/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 88 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 155 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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(1(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_2(x_1)) -> 2(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)) encArg(cons_4(x_1)) -> 4(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(1(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_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)) encArg(cons_4(x_1)) -> 4(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: 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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(1(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_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)) encArg(cons_4(x_1)) -> 4(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: 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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(1(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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)) encArg(cons_4(x_1)) -> 4(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: 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 4. The certificate found is represented by the following graph. "[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, 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, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649] {(98,99,[0_1|0, 2_1|0, 5_1|0, 1_1|0, 3_1|0, 4_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]), (98,100,[0_1|1, 2_1|1, 5_1|1, 1_1|1, 3_1|1, 4_1|1]), (98,101,[1_1|2]), (98,110,[0_1|2]), 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1_1|1, 3_1|1, 4_1|1]), (100,101,[1_1|2]), (100,110,[0_1|2]), (100,119,[0_1|2]), (100,128,[1_1|2]), (100,137,[1_1|2]), (100,146,[1_1|2]), (100,155,[0_1|2]), (100,164,[0_1|2]), (100,173,[0_1|2]), (100,182,[1_1|2]), (100,191,[0_1|2]), (100,200,[1_1|2]), (100,209,[3_1|2]), (100,218,[2_1|2]), (100,227,[4_1|2]), (100,236,[2_1|2]), (100,245,[2_1|2]), (100,254,[2_1|2]), (100,263,[3_1|2]), (100,272,[2_1|2]), (100,281,[0_1|2]), (100,290,[1_1|2]), (100,299,[1_1|2]), (100,308,[1_1|2]), (100,317,[3_1|2]), (100,326,[5_1|2]), (100,335,[5_1|2]), (100,344,[5_1|2]), (100,353,[5_1|2]), (100,362,[1_1|2]), (100,371,[5_1|2]), (100,380,[1_1|2]), (100,389,[5_1|2]), (100,398,[1_1|2]), (100,407,[5_1|2]), (100,416,[5_1|2]), (100,425,[1_1|2]), (100,434,[1_1|2]), (100,443,[1_1|2]), (100,452,[1_1|2]), (100,461,[1_1|2]), (100,470,[3_1|2]), (100,479,[1_1|2]), (100,488,[2_1|2]), (100,497,[1_1|2]), (100,506,[2_1|2]), (100,515,[3_1|2]), (100,524,[2_1|2]), (100,533,[4_1|2]), (100,542,[1_1|3]), (101,102,[1_1|2]), 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(506,507,[1_1|2]), (507,508,[0_1|2]), (508,509,[4_1|2]), (509,510,[2_1|2]), (510,511,[3_1|2]), (511,512,[4_1|2]), (512,513,[1_1|2]), (513,514,[1_1|2]), (514,100,[4_1|2]), (514,326,[4_1|2]), (514,335,[4_1|2]), (514,344,[4_1|2]), (514,353,[4_1|2]), (514,371,[4_1|2]), (514,389,[4_1|2]), (514,407,[4_1|2]), (514,416,[4_1|2]), (514,237,[4_1|2]), (514,488,[2_1|2]), (514,497,[1_1|2]), (514,506,[2_1|2]), (514,515,[3_1|2]), (514,524,[2_1|2]), (514,533,[4_1|2]), (515,516,[5_1|2]), (516,517,[1_1|2]), (517,518,[1_1|2]), (518,519,[1_1|2]), (519,520,[4_1|2]), (520,521,[0_1|2]), (521,522,[2_1|2]), (522,523,[2_1|2]), (522,236,[2_1|2]), (522,605,[2_1|3]), (523,100,[0_1|2]), (523,110,[0_1|2]), (523,119,[0_1|2]), (523,155,[0_1|2]), (523,164,[0_1|2]), (523,173,[0_1|2]), (523,191,[0_1|2]), (523,281,[0_1|2]), (523,156,[0_1|2]), (523,192,[0_1|2]), (523,391,[0_1|2]), (523,101,[1_1|2]), (523,128,[1_1|2]), (523,137,[1_1|2]), (523,146,[1_1|2]), (523,182,[1_1|2]), (523,200,[1_1|2]), (523,209,[3_1|2]), (523,569,[1_1|3]), (524,525,[0_1|2]), (525,526,[4_1|2]), (526,527,[5_1|2]), (527,528,[3_1|2]), (528,529,[4_1|2]), (529,530,[4_1|2]), (530,531,[3_1|2]), (531,532,[3_1|2]), (531,470,[3_1|2]), (532,100,[3_1|2]), (532,209,[3_1|2]), (532,263,[3_1|2]), (532,317,[3_1|2]), (532,470,[3_1|2]), (532,515,[3_1|2]), (532,174,[3_1|2]), (532,282,[3_1|2]), (532,479,[1_1|2]), (533,534,[1_1|2]), (534,535,[4_1|2]), (535,536,[1_1|2]), (536,537,[2_1|2]), (537,538,[1_1|2]), (538,539,[0_1|2]), (539,540,[4_1|2]), (540,541,[4_1|2]), (541,100,[1_1|2]), (541,218,[1_1|2]), (541,236,[1_1|2]), (541,245,[1_1|2]), (541,254,[1_1|2]), (541,272,[1_1|2]), (541,488,[1_1|2]), (541,506,[1_1|2]), (541,524,[1_1|2]), (541,111,[1_1|2]), (541,165,[1_1|2]), (541,193,[1_1|2]), (541,425,[1_1|2]), (541,434,[1_1|2]), (541,443,[1_1|2]), (541,452,[1_1|2]), (541,461,[1_1|2]), (542,543,[1_1|3]), (543,544,[2_1|3]), (544,545,[3_1|3]), (545,546,[1_1|3]), (546,547,[4_1|3]), (547,548,[3_1|3]), (548,549,[1_1|3]), (549,550,[5_1|3]), (550,156,[4_1|3]), (550,192,[4_1|3]), (551,552,[1_1|3]), (552,553,[4_1|3]), (553,554,[1_1|3]), (554,555,[2_1|3]), (555,556,[1_1|3]), (556,557,[0_1|3]), (557,558,[4_1|3]), (558,559,[4_1|3]), (559,193,[1_1|3]), (560,561,[1_1|3]), (561,562,[4_1|3]), (562,563,[5_1|3]), (563,564,[5_1|3]), (564,565,[0_1|3]), (565,566,[1_1|3]), (566,567,[3_1|3]), (567,568,[1_1|3]), (568,392,[2_1|3]), (569,570,[1_1|3]), (570,571,[2_1|3]), (571,572,[3_1|3]), (572,573,[1_1|3]), (573,574,[4_1|3]), (574,575,[3_1|3]), (575,576,[1_1|3]), (576,577,[5_1|3]), (577,110,[4_1|3]), (577,119,[4_1|3]), (577,155,[4_1|3]), (577,164,[4_1|3]), (577,173,[4_1|3]), (577,191,[4_1|3]), (577,281,[4_1|3]), (577,156,[4_1|3]), (577,192,[4_1|3]), (577,391,[4_1|3]), (578,579,[2_1|3]), (579,580,[1_1|3]), (580,581,[1_1|3]), (581,582,[1_1|3]), (582,583,[3_1|3]), (583,584,[3_1|3]), (584,585,[4_1|3]), (585,586,[1_1|3]), (586,237,[4_1|3]), (587,588,[1_1|3]), (588,589,[2_1|3]), (589,590,[3_1|3]), (590,591,[1_1|3]), (591,592,[4_1|3]), (592,593,[3_1|3]), (593,594,[1_1|3]), (594,595,[5_1|3]), (595,391,[4_1|3]), (596,597,[1_1|3]), (597,598,[2_1|3]), (598,599,[3_1|3]), (599,600,[1_1|3]), (600,601,[4_1|3]), (601,602,[3_1|3]), (602,603,[1_1|3]), (603,604,[5_1|3]), (604,401,[4_1|3]), (605,606,[5_1|3]), (606,607,[2_1|3]), (607,608,[2_1|3]), (608,609,[2_1|3]), (609,610,[1_1|3]), (610,611,[3_1|3]), (611,612,[1_1|3]), (612,613,[1_1|3]), (613,346,[1_1|3]), (614,615,[0_1|3]), (615,616,[1_1|3]), (616,617,[1_1|3]), (617,618,[1_1|3]), (618,619,[0_1|3]), (618,641,[1_1|4]), (619,620,[0_1|3]), (620,621,[1_1|3]), (621,622,[4_1|3]), (622,193,[1_1|3]), (623,624,[1_1|3]), (624,625,[2_1|3]), (625,626,[3_1|3]), (626,627,[1_1|3]), (627,628,[4_1|3]), (628,629,[3_1|3]), (629,630,[1_1|3]), (630,631,[5_1|3]), (631,494,[4_1|3]), (632,633,[1_1|3]), (633,634,[2_1|3]), (634,635,[3_1|3]), (635,636,[1_1|3]), (636,637,[4_1|3]), (637,638,[3_1|3]), (638,639,[1_1|3]), (639,640,[5_1|3]), (640,501,[4_1|3]), (641,642,[1_1|4]), (642,643,[2_1|4]), (643,644,[3_1|4]), (644,645,[1_1|4]), (645,646,[4_1|4]), (646,647,[3_1|4]), (647,648,[1_1|4]), (648,649,[5_1|4]), (649,620,[4_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1)