(1(1(5(5(4(1(4(5(5(0(0(x1))))))))))))))))))) 5(5(4(3(2(1(3(3(1(4(1(1(3(3(3(4(0(1(2(x1))))))))))))))))))) -> 4(0(1(3(1(2(4(5(3(5(0(0(3(2(4(5(5(3(x1)))))))))))))))))) 2(0(4(4(2(4(3(3(2(0(0(3(3(0(1(0(2(2(2(5(x1)))))))))))))))))))) -> 2(3(2(3(5(1(1(0(1(3(0(4(3(2(4(2(5(2(5(x1))))))))))))))))))) 2(2(2(3(0(3(1(4(4(1(1(0(5(1(0(0(5(4(1(2(x1)))))))))))))))))))) -> 2(3(4(3(5(5(2(1(1(2(5(3(5(0(4(1(2(2(1(4(x1)))))))))))))))))))) 4(4(0(2(4(0(0(2(3(0(1(4(1(0(0(0(3(0(2(3(x1)))))))))))))))))))) -> 5(2(4(4(0(4(1(5(0(5(3(3(1(4(1(0(2(4(2(x1))))))))))))))))))) 0(3(4(1(5(3(4(5(1(5(0(2(3(1(1(0(1(0(5(4(2(x1))))))))))))))))))))) -> 0(5(5(0(2(4(1(0(1(4(1(5(3(0(1(5(2(5(2(x1))))))))))))))))))) 0(4(5(3(5(2(4(4(1(4(2(0(5(5(1(0(1(5(0(4(3(x1))))))))))))))))))))) -> 4(4(2(2(3(2(0(5(5(0(2(4(0(0(0(1(3(5(2(3(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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(x1)))) -> 0(0(1(x1))) 3(0(2(2(x1)))) -> 3(4(3(2(x1)))) 5(2(1(5(x1)))) -> 4(2(5(x1))) 5(2(3(0(x1)))) -> 2(0(0(x1))) 4(3(5(3(2(5(x1)))))) -> 2(3(1(2(5(x1))))) 3(1(0(0(0(1(1(x1))))))) -> 5(3(2(0(2(0(1(x1))))))) 3(2(5(3(1(0(3(x1))))))) -> 3(5(3(4(0(1(x1)))))) 5(2(2(2(4(0(3(x1))))))) -> 4(5(4(0(1(5(x1)))))) 0(1(2(5(1(5(3(0(x1)))))))) -> 0(0(5(2(5(4(4(x1))))))) 3(0(2(2(2(1(0(3(x1)))))))) -> 3(5(2(3(0(0(4(x1))))))) 3(3(1(5(0(3(2(4(x1)))))))) -> 4(3(3(1(0(4(4(x1))))))) 1(2(4(5(4(0(5(2(1(x1))))))))) -> 1(5(0(5(4(4(0(4(1(x1))))))))) 2(3(1(3(1(3(1(5(3(x1))))))))) -> 5(4(1(0(4(1(3(x1))))))) 3(5(5(4(4(1(5(1(4(x1))))))))) -> 3(2(5(0(4(0(0(5(4(x1))))))))) 1(0(1(5(3(3(4(5(5(1(x1)))))))))) -> 1(1(5(4(3(5(1(1(5(1(x1)))))))))) 2(3(0(3(0(5(3(2(1(3(x1)))))))))) -> 2(5(0(2(2(5(0(1(x1)))))))) 5(5(1(1(4(5(0(5(1(3(x1)))))))))) -> 5(1(1(2(5(3(3(4(3(3(x1)))))))))) 5(0(5(3(0(5(0(4(0(5(5(x1))))))))))) -> 3(4(4(3(4(0(3(5(3(5(x1)))))))))) 5(3(2(5(5(5(4(3(3(4(5(x1))))))))))) -> 3(0(2(3(2(5(3(0(2(4(x1)))))))))) 5(4(2(1(4(4(1(3(2(1(2(x1))))))))))) -> 5(4(1(2(3(1(3(0(0(0(1(x1))))))))))) 0(0(2(1(0(1(2(1(0(3(2(2(x1)))))))))))) -> 0(1(2(5(1(4(5(2(1(4(1(x1))))))))))) 5(3(2(0(5(4(5(2(2(3(4(5(x1)))))))))))) -> 2(4(3(5(0(2(0(2(0(4(4(x1))))))))))) 0(1(3(3(0(4(1(0(2(2(4(3(2(x1))))))))))))) -> 1(3(4(2(4(1(3(3(4(2(1(x1))))))))))) 2(4(4(5(1(5(1(0(2(5(4(1(2(x1))))))))))))) -> 5(1(5(2(1(2(3(2(0(3(1(2(3(3(x1)))))))))))))) 0(0(0(4(0(0(3(2(0(5(4(4(2(1(2(x1))))))))))))))) -> 0(4(4(2(1(0(2(1(5(0(2(3(4(2(x1)))))))))))))) 3(0(2(5(1(1(4(1(3(4(2(5(3(1(0(x1))))))))))))))) -> 3(5(5(4(0(5(2(1(5(0(4(0(4(4(3(0(x1)))))))))))))))) 5(3(4(3(2(1(4(2(2(1(3(3(1(4(4(x1))))))))))))))) -> 1(1(5(3(4(5(4(3(3(3(3(2(0(2(4(x1))))))))))))))) 4(3(3(4(2(0(0(1(3(3(0(1(3(1(2(1(x1)))))))))))))))) -> 4(3(3(1(1(4(4(4(5(2(0(5(0(5(4(1(x1)))))))))))))))) 0(3(3(4(2(4(3(1(4(1(0(4(1(4(0(4(1(1(x1)))))))))))))))))) -> 0(2(2(1(1(2(3(1(3(2(2(2(5(5(5(0(3(4(2(x1))))))))))))))))))) 3(1(3(2(4(2(3(3(1(5(4(0(3(1(0(4(1(0(x1)))))))))))))))))) -> 5(1(3(5(0(2(3(2(0(4(0(1(2(2(2(2(0(x1))))))))))))))))) 5(5(5(3(2(1(0(4(1(3(4(4(3(1(1(4(3(2(x1)))))))))))))))))) -> 5(5(4(3(2(2(3(0(3(4(4(5(5(4(4(3(0(3(x1)))))))))))))))))) 1(1(1(0(3(0(1(4(3(1(1(2(5(2(1(2(1(1(0(x1))))))))))))))))))) -> 1(5(5(2(3(1(3(3(2(0(1(5(5(4(1(2(4(x1))))))))))))))))) 1(1(5(3(2(0(4(1(3(4(5(3(0(2(5(2(3(3(0(x1))))))))))))))))))) -> 0(1(3(4(3(5(0(2(2(0(2(4(5(0(3(3(2(0(x1)))))))))))))))))) 5(2(2(2(1(1(4(0(4(3(5(3(4(4(4(4(0(5(0(x1))))))))))))))))))) -> 3(4(5(0(2(4(3(5(1(1(5(5(4(1(4(5(5(0(0(x1))))))))))))))))))) 5(5(4(3(2(1(3(3(1(4(1(1(3(3(3(4(0(1(2(x1))))))))))))))))))) -> 4(0(1(3(1(2(4(5(3(5(0(0(3(2(4(5(5(3(x1)))))))))))))))))) 2(0(4(4(2(4(3(3(2(0(0(3(3(0(1(0(2(2(2(5(x1)))))))))))))))))))) -> 2(3(2(3(5(1(1(0(1(3(0(4(3(2(4(2(5(2(5(x1))))))))))))))))))) 2(2(2(3(0(3(1(4(4(1(1(0(5(1(0(0(5(4(1(2(x1)))))))))))))))))))) -> 2(3(4(3(5(5(2(1(1(2(5(3(5(0(4(1(2(2(1(4(x1)))))))))))))))))))) 4(4(0(2(4(0(0(2(3(0(1(4(1(0(0(0(3(0(2(3(x1)))))))))))))))))))) -> 5(2(4(4(0(4(1(5(0(5(3(3(1(4(1(0(2(4(2(x1))))))))))))))))))) 0(3(4(1(5(3(4(5(1(5(0(2(3(1(1(0(1(0(5(4(2(x1))))))))))))))))))))) -> 0(5(5(0(2(4(1(0(1(4(1(5(3(0(1(5(2(5(2(x1))))))))))))))))))) 0(4(5(3(5(2(4(4(1(4(2(0(5(5(1(0(1(5(0(4(3(x1))))))))))))))))))))) -> 4(4(2(2(3(2(0(5(5(0(2(4(0(0(0(1(3(5(2(3(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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. "[85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 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, 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] {(85,86,[0_1|0, 3_1|0, 5_1|0, 4_1|0, 1_1|0, 2_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]), (85,87,[0_1|1, 3_1|1, 5_1|1, 4_1|1, 1_1|1, 2_1|1]), (85,88,[0_1|2]), (85,90,[0_1|2]), (85,96,[1_1|2]), (85,106,[0_1|2]), (85,116,[0_1|2]), (85,129,[0_1|2]), (85,147,[0_1|2]), (85,165,[4_1|2]), (85,184,[3_1|2]), (85,187,[3_1|2]), (85,193,[3_1|2]), (85,208,[5_1|2]), (85,214,[5_1|2]), (85,230,[3_1|2]), (85,235,[4_1|2]), (85,241,[3_1|2]), (85,249,[4_1|2]), (85,251,[2_1|2]), (85,253,[4_1|2]), (85,258,[3_1|2]), (85,276,[5_1|2]), (85,285,[5_1|2]), (85,302,[4_1|2]), (85,319,[3_1|2]), (85,328,[3_1|2]), (85,337,[2_1|2]), (85,347,[1_1|2]), (85,361,[5_1|2]), (85,371,[2_1|2]), (85,375,[4_1|2]), (85,390,[5_1|2]), (85,408,[1_1|2]), (85,416,[1_1|2]), (85,425,[1_1|2]), (85,441,[0_1|2]), (85,458,[5_1|2]), (85,464,[2_1|2]), (85,471,[5_1|2]), (85,484,[2_1|2]), (85,502,[2_1|2]), (86,86,[cons_0_1|0, cons_3_1|0, cons_5_1|0, cons_4_1|0, cons_1_1|0, cons_2_1|0]), (87,86,[encArg_1|1]), (87,87,[0_1|1, 3_1|1, 5_1|1, 4_1|1, 1_1|1, 2_1|1]), (87,88,[0_1|2]), (87,90,[0_1|2]), (87,96,[1_1|2]), (87,106,[0_1|2]), (87,116,[0_1|2]), (87,129,[0_1|2]), (87,147,[0_1|2]), (87,165,[4_1|2]), (87,184,[3_1|2]), (87,187,[3_1|2]), (87,193,[3_1|2]), (87,208,[5_1|2]), (87,214,[5_1|2]), (87,230,[3_1|2]), (87,235,[4_1|2]), (87,241,[3_1|2]), (87,249,[4_1|2]), (87,251,[2_1|2]), (87,253,[4_1|2]), (87,258,[3_1|2]), (87,276,[5_1|2]), (87,285,[5_1|2]), (87,302,[4_1|2]), (87,319,[3_1|2]), (87,328,[3_1|2]), (87,337,[2_1|2]), (87,347,[1_1|2]), (87,361,[5_1|2]), (87,371,[2_1|2]), (87,375,[4_1|2]), (87,390,[5_1|2]), (87,408,[1_1|2]), (87,416,[1_1|2]), (87,425,[1_1|2]), (87,441,[0_1|2]), (87,458,[5_1|2]), (87,464,[2_1|2]), 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(374,302,[4_1|2]), (374,319,[3_1|2]), (374,328,[3_1|2]), (374,337,[2_1|2]), (374,347,[1_1|2]), (375,376,[3_1|2]), (376,377,[3_1|2]), (377,378,[1_1|2]), (378,379,[1_1|2]), (379,380,[4_1|2]), (380,381,[4_1|2]), (381,382,[4_1|2]), (382,383,[5_1|2]), (383,384,[2_1|2]), (384,385,[0_1|2]), (385,386,[5_1|2]), (386,387,[0_1|2]), (387,388,[5_1|2]), (388,389,[4_1|2]), (389,87,[1_1|2]), (389,96,[1_1|2]), (389,347,[1_1|2]), (389,408,[1_1|2]), (389,416,[1_1|2]), (389,425,[1_1|2]), (389,441,[0_1|2]), (390,391,[2_1|2]), (391,392,[4_1|2]), (392,393,[4_1|2]), (393,394,[0_1|2]), (394,395,[4_1|2]), (395,396,[1_1|2]), (396,397,[5_1|2]), (397,398,[0_1|2]), (398,399,[5_1|2]), (399,400,[3_1|2]), (400,401,[3_1|2]), (401,402,[1_1|2]), (402,403,[4_1|2]), (403,404,[1_1|2]), (404,405,[0_1|2]), (405,406,[2_1|2]), (406,407,[4_1|2]), (407,87,[2_1|2]), (407,184,[2_1|2]), (407,187,[2_1|2]), (407,193,[2_1|2]), (407,230,[2_1|2]), (407,241,[2_1|2]), (407,258,[2_1|2]), (407,319,[2_1|2]), (407,328,[2_1|2]), (407,372,[2_1|2]), (407,485,[2_1|2]), (407,503,[2_1|2]), (407,331,[2_1|2]), (407,458,[5_1|2]), (407,464,[2_1|2]), (407,471,[5_1|2]), (407,484,[2_1|2]), (407,502,[2_1|2]), (408,409,[5_1|2]), (409,410,[0_1|2]), (410,411,[5_1|2]), (411,412,[4_1|2]), (412,413,[4_1|2]), (413,414,[0_1|2]), (414,415,[4_1|2]), (415,87,[1_1|2]), (415,96,[1_1|2]), (415,347,[1_1|2]), (415,408,[1_1|2]), (415,416,[1_1|2]), (415,425,[1_1|2]), (415,441,[0_1|2]), (416,417,[1_1|2]), (417,418,[5_1|2]), (418,419,[4_1|2]), (419,420,[3_1|2]), (420,421,[5_1|2]), (421,422,[1_1|2]), (422,423,[1_1|2]), (423,424,[5_1|2]), (424,87,[1_1|2]), (424,96,[1_1|2]), (424,347,[1_1|2]), (424,408,[1_1|2]), (424,416,[1_1|2]), (424,425,[1_1|2]), (424,215,[1_1|2]), (424,277,[1_1|2]), (424,472,[1_1|2]), (424,441,[0_1|2]), (425,426,[5_1|2]), (426,427,[5_1|2]), (427,428,[2_1|2]), (428,429,[3_1|2]), (429,430,[1_1|2]), (430,431,[3_1|2]), (431,432,[3_1|2]), (432,433,[2_1|2]), (433,434,[0_1|2]), (434,435,[1_1|2]), (435,436,[5_1|2]), (436,437,[5_1|2]), (437,438,[4_1|2]), (438,439,[1_1|2]), (438,408,[1_1|2]), (439,440,[2_1|2]), (439,471,[5_1|2]), (440,87,[4_1|2]), (440,88,[4_1|2]), (440,90,[4_1|2]), (440,106,[4_1|2]), (440,116,[4_1|2]), (440,129,[4_1|2]), (440,147,[4_1|2]), (440,441,[4_1|2]), (440,371,[2_1|2]), (440,375,[4_1|2]), (440,390,[5_1|2]), (441,442,[1_1|2]), (442,443,[3_1|2]), (443,444,[4_1|2]), (444,445,[3_1|2]), (445,446,[5_1|2]), (446,447,[0_1|2]), (447,448,[2_1|2]), (448,449,[2_1|2]), (449,450,[0_1|2]), (450,451,[2_1|2]), (451,452,[4_1|2]), (452,453,[5_1|2]), (453,454,[0_1|2]), (454,455,[3_1|2]), (455,456,[3_1|2]), (456,457,[2_1|2]), (456,484,[2_1|2]), (457,87,[0_1|2]), (457,88,[0_1|2]), (457,90,[0_1|2]), (457,106,[0_1|2]), (457,116,[0_1|2]), (457,129,[0_1|2]), (457,147,[0_1|2]), (457,441,[0_1|2]), (457,329,[0_1|2]), (457,96,[1_1|2]), (457,165,[4_1|2]), (458,459,[4_1|2]), (459,460,[1_1|2]), (460,461,[0_1|2]), (461,462,[4_1|2]), (462,463,[1_1|2]), (463,87,[3_1|2]), (463,184,[3_1|2]), (463,187,[3_1|2]), (463,193,[3_1|2]), (463,230,[3_1|2]), (463,241,[3_1|2]), (463,258,[3_1|2]), (463,319,[3_1|2]), (463,328,[3_1|2]), (463,209,[3_1|2]), (463,208,[5_1|2]), (463,214,[5_1|2]), (463,235,[4_1|2]), (463,527,[3_1|3]), (464,465,[5_1|2]), (465,466,[0_1|2]), (466,467,[2_1|2]), (467,468,[2_1|2]), (468,469,[5_1|2]), (469,470,[0_1|2]), (469,88,[0_1|2]), (469,90,[0_1|2]), (469,96,[1_1|2]), (470,87,[1_1|2]), (470,184,[1_1|2]), (470,187,[1_1|2]), (470,193,[1_1|2]), (470,230,[1_1|2]), (470,241,[1_1|2]), (470,258,[1_1|2]), (470,319,[1_1|2]), (470,328,[1_1|2]), (470,97,[1_1|2]), (470,408,[1_1|2]), (470,416,[1_1|2]), (470,425,[1_1|2]), (470,441,[0_1|2]), (471,472,[1_1|2]), (472,473,[5_1|2]), (473,474,[2_1|2]), (474,475,[1_1|2]), (475,476,[2_1|2]), (476,477,[3_1|2]), (477,478,[2_1|2]), (478,479,[0_1|2]), (479,480,[3_1|2]), (480,481,[1_1|2]), (481,482,[2_1|2]), (482,483,[3_1|2]), (482,235,[4_1|2]), (483,87,[3_1|2]), (483,251,[3_1|2]), (483,337,[3_1|2]), (483,371,[3_1|2]), (483,464,[3_1|2]), (483,484,[3_1|2]), (483,502,[3_1|2]), (483,364,[3_1|2]), (483,184,[3_1|2]), (483,187,[3_1|2]), (483,193,[3_1|2]), (483,208,[5_1|2]), (483,214,[5_1|2]), (483,230,[3_1|2]), (483,235,[4_1|2]), (483,241,[3_1|2]), (483,527,[3_1|3]), (484,485,[3_1|2]), (485,486,[2_1|2]), (486,487,[3_1|2]), (487,488,[5_1|2]), (488,489,[1_1|2]), (489,490,[1_1|2]), (490,491,[0_1|2]), (491,492,[1_1|2]), (492,493,[3_1|2]), (493,494,[0_1|2]), (494,495,[4_1|2]), (495,496,[3_1|2]), (496,497,[2_1|2]), (497,498,[4_1|2]), (498,499,[2_1|2]), (499,500,[5_1|2]), (499,521,[2_1|3]), (500,501,[2_1|2]), (501,87,[5_1|2]), (501,208,[5_1|2]), (501,214,[5_1|2]), (501,276,[5_1|2]), (501,285,[5_1|2]), (501,361,[5_1|2]), (501,390,[5_1|2]), (501,458,[5_1|2]), (501,471,[5_1|2]), (501,465,[5_1|2]), (501,249,[4_1|2]), (501,251,[2_1|2]), (501,253,[4_1|2]), (501,258,[3_1|2]), (501,302,[4_1|2]), (501,319,[3_1|2]), (501,328,[3_1|2]), (501,337,[2_1|2]), (501,347,[1_1|2]), (502,503,[3_1|2]), (503,504,[4_1|2]), (504,505,[3_1|2]), (505,506,[5_1|2]), (506,507,[5_1|2]), (507,508,[2_1|2]), (508,509,[1_1|2]), (509,510,[1_1|2]), (510,511,[2_1|2]), (511,512,[5_1|2]), (512,513,[3_1|2]), (513,514,[5_1|2]), (514,515,[0_1|2]), (515,516,[4_1|2]), (516,517,[1_1|2]), (517,518,[2_1|2]), (518,519,[2_1|2]), (519,520,[1_1|2]), (520,87,[4_1|2]), (520,251,[4_1|2]), (520,337,[4_1|2]), (520,371,[4_1|2, 2_1|2]), (520,464,[4_1|2]), (520,484,[4_1|2]), (520,502,[4_1|2]), (520,364,[4_1|2]), (520,375,[4_1|2]), (520,390,[5_1|2]), (521,522,[0_1|3]), (522,329,[0_1|3]), (523,524,[2_1|3]), (524,409,[5_1|3]), (524,426,[5_1|3]), (525,526,[0_1|3]), (526,88,[0_1|3]), (526,90,[0_1|3]), (526,106,[0_1|3]), (526,116,[0_1|3]), (526,129,[0_1|3]), (526,147,[0_1|3]), (526,441,[0_1|3]), (526,329,[0_1|3]), (527,528,[4_1|3]), (528,529,[3_1|3]), (529,131,[2_1|3]), (530,531,[0_1|3]), (530,116,[0_1|2]), (531,191,[0_1|3]), (532,533,[2_1|3]), (533,201,[5_1|3]), (534,535,[4_1|3]), (535,536,[3_1|3]), (536,371,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)