/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 177 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) 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(2(3(0(2(0(0(0(2(0(x1)))))))))) 1(1(3(3(1(x1))))) -> 5(3(3(2(3(0(2(2(2(0(x1)))))))))) 1(3(3(4(4(x1))))) -> 0(3(5(5(3(0(2(0(0(1(x1)))))))))) 3(1(3(5(2(x1))))) -> 5(3(1(5(3(0(0(2(2(2(x1)))))))))) 4(2(3(3(3(x1))))) -> 5(0(3(0(2(0(0(2(0(0(x1)))))))))) 0(1(2(1(1(4(x1)))))) -> 0(0(4(2(3(0(2(2(0(4(x1)))))))))) 2(4(4(3(4(2(x1)))))) -> 2(4(5(0(3(0(0(2(2(0(x1)))))))))) 3(3(0(3(3(3(x1)))))) -> 5(1(4(5(4(0(3(0(2(0(x1)))))))))) 3(5(2(5(4(1(x1)))))) -> 3(0(2(0(0(4(2(3(0(0(x1)))))))))) 0(1(3(1(3(1(0(x1))))))) -> 2(3(0(2(0(1(4(3(1(0(x1)))))))))) 0(1(3(3(3(5(3(x1))))))) -> 0(0(0(4(0(2(4(2(3(0(x1)))))))))) 0(1(3(3(4(3(4(x1))))))) -> 0(3(0(2(0(4(2(0(2(4(x1)))))))))) 0(2(3(3(5(1(3(x1))))))) -> 2(3(2(0(2(0(0(0(5(3(x1)))))))))) 1(1(1(3(1(2(1(x1))))))) -> 1(2(5(3(0(2(2(2(5(1(x1)))))))))) 1(1(3(1(2(3(3(x1))))))) -> 3(0(2(1(5(3(2(5(3(0(x1)))))))))) 1(1(3(3(4(0(1(x1))))))) -> 3(2(1(0(5(1(2(3(2(2(x1)))))))))) 1(2(4(3(4(3(3(x1))))))) -> 3(0(2(1(0(1(1(5(5(3(x1)))))))))) 1(3(0(5(4(1(3(x1))))))) -> 1(3(2(4(0(0(2(2(5(3(x1)))))))))) 1(3(3(0(3(2(4(x1))))))) -> 1(5(2(3(2(4(0(0(0(2(x1)))))))))) 1(3(3(3(3(1(3(x1))))))) -> 1(3(0(2(4(1(4(2(2(3(x1)))))))))) 1(3(3(3(3(4(4(x1))))))) -> 1(0(0(4(5(5(5(1(1(4(x1)))))))))) 1(3(3(4(1(1(2(x1))))))) -> 3(1(5(2(4(3(0(2(2(0(x1)))))))))) 1(3(3(5(1(1(1(x1))))))) -> 1(0(0(3(1(0(0(2(2(2(x1)))))))))) 1(3(4(4(1(3(5(x1))))))) -> 1(3(3(3(0(2(0(2(1(5(x1)))))))))) 1(3(5(0(2(3(4(x1))))))) -> 3(3(4(5(5(5(0(3(5(4(x1)))))))))) 1(3(5(1(1(5(1(x1))))))) -> 1(0(0(4(4(0(0(4(0(0(x1)))))))))) 1(3(5(1(3(4(3(x1))))))) -> 2(5(0(0(5(5(5(1(1(3(x1)))))))))) 2(1(3(3(3(3(1(x1))))))) -> 2(3(0(2(0(4(0(0(3(1(x1)))))))))) 2(3(1(1(1(1(0(x1))))))) -> 2(1(5(0(3(5(5(4(0(0(x1)))))))))) 2(3(1(3(3(1(1(x1))))))) -> 2(2(2(2(2(0(5(5(5(1(x1)))))))))) 2(3(3(5(1(3(3(x1))))))) -> 2(4(4(5(5(3(4(3(2(3(x1)))))))))) 2(4(3(3(4(2(5(x1))))))) -> 0(4(2(4(0(2(5(1(0(5(x1)))))))))) 2(5(2(0(3(5(1(x1))))))) -> 2(5(2(0(4(0(2(0(0(2(x1)))))))))) 3(1(3(1(1(1(4(x1))))))) -> 5(3(5(3(0(0(4(0(1(4(x1)))))))))) 3(1(3(1(4(4(0(x1))))))) -> 3(2(1(4(0(2(0(0(4(2(x1)))))))))) 3(2(5(4(4(2(4(x1))))))) -> 5(3(1(0(2(2(2(0(0(4(x1)))))))))) 3(3(3(3(4(5(2(x1))))))) -> 5(5(1(1(2(4(0(0(2(1(x1)))))))))) 3(3(5(1(5(1(1(x1))))))) -> 5(3(0(2(4(5(5(4(3(1(x1)))))))))) 3(3(5(2(5(0(4(x1))))))) -> 5(5(1(0(1(5(5(3(0(2(x1)))))))))) 3(3(5(3(0(1(1(x1))))))) -> 5(1(4(5(5(3(5(4(5(1(x1)))))))))) 3(4(1(1(3(1(1(x1))))))) -> 1(2(2(4(5(3(5(5(5(1(x1)))))))))) 3(4(2(0(4(1(1(x1))))))) -> 5(3(3(0(2(0(2(5(5(1(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 5(5(1(0(0(5(3(5(5(3(x1)))))))))) 4(1(1(1(4(5(1(x1))))))) -> 5(3(0(4(3(0(2(4(0(2(x1)))))))))) 4(2(1(3(3(3(4(x1))))))) -> 5(4(2(0(0(0(5(1(4(5(x1)))))))))) 4(3(3(5(1(4(1(x1))))))) -> 5(3(5(0(0(5(3(2(0(4(x1)))))))))) 4(4(0(4(3(3(3(x1))))))) -> 4(5(3(0(2(2(3(2(1(3(x1)))))))))) 4(4(3(3(3(4(1(x1))))))) -> 4(0(1(0(2(1(0(2(0(2(x1)))))))))) 4(5(1(5(2(4(3(x1))))))) -> 4(3(3(0(2(0(0(0(0(3(x1)))))))))) 4(5(2(0(3(2(5(x1))))))) -> 5(0(3(4(5(4(0(0(0(5(x1)))))))))) 4(5(2(5(4(1(3(x1))))))) -> 5(0(3(2(1(4(3(0(2(4(x1)))))))))) 5(0(1(3(2(5(4(x1))))))) -> 4(2(3(0(0(0(0(2(5(1(x1)))))))))) 5(1(1(1(1(5(0(x1))))))) -> 3(4(5(5(4(2(0(0(5(0(x1)))))))))) 5(1(1(2(1(1(1(x1))))))) -> 0(5(0(0(5(1(2(0(2(1(x1)))))))))) 5(2(1(3(5(2(2(x1))))))) -> 4(2(1(2(3(0(0(0(0(0(x1)))))))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) 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(2(3(0(2(0(0(0(2(0(x1)))))))))) 1(1(3(3(1(x1))))) -> 5(3(3(2(3(0(2(2(2(0(x1)))))))))) 1(3(3(4(4(x1))))) -> 0(3(5(5(3(0(2(0(0(1(x1)))))))))) 3(1(3(5(2(x1))))) -> 5(3(1(5(3(0(0(2(2(2(x1)))))))))) 4(2(3(3(3(x1))))) -> 5(0(3(0(2(0(0(2(0(0(x1)))))))))) 0(1(2(1(1(4(x1)))))) -> 0(0(4(2(3(0(2(2(0(4(x1)))))))))) 2(4(4(3(4(2(x1)))))) -> 2(4(5(0(3(0(0(2(2(0(x1)))))))))) 3(3(0(3(3(3(x1)))))) -> 5(1(4(5(4(0(3(0(2(0(x1)))))))))) 3(5(2(5(4(1(x1)))))) -> 3(0(2(0(0(4(2(3(0(0(x1)))))))))) 0(1(3(1(3(1(0(x1))))))) -> 2(3(0(2(0(1(4(3(1(0(x1)))))))))) 0(1(3(3(3(5(3(x1))))))) -> 0(0(0(4(0(2(4(2(3(0(x1)))))))))) 0(1(3(3(4(3(4(x1))))))) -> 0(3(0(2(0(4(2(0(2(4(x1)))))))))) 0(2(3(3(5(1(3(x1))))))) -> 2(3(2(0(2(0(0(0(5(3(x1)))))))))) 1(1(1(3(1(2(1(x1))))))) -> 1(2(5(3(0(2(2(2(5(1(x1)))))))))) 1(1(3(1(2(3(3(x1))))))) -> 3(0(2(1(5(3(2(5(3(0(x1)))))))))) 1(1(3(3(4(0(1(x1))))))) -> 3(2(1(0(5(1(2(3(2(2(x1)))))))))) 1(2(4(3(4(3(3(x1))))))) -> 3(0(2(1(0(1(1(5(5(3(x1)))))))))) 1(3(0(5(4(1(3(x1))))))) -> 1(3(2(4(0(0(2(2(5(3(x1)))))))))) 1(3(3(0(3(2(4(x1))))))) -> 1(5(2(3(2(4(0(0(0(2(x1)))))))))) 1(3(3(3(3(1(3(x1))))))) -> 1(3(0(2(4(1(4(2(2(3(x1)))))))))) 1(3(3(3(3(4(4(x1))))))) -> 1(0(0(4(5(5(5(1(1(4(x1)))))))))) 1(3(3(4(1(1(2(x1))))))) -> 3(1(5(2(4(3(0(2(2(0(x1)))))))))) 1(3(3(5(1(1(1(x1))))))) -> 1(0(0(3(1(0(0(2(2(2(x1)))))))))) 1(3(4(4(1(3(5(x1))))))) -> 1(3(3(3(0(2(0(2(1(5(x1)))))))))) 1(3(5(0(2(3(4(x1))))))) -> 3(3(4(5(5(5(0(3(5(4(x1)))))))))) 1(3(5(1(1(5(1(x1))))))) -> 1(0(0(4(4(0(0(4(0(0(x1)))))))))) 1(3(5(1(3(4(3(x1))))))) -> 2(5(0(0(5(5(5(1(1(3(x1)))))))))) 2(1(3(3(3(3(1(x1))))))) -> 2(3(0(2(0(4(0(0(3(1(x1)))))))))) 2(3(1(1(1(1(0(x1))))))) -> 2(1(5(0(3(5(5(4(0(0(x1)))))))))) 2(3(1(3(3(1(1(x1))))))) -> 2(2(2(2(2(0(5(5(5(1(x1)))))))))) 2(3(3(5(1(3(3(x1))))))) -> 2(4(4(5(5(3(4(3(2(3(x1)))))))))) 2(4(3(3(4(2(5(x1))))))) -> 0(4(2(4(0(2(5(1(0(5(x1)))))))))) 2(5(2(0(3(5(1(x1))))))) -> 2(5(2(0(4(0(2(0(0(2(x1)))))))))) 3(1(3(1(1(1(4(x1))))))) -> 5(3(5(3(0(0(4(0(1(4(x1)))))))))) 3(1(3(1(4(4(0(x1))))))) -> 3(2(1(4(0(2(0(0(4(2(x1)))))))))) 3(2(5(4(4(2(4(x1))))))) -> 5(3(1(0(2(2(2(0(0(4(x1)))))))))) 3(3(3(3(4(5(2(x1))))))) -> 5(5(1(1(2(4(0(0(2(1(x1)))))))))) 3(3(5(1(5(1(1(x1))))))) -> 5(3(0(2(4(5(5(4(3(1(x1)))))))))) 3(3(5(2(5(0(4(x1))))))) -> 5(5(1(0(1(5(5(3(0(2(x1)))))))))) 3(3(5(3(0(1(1(x1))))))) -> 5(1(4(5(5(3(5(4(5(1(x1)))))))))) 3(4(1(1(3(1(1(x1))))))) -> 1(2(2(4(5(3(5(5(5(1(x1)))))))))) 3(4(2(0(4(1(1(x1))))))) -> 5(3(3(0(2(0(2(5(5(1(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 5(5(1(0(0(5(3(5(5(3(x1)))))))))) 4(1(1(1(4(5(1(x1))))))) -> 5(3(0(4(3(0(2(4(0(2(x1)))))))))) 4(2(1(3(3(3(4(x1))))))) -> 5(4(2(0(0(0(5(1(4(5(x1)))))))))) 4(3(3(5(1(4(1(x1))))))) -> 5(3(5(0(0(5(3(2(0(4(x1)))))))))) 4(4(0(4(3(3(3(x1))))))) -> 4(5(3(0(2(2(3(2(1(3(x1)))))))))) 4(4(3(3(3(4(1(x1))))))) -> 4(0(1(0(2(1(0(2(0(2(x1)))))))))) 4(5(1(5(2(4(3(x1))))))) -> 4(3(3(0(2(0(0(0(0(3(x1)))))))))) 4(5(2(0(3(2(5(x1))))))) -> 5(0(3(4(5(4(0(0(0(5(x1)))))))))) 4(5(2(5(4(1(3(x1))))))) -> 5(0(3(2(1(4(3(0(2(4(x1)))))))))) 5(0(1(3(2(5(4(x1))))))) -> 4(2(3(0(0(0(0(2(5(1(x1)))))))))) 5(1(1(1(1(5(0(x1))))))) -> 3(4(5(5(4(2(0(0(5(0(x1)))))))))) 5(1(1(2(1(1(1(x1))))))) -> 0(5(0(0(5(1(2(0(2(1(x1)))))))))) 5(2(1(3(5(2(2(x1))))))) -> 4(2(1(2(3(0(0(0(0(0(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) 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(2(3(0(2(0(0(0(2(0(x1)))))))))) 1(1(3(3(1(x1))))) -> 5(3(3(2(3(0(2(2(2(0(x1)))))))))) 1(3(3(4(4(x1))))) -> 0(3(5(5(3(0(2(0(0(1(x1)))))))))) 3(1(3(5(2(x1))))) -> 5(3(1(5(3(0(0(2(2(2(x1)))))))))) 4(2(3(3(3(x1))))) -> 5(0(3(0(2(0(0(2(0(0(x1)))))))))) 0(1(2(1(1(4(x1)))))) -> 0(0(4(2(3(0(2(2(0(4(x1)))))))))) 2(4(4(3(4(2(x1)))))) -> 2(4(5(0(3(0(0(2(2(0(x1)))))))))) 3(3(0(3(3(3(x1)))))) -> 5(1(4(5(4(0(3(0(2(0(x1)))))))))) 3(5(2(5(4(1(x1)))))) -> 3(0(2(0(0(4(2(3(0(0(x1)))))))))) 0(1(3(1(3(1(0(x1))))))) -> 2(3(0(2(0(1(4(3(1(0(x1)))))))))) 0(1(3(3(3(5(3(x1))))))) -> 0(0(0(4(0(2(4(2(3(0(x1)))))))))) 0(1(3(3(4(3(4(x1))))))) -> 0(3(0(2(0(4(2(0(2(4(x1)))))))))) 0(2(3(3(5(1(3(x1))))))) -> 2(3(2(0(2(0(0(0(5(3(x1)))))))))) 1(1(1(3(1(2(1(x1))))))) -> 1(2(5(3(0(2(2(2(5(1(x1)))))))))) 1(1(3(1(2(3(3(x1))))))) -> 3(0(2(1(5(3(2(5(3(0(x1)))))))))) 1(1(3(3(4(0(1(x1))))))) -> 3(2(1(0(5(1(2(3(2(2(x1)))))))))) 1(2(4(3(4(3(3(x1))))))) -> 3(0(2(1(0(1(1(5(5(3(x1)))))))))) 1(3(0(5(4(1(3(x1))))))) -> 1(3(2(4(0(0(2(2(5(3(x1)))))))))) 1(3(3(0(3(2(4(x1))))))) -> 1(5(2(3(2(4(0(0(0(2(x1)))))))))) 1(3(3(3(3(1(3(x1))))))) -> 1(3(0(2(4(1(4(2(2(3(x1)))))))))) 1(3(3(3(3(4(4(x1))))))) -> 1(0(0(4(5(5(5(1(1(4(x1)))))))))) 1(3(3(4(1(1(2(x1))))))) -> 3(1(5(2(4(3(0(2(2(0(x1)))))))))) 1(3(3(5(1(1(1(x1))))))) -> 1(0(0(3(1(0(0(2(2(2(x1)))))))))) 1(3(4(4(1(3(5(x1))))))) -> 1(3(3(3(0(2(0(2(1(5(x1)))))))))) 1(3(5(0(2(3(4(x1))))))) -> 3(3(4(5(5(5(0(3(5(4(x1)))))))))) 1(3(5(1(1(5(1(x1))))))) -> 1(0(0(4(4(0(0(4(0(0(x1)))))))))) 1(3(5(1(3(4(3(x1))))))) -> 2(5(0(0(5(5(5(1(1(3(x1)))))))))) 2(1(3(3(3(3(1(x1))))))) -> 2(3(0(2(0(4(0(0(3(1(x1)))))))))) 2(3(1(1(1(1(0(x1))))))) -> 2(1(5(0(3(5(5(4(0(0(x1)))))))))) 2(3(1(3(3(1(1(x1))))))) -> 2(2(2(2(2(0(5(5(5(1(x1)))))))))) 2(3(3(5(1(3(3(x1))))))) -> 2(4(4(5(5(3(4(3(2(3(x1)))))))))) 2(4(3(3(4(2(5(x1))))))) -> 0(4(2(4(0(2(5(1(0(5(x1)))))))))) 2(5(2(0(3(5(1(x1))))))) -> 2(5(2(0(4(0(2(0(0(2(x1)))))))))) 3(1(3(1(1(1(4(x1))))))) -> 5(3(5(3(0(0(4(0(1(4(x1)))))))))) 3(1(3(1(4(4(0(x1))))))) -> 3(2(1(4(0(2(0(0(4(2(x1)))))))))) 3(2(5(4(4(2(4(x1))))))) -> 5(3(1(0(2(2(2(0(0(4(x1)))))))))) 3(3(3(3(4(5(2(x1))))))) -> 5(5(1(1(2(4(0(0(2(1(x1)))))))))) 3(3(5(1(5(1(1(x1))))))) -> 5(3(0(2(4(5(5(4(3(1(x1)))))))))) 3(3(5(2(5(0(4(x1))))))) -> 5(5(1(0(1(5(5(3(0(2(x1)))))))))) 3(3(5(3(0(1(1(x1))))))) -> 5(1(4(5(5(3(5(4(5(1(x1)))))))))) 3(4(1(1(3(1(1(x1))))))) -> 1(2(2(4(5(3(5(5(5(1(x1)))))))))) 3(4(2(0(4(1(1(x1))))))) -> 5(3(3(0(2(0(2(5(5(1(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 5(5(1(0(0(5(3(5(5(3(x1)))))))))) 4(1(1(1(4(5(1(x1))))))) -> 5(3(0(4(3(0(2(4(0(2(x1)))))))))) 4(2(1(3(3(3(4(x1))))))) -> 5(4(2(0(0(0(5(1(4(5(x1)))))))))) 4(3(3(5(1(4(1(x1))))))) -> 5(3(5(0(0(5(3(2(0(4(x1)))))))))) 4(4(0(4(3(3(3(x1))))))) -> 4(5(3(0(2(2(3(2(1(3(x1)))))))))) 4(4(3(3(3(4(1(x1))))))) -> 4(0(1(0(2(1(0(2(0(2(x1)))))))))) 4(5(1(5(2(4(3(x1))))))) -> 4(3(3(0(2(0(0(0(0(3(x1)))))))))) 4(5(2(0(3(2(5(x1))))))) -> 5(0(3(4(5(4(0(0(0(5(x1)))))))))) 4(5(2(5(4(1(3(x1))))))) -> 5(0(3(2(1(4(3(0(2(4(x1)))))))))) 5(0(1(3(2(5(4(x1))))))) -> 4(2(3(0(0(0(0(2(5(1(x1)))))))))) 5(1(1(1(1(5(0(x1))))))) -> 3(4(5(5(4(2(0(0(5(0(x1)))))))))) 5(1(1(2(1(1(1(x1))))))) -> 0(5(0(0(5(1(2(0(2(1(x1)))))))))) 5(2(1(3(5(2(2(x1))))))) -> 4(2(1(2(3(0(0(0(0(0(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) 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(2(3(0(2(0(0(0(2(0(x1)))))))))) 1(1(3(3(1(x1))))) -> 5(3(3(2(3(0(2(2(2(0(x1)))))))))) 1(3(3(4(4(x1))))) -> 0(3(5(5(3(0(2(0(0(1(x1)))))))))) 3(1(3(5(2(x1))))) -> 5(3(1(5(3(0(0(2(2(2(x1)))))))))) 4(2(3(3(3(x1))))) -> 5(0(3(0(2(0(0(2(0(0(x1)))))))))) 0(1(2(1(1(4(x1)))))) -> 0(0(4(2(3(0(2(2(0(4(x1)))))))))) 2(4(4(3(4(2(x1)))))) -> 2(4(5(0(3(0(0(2(2(0(x1)))))))))) 3(3(0(3(3(3(x1)))))) -> 5(1(4(5(4(0(3(0(2(0(x1)))))))))) 3(5(2(5(4(1(x1)))))) -> 3(0(2(0(0(4(2(3(0(0(x1)))))))))) 0(1(3(1(3(1(0(x1))))))) -> 2(3(0(2(0(1(4(3(1(0(x1)))))))))) 0(1(3(3(3(5(3(x1))))))) -> 0(0(0(4(0(2(4(2(3(0(x1)))))))))) 0(1(3(3(4(3(4(x1))))))) -> 0(3(0(2(0(4(2(0(2(4(x1)))))))))) 0(2(3(3(5(1(3(x1))))))) -> 2(3(2(0(2(0(0(0(5(3(x1)))))))))) 1(1(1(3(1(2(1(x1))))))) -> 1(2(5(3(0(2(2(2(5(1(x1)))))))))) 1(1(3(1(2(3(3(x1))))))) -> 3(0(2(1(5(3(2(5(3(0(x1)))))))))) 1(1(3(3(4(0(1(x1))))))) -> 3(2(1(0(5(1(2(3(2(2(x1)))))))))) 1(2(4(3(4(3(3(x1))))))) -> 3(0(2(1(0(1(1(5(5(3(x1)))))))))) 1(3(0(5(4(1(3(x1))))))) -> 1(3(2(4(0(0(2(2(5(3(x1)))))))))) 1(3(3(0(3(2(4(x1))))))) -> 1(5(2(3(2(4(0(0(0(2(x1)))))))))) 1(3(3(3(3(1(3(x1))))))) -> 1(3(0(2(4(1(4(2(2(3(x1)))))))))) 1(3(3(3(3(4(4(x1))))))) -> 1(0(0(4(5(5(5(1(1(4(x1)))))))))) 1(3(3(4(1(1(2(x1))))))) -> 3(1(5(2(4(3(0(2(2(0(x1)))))))))) 1(3(3(5(1(1(1(x1))))))) -> 1(0(0(3(1(0(0(2(2(2(x1)))))))))) 1(3(4(4(1(3(5(x1))))))) -> 1(3(3(3(0(2(0(2(1(5(x1)))))))))) 1(3(5(0(2(3(4(x1))))))) -> 3(3(4(5(5(5(0(3(5(4(x1)))))))))) 1(3(5(1(1(5(1(x1))))))) -> 1(0(0(4(4(0(0(4(0(0(x1)))))))))) 1(3(5(1(3(4(3(x1))))))) -> 2(5(0(0(5(5(5(1(1(3(x1)))))))))) 2(1(3(3(3(3(1(x1))))))) -> 2(3(0(2(0(4(0(0(3(1(x1)))))))))) 2(3(1(1(1(1(0(x1))))))) -> 2(1(5(0(3(5(5(4(0(0(x1)))))))))) 2(3(1(3(3(1(1(x1))))))) -> 2(2(2(2(2(0(5(5(5(1(x1)))))))))) 2(3(3(5(1(3(3(x1))))))) -> 2(4(4(5(5(3(4(3(2(3(x1)))))))))) 2(4(3(3(4(2(5(x1))))))) -> 0(4(2(4(0(2(5(1(0(5(x1)))))))))) 2(5(2(0(3(5(1(x1))))))) -> 2(5(2(0(4(0(2(0(0(2(x1)))))))))) 3(1(3(1(1(1(4(x1))))))) -> 5(3(5(3(0(0(4(0(1(4(x1)))))))))) 3(1(3(1(4(4(0(x1))))))) -> 3(2(1(4(0(2(0(0(4(2(x1)))))))))) 3(2(5(4(4(2(4(x1))))))) -> 5(3(1(0(2(2(2(0(0(4(x1)))))))))) 3(3(3(3(4(5(2(x1))))))) -> 5(5(1(1(2(4(0(0(2(1(x1)))))))))) 3(3(5(1(5(1(1(x1))))))) -> 5(3(0(2(4(5(5(4(3(1(x1)))))))))) 3(3(5(2(5(0(4(x1))))))) -> 5(5(1(0(1(5(5(3(0(2(x1)))))))))) 3(3(5(3(0(1(1(x1))))))) -> 5(1(4(5(5(3(5(4(5(1(x1)))))))))) 3(4(1(1(3(1(1(x1))))))) -> 1(2(2(4(5(3(5(5(5(1(x1)))))))))) 3(4(2(0(4(1(1(x1))))))) -> 5(3(3(0(2(0(2(5(5(1(x1)))))))))) 3(4(2(5(0(2(3(x1))))))) -> 5(5(1(0(0(5(3(5(5(3(x1)))))))))) 4(1(1(1(4(5(1(x1))))))) -> 5(3(0(4(3(0(2(4(0(2(x1)))))))))) 4(2(1(3(3(3(4(x1))))))) -> 5(4(2(0(0(0(5(1(4(5(x1)))))))))) 4(3(3(5(1(4(1(x1))))))) -> 5(3(5(0(0(5(3(2(0(4(x1)))))))))) 4(4(0(4(3(3(3(x1))))))) -> 4(5(3(0(2(2(3(2(1(3(x1)))))))))) 4(4(3(3(3(4(1(x1))))))) -> 4(0(1(0(2(1(0(2(0(2(x1)))))))))) 4(5(1(5(2(4(3(x1))))))) -> 4(3(3(0(2(0(0(0(0(3(x1)))))))))) 4(5(2(0(3(2(5(x1))))))) -> 5(0(3(4(5(4(0(0(0(5(x1)))))))))) 4(5(2(5(4(1(3(x1))))))) -> 5(0(3(2(1(4(3(0(2(4(x1)))))))))) 5(0(1(3(2(5(4(x1))))))) -> 4(2(3(0(0(0(0(2(5(1(x1)))))))))) 5(1(1(1(1(5(0(x1))))))) -> 3(4(5(5(4(2(0(0(5(0(x1)))))))))) 5(1(1(2(1(1(1(x1))))))) -> 0(5(0(0(5(1(2(0(2(1(x1)))))))))) 5(2(1(3(5(2(2(x1))))))) -> 4(2(1(2(3(0(0(0(0(0(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (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. "[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, 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, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624] {(109,110,[0_1|0, 1_1|0, 3_1|0, 4_1|0, 2_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_5_1|0, encode_4_1|0]), (109,111,[0_1|1, 1_1|1, 3_1|1, 4_1|1, 2_1|1, 5_1|1]), (109,112,[0_1|2]), (109,121,[0_1|2]), (109,130,[2_1|2]), (109,139,[0_1|2]), (109,148,[0_1|2]), (109,157,[2_1|2]), (109,166,[5_1|2]), (109,175,[3_1|2]), (109,184,[3_1|2]), (109,193,[1_1|2]), (109,202,[0_1|2]), (109,211,[3_1|2]), (109,220,[1_1|2]), (109,229,[1_1|2]), (109,238,[1_1|2]), (109,247,[1_1|2]), (109,256,[1_1|2]), (109,265,[1_1|2]), (109,274,[3_1|2]), (109,283,[1_1|2]), (109,292,[2_1|2]), (109,301,[3_1|2]), (109,310,[5_1|2]), (109,319,[5_1|2]), (109,328,[3_1|2]), (109,337,[5_1|2]), (109,346,[5_1|2]), (109,355,[5_1|2]), (109,364,[5_1|2]), (109,373,[5_1|2]), (109,382,[3_1|2]), (109,391,[5_1|2]), (109,400,[1_1|2]), (109,409,[5_1|2]), (109,418,[5_1|2]), (109,427,[5_1|2]), (109,436,[5_1|2]), (109,445,[5_1|2]), (109,454,[5_1|2]), (109,463,[4_1|2]), (109,472,[4_1|2]), (109,481,[4_1|2]), (109,490,[5_1|2]), (109,499,[5_1|2]), (109,508,[2_1|2]), (109,517,[0_1|2]), (109,526,[2_1|2]), (109,535,[2_1|2]), (109,544,[2_1|2]), (109,553,[2_1|2]), (109,562,[2_1|2]), (109,571,[4_1|2]), (109,580,[3_1|2]), (109,589,[0_1|2]), (109,598,[4_1|2]), (110,110,[cons_0_1|0, cons_1_1|0, cons_3_1|0, cons_4_1|0, cons_2_1|0, cons_5_1|0]), (111,110,[encArg_1|1]), (111,111,[0_1|1, 1_1|1, 3_1|1, 4_1|1, 2_1|1, 5_1|1]), (111,112,[0_1|2]), (111,121,[0_1|2]), (111,130,[2_1|2]), (111,139,[0_1|2]), (111,148,[0_1|2]), (111,157,[2_1|2]), (111,166,[5_1|2]), (111,175,[3_1|2]), (111,184,[3_1|2]), (111,193,[1_1|2]), (111,202,[0_1|2]), (111,211,[3_1|2]), (111,220,[1_1|2]), (111,229,[1_1|2]), (111,238,[1_1|2]), (111,247,[1_1|2]), (111,256,[1_1|2]), (111,265,[1_1|2]), (111,274,[3_1|2]), (111,283,[1_1|2]), (111,292,[2_1|2]), (111,301,[3_1|2]), (111,310,[5_1|2]), (111,319,[5_1|2]), (111,328,[3_1|2]), (111,337,[5_1|2]), (111,346,[5_1|2]), (111,355,[5_1|2]), (111,364,[5_1|2]), (111,373,[5_1|2]), (111,382,[3_1|2]), (111,391,[5_1|2]), (111,400,[1_1|2]), (111,409,[5_1|2]), (111,418,[5_1|2]), (111,427,[5_1|2]), (111,436,[5_1|2]), (111,445,[5_1|2]), (111,454,[5_1|2]), (111,463,[4_1|2]), (111,472,[4_1|2]), (111,481,[4_1|2]), (111,490,[5_1|2]), (111,499,[5_1|2]), (111,508,[2_1|2]), (111,517,[0_1|2]), (111,526,[2_1|2]), (111,535,[2_1|2]), (111,544,[2_1|2]), (111,553,[2_1|2]), (111,562,[2_1|2]), (111,571,[4_1|2]), (111,580,[3_1|2]), (111,589,[0_1|2]), (111,598,[4_1|2]), (112,113,[2_1|2]), (113,114,[3_1|2]), (114,115,[0_1|2]), (115,116,[2_1|2]), (116,117,[0_1|2]), (117,118,[0_1|2]), (118,119,[0_1|2]), (119,120,[2_1|2]), (120,111,[0_1|2]), (120,130,[0_1|2, 2_1|2]), (120,157,[0_1|2, 2_1|2]), (120,292,[0_1|2]), (120,508,[0_1|2]), (120,526,[0_1|2]), (120,535,[0_1|2]), (120,544,[0_1|2]), (120,553,[0_1|2]), (120,562,[0_1|2]), (120,194,[0_1|2]), (120,401,[0_1|2]), (120,112,[0_1|2]), (120,121,[0_1|2]), (120,139,[0_1|2]), (120,148,[0_1|2]), (121,122,[0_1|2]), (122,123,[4_1|2]), (123,124,[2_1|2]), (124,125,[3_1|2]), (125,126,[0_1|2]), (126,127,[2_1|2]), (127,128,[2_1|2]), (128,129,[0_1|2]), (129,111,[4_1|2]), (129,463,[4_1|2]), (129,472,[4_1|2]), (129,481,[4_1|2]), (129,571,[4_1|2]), (129,598,[4_1|2]), (129,427,[5_1|2]), (129,436,[5_1|2]), (129,445,[5_1|2]), (129,454,[5_1|2]), (129,490,[5_1|2]), (129,499,[5_1|2]), (130,131,[3_1|2]), (131,132,[0_1|2]), (132,133,[2_1|2]), (133,134,[0_1|2]), (134,135,[1_1|2]), (135,136,[4_1|2]), (136,137,[3_1|2]), (137,138,[1_1|2]), (138,111,[0_1|2]), (138,112,[0_1|2]), (138,121,[0_1|2]), (138,139,[0_1|2]), (138,148,[0_1|2]), (138,202,[0_1|2]), (138,517,[0_1|2]), (138,589,[0_1|2]), (138,239,[0_1|2]), (138,248,[0_1|2]), (138,284,[0_1|2]), (138,130,[2_1|2]), (138,157,[2_1|2]), (139,140,[0_1|2]), (140,141,[0_1|2]), (141,142,[4_1|2]), (142,143,[0_1|2]), (143,144,[2_1|2]), (144,145,[4_1|2]), (145,146,[2_1|2]), (146,147,[3_1|2]), (147,111,[0_1|2]), (147,175,[0_1|2]), (147,184,[0_1|2]), (147,211,[0_1|2]), (147,274,[0_1|2]), (147,301,[0_1|2]), (147,328,[0_1|2]), (147,382,[0_1|2]), (147,580,[0_1|2]), (147,167,[0_1|2]), (147,311,[0_1|2]), (147,320,[0_1|2]), (147,356,[0_1|2]), (147,392,[0_1|2]), (147,410,[0_1|2]), (147,446,[0_1|2]), (147,455,[0_1|2]), (147,112,[0_1|2]), (147,121,[0_1|2]), (147,130,[2_1|2]), (147,139,[0_1|2]), (147,148,[0_1|2]), (147,157,[2_1|2]), (148,149,[3_1|2]), (149,150,[0_1|2]), (150,151,[2_1|2]), (151,152,[0_1|2]), (152,153,[4_1|2]), (153,154,[2_1|2]), (154,155,[0_1|2]), (155,156,[2_1|2]), (155,508,[2_1|2]), (155,517,[0_1|2]), (156,111,[4_1|2]), (156,463,[4_1|2]), (156,472,[4_1|2]), (156,481,[4_1|2]), (156,571,[4_1|2]), (156,598,[4_1|2]), (156,581,[4_1|2]), (156,427,[5_1|2]), (156,436,[5_1|2]), (156,445,[5_1|2]), (156,454,[5_1|2]), (156,490,[5_1|2]), (156,499,[5_1|2]), (157,158,[3_1|2]), (158,159,[2_1|2]), (159,160,[0_1|2]), (160,161,[2_1|2]), (161,162,[0_1|2]), (162,163,[0_1|2]), (163,164,[0_1|2]), (164,165,[5_1|2]), (165,111,[3_1|2]), (165,175,[3_1|2]), (165,184,[3_1|2]), (165,211,[3_1|2]), (165,274,[3_1|2]), (165,301,[3_1|2]), (165,328,[3_1|2]), (165,382,[3_1|2]), (165,580,[3_1|2]), (165,230,[3_1|2]), (165,257,[3_1|2]), (165,266,[3_1|2]), (165,310,[5_1|2]), (165,319,[5_1|2]), (165,337,[5_1|2]), (165,346,[5_1|2]), (165,355,[5_1|2]), (165,364,[5_1|2]), (165,373,[5_1|2]), (165,391,[5_1|2]), (165,400,[1_1|2]), (165,409,[5_1|2]), (165,418,[5_1|2]), (166,167,[3_1|2]), (167,168,[3_1|2]), 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(577,578,[2_1|2]), (578,579,[5_1|2]), (578,580,[3_1|2]), (578,589,[0_1|2]), (579,111,[1_1|2]), (579,463,[1_1|2]), (579,472,[1_1|2]), (579,481,[1_1|2]), (579,571,[1_1|2]), (579,598,[1_1|2]), (579,437,[1_1|2]), (579,166,[5_1|2]), (579,175,[3_1|2]), (579,184,[3_1|2]), (579,193,[1_1|2]), (579,202,[0_1|2]), (579,211,[3_1|2]), (579,220,[1_1|2]), (579,229,[1_1|2]), (579,238,[1_1|2]), (579,247,[1_1|2]), (579,256,[1_1|2]), (579,265,[1_1|2]), (579,274,[3_1|2]), (579,283,[1_1|2]), (579,292,[2_1|2]), (579,301,[3_1|2]), (580,581,[4_1|2]), (581,582,[5_1|2]), (582,583,[5_1|2]), (583,584,[4_1|2]), (584,585,[2_1|2]), (585,586,[0_1|2]), (586,587,[0_1|2]), (587,588,[5_1|2]), (587,571,[4_1|2]), (588,111,[0_1|2]), (588,112,[0_1|2]), (588,121,[0_1|2]), (588,139,[0_1|2]), (588,148,[0_1|2]), (588,202,[0_1|2]), (588,517,[0_1|2]), (588,589,[0_1|2]), (588,428,[0_1|2]), (588,491,[0_1|2]), (588,500,[0_1|2]), (588,130,[2_1|2]), (588,157,[2_1|2]), (589,590,[5_1|2]), (590,591,[0_1|2]), (591,592,[0_1|2]), (592,593,[5_1|2]), (593,594,[1_1|2]), (594,595,[2_1|2]), (595,596,[0_1|2]), (596,597,[2_1|2]), (596,526,[2_1|2]), (597,111,[1_1|2]), (597,193,[1_1|2]), (597,220,[1_1|2]), (597,229,[1_1|2]), (597,238,[1_1|2]), (597,247,[1_1|2]), (597,256,[1_1|2]), (597,265,[1_1|2]), (597,283,[1_1|2]), (597,400,[1_1|2]), (597,166,[5_1|2]), (597,175,[3_1|2]), (597,184,[3_1|2]), (597,202,[0_1|2]), (597,211,[3_1|2]), (597,274,[3_1|2]), (597,292,[2_1|2]), (597,301,[3_1|2]), (598,599,[2_1|2]), (599,600,[1_1|2]), (600,601,[2_1|2]), (601,602,[3_1|2]), (602,603,[0_1|2]), (603,604,[0_1|2]), (604,605,[0_1|2]), (605,606,[0_1|2]), (606,111,[0_1|2]), (606,130,[0_1|2, 2_1|2]), (606,157,[0_1|2, 2_1|2]), (606,292,[0_1|2]), (606,508,[0_1|2]), (606,526,[0_1|2]), (606,535,[0_1|2]), (606,544,[0_1|2]), (606,553,[0_1|2]), (606,562,[0_1|2]), (606,545,[0_1|2]), (606,112,[0_1|2]), (606,121,[0_1|2]), (606,139,[0_1|2]), (606,148,[0_1|2]), (607,608,[2_1|3]), (608,609,[3_1|3]), (609,610,[0_1|3]), (610,611,[2_1|3]), (611,612,[0_1|3]), (612,613,[0_1|3]), (613,614,[0_1|3]), (614,615,[2_1|3]), (615,194,[0_1|3]), (615,401,[0_1|3]), (616,617,[3_1|3]), (617,618,[3_1|3]), (618,619,[2_1|3]), (619,620,[3_1|3]), (620,621,[0_1|3]), (621,622,[2_1|3]), (622,623,[2_1|3]), (623,624,[2_1|3]), (624,212,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)