/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 43 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 102 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: 3(4(5(x1))) -> 3(5(0(3(1(0(3(0(4(0(x1)))))))))) 4(2(0(x1))) -> 3(1(3(0(1(0(3(0(3(0(x1)))))))))) 4(5(1(x1))) -> 0(3(0(3(5(0(5(0(5(4(x1)))))))))) 0(2(2(2(2(x1))))) -> 0(0(3(1(0(5(2(5(4(2(x1)))))))))) 2(2(2(5(0(x1))))) -> 2(3(3(3(0(2(3(5(0(5(x1)))))))))) 2(2(5(2(5(x1))))) -> 5(5(4(1(3(1(4(0(5(5(x1)))))))))) 3(2(2(0(2(x1))))) -> 4(2(1(3(5(3(5(5(4(3(x1)))))))))) 3(4(5(1(4(x1))))) -> 3(5(0(1(0(4(5(5(1(1(x1)))))))))) 4(1(4(1(5(x1))))) -> 5(5(5(3(1(4(0(4(1(5(x1)))))))))) 4(5(1(2(2(x1))))) -> 3(1(5(4(0(0(0(2(3(4(x1)))))))))) 5(4(5(2(2(x1))))) -> 5(1(2(5(5(3(4(0(0(1(x1)))))))))) 0(1(4(1(2(2(x1)))))) -> 0(0(4(2(0(5(5(5(5(4(x1)))))))))) 0(1(5(1(4(1(x1)))))) -> 4(5(5(5(4(5(4(0(0(0(x1)))))))))) 0(2(1(5(0(2(x1)))))) -> 3(3(5(5(5(0(1(3(3(2(x1)))))))))) 0(2(2(1(5(2(x1)))))) -> 0(5(0(0(2(1(2(5(3(3(x1)))))))))) 0(4(5(4(4(1(x1)))))) -> 3(2(0(4(5(5(2(0(5(3(x1)))))))))) 1(3(2(4(5(4(x1)))))) -> 1(2(0(5(4(2(1(1(0(4(x1)))))))))) 2(1(2(4(5(2(x1)))))) -> 0(5(5(1(4(3(1(2(2(2(x1)))))))))) 2(4(5(2(3(2(x1)))))) -> 2(0(4(3(5(2(0(2(3(3(x1)))))))))) 3(4(1(4(5(2(x1)))))) -> 5(3(5(5(5(5(0(3(1(3(x1)))))))))) 4(2(0(1(5(4(x1)))))) -> 0(1(5(1(1(0(3(4(1(1(x1)))))))))) 4(4(1(4(1(0(x1)))))) -> 0(2(0(0(2(5(4(3(1(0(x1)))))))))) 4(5(4(5(2(5(x1)))))) -> 0(5(0(0(3(3(1(5(4(3(x1)))))))))) 5(2(5(1(5(1(x1)))))) -> 5(0(4(3(1(0(4(5(0(5(x1)))))))))) 5(4(5(0(3(2(x1)))))) -> 5(5(1(1(3(5(1(1(5(2(x1)))))))))) 0(2(3(4(4(1(3(x1))))))) -> 0(3(1(1(3(0(4(1(3(3(x1)))))))))) 0(2(4(1(4(1(5(x1))))))) -> 0(4(1(1(2(0(0(5(5(2(x1)))))))))) 1(1(5(2(3(4(5(x1))))))) -> 4(3(1(0(2(1(1(1(4(0(x1)))))))))) 1(4(0(1(2(4(1(x1))))))) -> 1(0(5(3(4(0(2(3(0(2(x1)))))))))) 2(1(5(2(3(2(2(x1))))))) -> 2(1(5(3(3(1(2(5(5(5(x1)))))))))) 2(4(2(0(0(4(5(x1))))))) -> 3(5(5(1(3(4(5(3(2(0(x1)))))))))) 2(5(4(1(5(1(5(x1))))))) -> 2(2(0(4(4(0(4(0(2(5(x1)))))))))) 2(5(5(3(1(4(5(x1))))))) -> 2(3(0(4(0(0(0(1(0(2(x1)))))))))) 4(1(0(1(2(4(5(x1))))))) -> 5(5(4(2(3(4(3(5(4(5(x1)))))))))) 4(2(2(2(2(5(2(x1))))))) -> 4(3(3(0(2(4(1(0(5(1(x1)))))))))) 4(2(3(1(4(5(1(x1))))))) -> 0(4(1(0(5(1(2(1(4(1(x1)))))))))) 4(2(4(5(2(5(4(x1))))))) -> 0(3(3(5(4(2(2(3(2(0(x1)))))))))) 4(4(1(2(4(2(2(x1))))))) -> 0(3(3(3(0(3(1(5(4(0(x1)))))))))) 4(4(4(2(3(2(5(x1))))))) -> 0(1(3(4(0(4(1(5(0(5(x1)))))))))) 4(5(2(4(4(1(1(x1))))))) -> 4(5(2(5(0(3(2(0(5(0(x1)))))))))) 5(0(0(4(5(4(2(x1))))))) -> 5(1(3(5(3(3(2(0(4(2(x1)))))))))) 5(4(5(2(0(4(5(x1))))))) -> 5(0(0(1(1(1(3(4(1(5(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) 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)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_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)) ---------------------------------------- (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: 3(4(5(x1))) -> 3(5(0(3(1(0(3(0(4(0(x1)))))))))) 4(2(0(x1))) -> 3(1(3(0(1(0(3(0(3(0(x1)))))))))) 4(5(1(x1))) -> 0(3(0(3(5(0(5(0(5(4(x1)))))))))) 0(2(2(2(2(x1))))) -> 0(0(3(1(0(5(2(5(4(2(x1)))))))))) 2(2(2(5(0(x1))))) -> 2(3(3(3(0(2(3(5(0(5(x1)))))))))) 2(2(5(2(5(x1))))) -> 5(5(4(1(3(1(4(0(5(5(x1)))))))))) 3(2(2(0(2(x1))))) -> 4(2(1(3(5(3(5(5(4(3(x1)))))))))) 3(4(5(1(4(x1))))) -> 3(5(0(1(0(4(5(5(1(1(x1)))))))))) 4(1(4(1(5(x1))))) -> 5(5(5(3(1(4(0(4(1(5(x1)))))))))) 4(5(1(2(2(x1))))) -> 3(1(5(4(0(0(0(2(3(4(x1)))))))))) 5(4(5(2(2(x1))))) -> 5(1(2(5(5(3(4(0(0(1(x1)))))))))) 0(1(4(1(2(2(x1)))))) -> 0(0(4(2(0(5(5(5(5(4(x1)))))))))) 0(1(5(1(4(1(x1)))))) -> 4(5(5(5(4(5(4(0(0(0(x1)))))))))) 0(2(1(5(0(2(x1)))))) -> 3(3(5(5(5(0(1(3(3(2(x1)))))))))) 0(2(2(1(5(2(x1)))))) -> 0(5(0(0(2(1(2(5(3(3(x1)))))))))) 0(4(5(4(4(1(x1)))))) -> 3(2(0(4(5(5(2(0(5(3(x1)))))))))) 1(3(2(4(5(4(x1)))))) -> 1(2(0(5(4(2(1(1(0(4(x1)))))))))) 2(1(2(4(5(2(x1)))))) -> 0(5(5(1(4(3(1(2(2(2(x1)))))))))) 2(4(5(2(3(2(x1)))))) -> 2(0(4(3(5(2(0(2(3(3(x1)))))))))) 3(4(1(4(5(2(x1)))))) -> 5(3(5(5(5(5(0(3(1(3(x1)))))))))) 4(2(0(1(5(4(x1)))))) -> 0(1(5(1(1(0(3(4(1(1(x1)))))))))) 4(4(1(4(1(0(x1)))))) -> 0(2(0(0(2(5(4(3(1(0(x1)))))))))) 4(5(4(5(2(5(x1)))))) -> 0(5(0(0(3(3(1(5(4(3(x1)))))))))) 5(2(5(1(5(1(x1)))))) -> 5(0(4(3(1(0(4(5(0(5(x1)))))))))) 5(4(5(0(3(2(x1)))))) -> 5(5(1(1(3(5(1(1(5(2(x1)))))))))) 0(2(3(4(4(1(3(x1))))))) -> 0(3(1(1(3(0(4(1(3(3(x1)))))))))) 0(2(4(1(4(1(5(x1))))))) -> 0(4(1(1(2(0(0(5(5(2(x1)))))))))) 1(1(5(2(3(4(5(x1))))))) -> 4(3(1(0(2(1(1(1(4(0(x1)))))))))) 1(4(0(1(2(4(1(x1))))))) -> 1(0(5(3(4(0(2(3(0(2(x1)))))))))) 2(1(5(2(3(2(2(x1))))))) -> 2(1(5(3(3(1(2(5(5(5(x1)))))))))) 2(4(2(0(0(4(5(x1))))))) -> 3(5(5(1(3(4(5(3(2(0(x1)))))))))) 2(5(4(1(5(1(5(x1))))))) -> 2(2(0(4(4(0(4(0(2(5(x1)))))))))) 2(5(5(3(1(4(5(x1))))))) -> 2(3(0(4(0(0(0(1(0(2(x1)))))))))) 4(1(0(1(2(4(5(x1))))))) -> 5(5(4(2(3(4(3(5(4(5(x1)))))))))) 4(2(2(2(2(5(2(x1))))))) -> 4(3(3(0(2(4(1(0(5(1(x1)))))))))) 4(2(3(1(4(5(1(x1))))))) -> 0(4(1(0(5(1(2(1(4(1(x1)))))))))) 4(2(4(5(2(5(4(x1))))))) -> 0(3(3(5(4(2(2(3(2(0(x1)))))))))) 4(4(1(2(4(2(2(x1))))))) -> 0(3(3(3(0(3(1(5(4(0(x1)))))))))) 4(4(4(2(3(2(5(x1))))))) -> 0(1(3(4(0(4(1(5(0(5(x1)))))))))) 4(5(2(4(4(1(1(x1))))))) -> 4(5(2(5(0(3(2(0(5(0(x1)))))))))) 5(0(0(4(5(4(2(x1))))))) -> 5(1(3(5(3(3(2(0(4(2(x1)))))))))) 5(4(5(2(0(4(5(x1))))))) -> 5(0(0(1(1(1(3(4(1(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) 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)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_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)) 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: 3(4(5(x1))) -> 3(5(0(3(1(0(3(0(4(0(x1)))))))))) 4(2(0(x1))) -> 3(1(3(0(1(0(3(0(3(0(x1)))))))))) 4(5(1(x1))) -> 0(3(0(3(5(0(5(0(5(4(x1)))))))))) 0(2(2(2(2(x1))))) -> 0(0(3(1(0(5(2(5(4(2(x1)))))))))) 2(2(2(5(0(x1))))) -> 2(3(3(3(0(2(3(5(0(5(x1)))))))))) 2(2(5(2(5(x1))))) -> 5(5(4(1(3(1(4(0(5(5(x1)))))))))) 3(2(2(0(2(x1))))) -> 4(2(1(3(5(3(5(5(4(3(x1)))))))))) 3(4(5(1(4(x1))))) -> 3(5(0(1(0(4(5(5(1(1(x1)))))))))) 4(1(4(1(5(x1))))) -> 5(5(5(3(1(4(0(4(1(5(x1)))))))))) 4(5(1(2(2(x1))))) -> 3(1(5(4(0(0(0(2(3(4(x1)))))))))) 5(4(5(2(2(x1))))) -> 5(1(2(5(5(3(4(0(0(1(x1)))))))))) 0(1(4(1(2(2(x1)))))) -> 0(0(4(2(0(5(5(5(5(4(x1)))))))))) 0(1(5(1(4(1(x1)))))) -> 4(5(5(5(4(5(4(0(0(0(x1)))))))))) 0(2(1(5(0(2(x1)))))) -> 3(3(5(5(5(0(1(3(3(2(x1)))))))))) 0(2(2(1(5(2(x1)))))) -> 0(5(0(0(2(1(2(5(3(3(x1)))))))))) 0(4(5(4(4(1(x1)))))) -> 3(2(0(4(5(5(2(0(5(3(x1)))))))))) 1(3(2(4(5(4(x1)))))) -> 1(2(0(5(4(2(1(1(0(4(x1)))))))))) 2(1(2(4(5(2(x1)))))) -> 0(5(5(1(4(3(1(2(2(2(x1)))))))))) 2(4(5(2(3(2(x1)))))) -> 2(0(4(3(5(2(0(2(3(3(x1)))))))))) 3(4(1(4(5(2(x1)))))) -> 5(3(5(5(5(5(0(3(1(3(x1)))))))))) 4(2(0(1(5(4(x1)))))) -> 0(1(5(1(1(0(3(4(1(1(x1)))))))))) 4(4(1(4(1(0(x1)))))) -> 0(2(0(0(2(5(4(3(1(0(x1)))))))))) 4(5(4(5(2(5(x1)))))) -> 0(5(0(0(3(3(1(5(4(3(x1)))))))))) 5(2(5(1(5(1(x1)))))) -> 5(0(4(3(1(0(4(5(0(5(x1)))))))))) 5(4(5(0(3(2(x1)))))) -> 5(5(1(1(3(5(1(1(5(2(x1)))))))))) 0(2(3(4(4(1(3(x1))))))) -> 0(3(1(1(3(0(4(1(3(3(x1)))))))))) 0(2(4(1(4(1(5(x1))))))) -> 0(4(1(1(2(0(0(5(5(2(x1)))))))))) 1(1(5(2(3(4(5(x1))))))) -> 4(3(1(0(2(1(1(1(4(0(x1)))))))))) 1(4(0(1(2(4(1(x1))))))) -> 1(0(5(3(4(0(2(3(0(2(x1)))))))))) 2(1(5(2(3(2(2(x1))))))) -> 2(1(5(3(3(1(2(5(5(5(x1)))))))))) 2(4(2(0(0(4(5(x1))))))) -> 3(5(5(1(3(4(5(3(2(0(x1)))))))))) 2(5(4(1(5(1(5(x1))))))) -> 2(2(0(4(4(0(4(0(2(5(x1)))))))))) 2(5(5(3(1(4(5(x1))))))) -> 2(3(0(4(0(0(0(1(0(2(x1)))))))))) 4(1(0(1(2(4(5(x1))))))) -> 5(5(4(2(3(4(3(5(4(5(x1)))))))))) 4(2(2(2(2(5(2(x1))))))) -> 4(3(3(0(2(4(1(0(5(1(x1)))))))))) 4(2(3(1(4(5(1(x1))))))) -> 0(4(1(0(5(1(2(1(4(1(x1)))))))))) 4(2(4(5(2(5(4(x1))))))) -> 0(3(3(5(4(2(2(3(2(0(x1)))))))))) 4(4(1(2(4(2(2(x1))))))) -> 0(3(3(3(0(3(1(5(4(0(x1)))))))))) 4(4(4(2(3(2(5(x1))))))) -> 0(1(3(4(0(4(1(5(0(5(x1)))))))))) 4(5(2(4(4(1(1(x1))))))) -> 4(5(2(5(0(3(2(0(5(0(x1)))))))))) 5(0(0(4(5(4(2(x1))))))) -> 5(1(3(5(3(3(2(0(4(2(x1)))))))))) 5(4(5(2(0(4(5(x1))))))) -> 5(0(0(1(1(1(3(4(1(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) 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)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_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)) 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: 3(4(5(x1))) -> 3(5(0(3(1(0(3(0(4(0(x1)))))))))) 4(2(0(x1))) -> 3(1(3(0(1(0(3(0(3(0(x1)))))))))) 4(5(1(x1))) -> 0(3(0(3(5(0(5(0(5(4(x1)))))))))) 0(2(2(2(2(x1))))) -> 0(0(3(1(0(5(2(5(4(2(x1)))))))))) 2(2(2(5(0(x1))))) -> 2(3(3(3(0(2(3(5(0(5(x1)))))))))) 2(2(5(2(5(x1))))) -> 5(5(4(1(3(1(4(0(5(5(x1)))))))))) 3(2(2(0(2(x1))))) -> 4(2(1(3(5(3(5(5(4(3(x1)))))))))) 3(4(5(1(4(x1))))) -> 3(5(0(1(0(4(5(5(1(1(x1)))))))))) 4(1(4(1(5(x1))))) -> 5(5(5(3(1(4(0(4(1(5(x1)))))))))) 4(5(1(2(2(x1))))) -> 3(1(5(4(0(0(0(2(3(4(x1)))))))))) 5(4(5(2(2(x1))))) -> 5(1(2(5(5(3(4(0(0(1(x1)))))))))) 0(1(4(1(2(2(x1)))))) -> 0(0(4(2(0(5(5(5(5(4(x1)))))))))) 0(1(5(1(4(1(x1)))))) -> 4(5(5(5(4(5(4(0(0(0(x1)))))))))) 0(2(1(5(0(2(x1)))))) -> 3(3(5(5(5(0(1(3(3(2(x1)))))))))) 0(2(2(1(5(2(x1)))))) -> 0(5(0(0(2(1(2(5(3(3(x1)))))))))) 0(4(5(4(4(1(x1)))))) -> 3(2(0(4(5(5(2(0(5(3(x1)))))))))) 1(3(2(4(5(4(x1)))))) -> 1(2(0(5(4(2(1(1(0(4(x1)))))))))) 2(1(2(4(5(2(x1)))))) -> 0(5(5(1(4(3(1(2(2(2(x1)))))))))) 2(4(5(2(3(2(x1)))))) -> 2(0(4(3(5(2(0(2(3(3(x1)))))))))) 3(4(1(4(5(2(x1)))))) -> 5(3(5(5(5(5(0(3(1(3(x1)))))))))) 4(2(0(1(5(4(x1)))))) -> 0(1(5(1(1(0(3(4(1(1(x1)))))))))) 4(4(1(4(1(0(x1)))))) -> 0(2(0(0(2(5(4(3(1(0(x1)))))))))) 4(5(4(5(2(5(x1)))))) -> 0(5(0(0(3(3(1(5(4(3(x1)))))))))) 5(2(5(1(5(1(x1)))))) -> 5(0(4(3(1(0(4(5(0(5(x1)))))))))) 5(4(5(0(3(2(x1)))))) -> 5(5(1(1(3(5(1(1(5(2(x1)))))))))) 0(2(3(4(4(1(3(x1))))))) -> 0(3(1(1(3(0(4(1(3(3(x1)))))))))) 0(2(4(1(4(1(5(x1))))))) -> 0(4(1(1(2(0(0(5(5(2(x1)))))))))) 1(1(5(2(3(4(5(x1))))))) -> 4(3(1(0(2(1(1(1(4(0(x1)))))))))) 1(4(0(1(2(4(1(x1))))))) -> 1(0(5(3(4(0(2(3(0(2(x1)))))))))) 2(1(5(2(3(2(2(x1))))))) -> 2(1(5(3(3(1(2(5(5(5(x1)))))))))) 2(4(2(0(0(4(5(x1))))))) -> 3(5(5(1(3(4(5(3(2(0(x1)))))))))) 2(5(4(1(5(1(5(x1))))))) -> 2(2(0(4(4(0(4(0(2(5(x1)))))))))) 2(5(5(3(1(4(5(x1))))))) -> 2(3(0(4(0(0(0(1(0(2(x1)))))))))) 4(1(0(1(2(4(5(x1))))))) -> 5(5(4(2(3(4(3(5(4(5(x1)))))))))) 4(2(2(2(2(5(2(x1))))))) -> 4(3(3(0(2(4(1(0(5(1(x1)))))))))) 4(2(3(1(4(5(1(x1))))))) -> 0(4(1(0(5(1(2(1(4(1(x1)))))))))) 4(2(4(5(2(5(4(x1))))))) -> 0(3(3(5(4(2(2(3(2(0(x1)))))))))) 4(4(1(2(4(2(2(x1))))))) -> 0(3(3(3(0(3(1(5(4(0(x1)))))))))) 4(4(4(2(3(2(5(x1))))))) -> 0(1(3(4(0(4(1(5(0(5(x1)))))))))) 4(5(2(4(4(1(1(x1))))))) -> 4(5(2(5(0(3(2(0(5(0(x1)))))))))) 5(0(0(4(5(4(2(x1))))))) -> 5(1(3(5(3(3(2(0(4(2(x1)))))))))) 5(4(5(2(0(4(5(x1))))))) -> 5(0(0(1(1(1(3(4(1(5(x1)))))))))) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) 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)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_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)) 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. "[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, 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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] {(137,138,[3_1|0, 4_1|0, 0_1|0, 2_1|0, 5_1|0, 1_1|0, encArg_1|0, encode_3_1|0, encode_4_1|0, encode_5_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0]), (137,139,[3_1|1, 4_1|1, 0_1|1, 2_1|1, 5_1|1, 1_1|1]), (137,140,[3_1|2]), (137,149,[3_1|2]), (137,158,[5_1|2]), (137,167,[4_1|2]), (137,176,[3_1|2]), (137,185,[0_1|2]), (137,194,[4_1|2]), (137,203,[0_1|2]), (137,212,[0_1|2]), (137,221,[0_1|2]), (137,230,[3_1|2]), (137,239,[0_1|2]), (137,248,[4_1|2]), (137,257,[5_1|2]), (137,266,[5_1|2]), (137,275,[0_1|2]), (137,284,[0_1|2]), (137,293,[0_1|2]), (137,302,[0_1|2]), (137,311,[0_1|2]), 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(382,284,[5_1|2]), (382,293,[5_1|2]), (382,302,[5_1|2]), (382,311,[5_1|2]), (382,329,[5_1|2]), (382,338,[5_1|2]), (382,347,[5_1|2]), (382,392,[5_1|2]), (382,456,[5_1|2]), (382,474,[5_1|2]), (382,446,[5_1|2]), (382,455,[5_1|2]), (382,464,[5_1|2]), (382,473,[5_1|2]), (382,482,[5_1|2]), (383,384,[5_1|2]), (384,385,[4_1|2]), (385,386,[1_1|2]), (386,387,[3_1|2]), (387,388,[1_1|2]), (388,389,[4_1|2]), (389,390,[0_1|2]), (390,391,[5_1|2]), (391,139,[5_1|2]), (391,158,[5_1|2]), (391,257,[5_1|2]), (391,266,[5_1|2]), (391,383,[5_1|2]), (391,446,[5_1|2]), (391,455,[5_1|2]), (391,464,[5_1|2]), (391,473,[5_1|2]), (391,482,[5_1|2]), (392,393,[5_1|2]), (393,394,[5_1|2]), (394,395,[1_1|2]), (395,396,[4_1|2]), (396,397,[3_1|2]), (397,398,[1_1|2]), (398,399,[2_1|2]), (398,374,[2_1|2]), (398,599,[2_1|3]), (399,400,[2_1|2]), (399,374,[2_1|2]), (399,383,[5_1|2]), (400,139,[2_1|2]), (400,374,[2_1|2]), (400,401,[2_1|2]), (400,410,[2_1|2]), (400,428,[2_1|2]), (400,437,[2_1|2]), (400,250,[2_1|2]), (400,383,[5_1|2]), (400,392,[0_1|2]), (400,419,[3_1|2]), (401,402,[1_1|2]), (402,403,[5_1|2]), (403,404,[3_1|2]), (404,405,[3_1|2]), (405,406,[1_1|2]), (406,407,[2_1|2]), (407,408,[5_1|2]), (408,409,[5_1|2]), (409,139,[5_1|2]), (409,374,[5_1|2]), (409,401,[5_1|2]), (409,410,[5_1|2]), (409,428,[5_1|2]), (409,437,[5_1|2]), (409,429,[5_1|2]), (409,446,[5_1|2]), (409,455,[5_1|2]), (409,464,[5_1|2]), (409,473,[5_1|2]), (409,482,[5_1|2]), (410,411,[0_1|2]), (411,412,[4_1|2]), (412,413,[3_1|2]), (413,414,[5_1|2]), (414,415,[2_1|2]), (415,416,[0_1|2]), (416,417,[2_1|2]), (417,418,[3_1|2]), (418,139,[3_1|2]), (418,374,[3_1|2]), (418,401,[3_1|2]), (418,410,[3_1|2]), (418,428,[3_1|2]), (418,437,[3_1|2]), (418,366,[3_1|2]), (418,140,[3_1|2]), (418,149,[3_1|2]), (418,158,[5_1|2]), (418,167,[4_1|2]), (418,518,[3_1|3]), (419,420,[5_1|2]), (420,421,[5_1|2]), (421,422,[1_1|2]), (422,423,[3_1|2]), (422,608,[3_1|3]), (423,424,[4_1|2]), (424,425,[5_1|2]), (425,426,[3_1|2]), (426,427,[2_1|2]), (427,139,[0_1|2]), (427,158,[0_1|2]), (427,257,[0_1|2]), (427,266,[0_1|2]), (427,383,[0_1|2]), (427,446,[0_1|2]), (427,455,[0_1|2]), (427,464,[0_1|2]), (427,473,[0_1|2]), (427,482,[0_1|2]), (427,249,[0_1|2]), (427,357,[0_1|2]), (427,302,[0_1|2]), (427,311,[0_1|2]), (427,320,[3_1|2]), (427,329,[0_1|2]), (427,338,[0_1|2]), (427,347,[0_1|2]), (427,356,[4_1|2]), (427,365,[3_1|2]), (428,429,[2_1|2]), (429,430,[0_1|2]), (430,431,[4_1|2]), (431,432,[4_1|2]), (432,433,[0_1|2]), (433,434,[4_1|2]), (434,435,[0_1|2]), (435,436,[2_1|2]), (435,428,[2_1|2]), (435,437,[2_1|2]), (436,139,[5_1|2]), (436,158,[5_1|2]), (436,257,[5_1|2]), (436,266,[5_1|2]), (436,383,[5_1|2]), (436,446,[5_1|2]), (436,455,[5_1|2]), (436,464,[5_1|2]), (436,473,[5_1|2]), (436,482,[5_1|2]), (437,438,[3_1|2]), (438,439,[0_1|2]), (439,440,[4_1|2]), (440,441,[0_1|2]), (441,442,[0_1|2]), (442,443,[0_1|2]), (443,444,[1_1|2]), (444,445,[0_1|2]), (444,302,[0_1|2]), (444,311,[0_1|2]), (444,320,[3_1|2]), (444,329,[0_1|2]), (444,338,[0_1|2]), (445,139,[2_1|2]), (445,158,[2_1|2]), (445,257,[2_1|2]), (445,266,[2_1|2]), (445,383,[2_1|2, 5_1|2]), (445,446,[2_1|2]), (445,455,[2_1|2]), (445,464,[2_1|2]), (445,473,[2_1|2]), (445,482,[2_1|2]), (445,249,[2_1|2]), (445,357,[2_1|2]), (445,374,[2_1|2]), (445,392,[0_1|2]), (445,401,[2_1|2]), (445,410,[2_1|2]), (445,419,[3_1|2]), (445,428,[2_1|2]), (445,437,[2_1|2]), (446,447,[1_1|2]), (447,448,[2_1|2]), (448,449,[5_1|2]), (449,450,[5_1|2]), (450,451,[3_1|2]), (451,452,[4_1|2]), (452,453,[0_1|2]), (453,454,[0_1|2]), (453,347,[0_1|2]), (453,356,[4_1|2]), (454,139,[1_1|2]), (454,374,[1_1|2]), (454,401,[1_1|2]), (454,410,[1_1|2]), (454,428,[1_1|2]), (454,437,[1_1|2]), (454,429,[1_1|2]), (454,491,[1_1|2]), (454,500,[4_1|2]), (454,509,[1_1|2]), (455,456,[0_1|2]), (456,457,[0_1|2]), (457,458,[1_1|2]), (458,459,[1_1|2]), (459,460,[1_1|2]), (460,461,[3_1|2]), (461,462,[4_1|2]), (462,463,[1_1|2]), (463,139,[5_1|2]), (463,158,[5_1|2]), (463,257,[5_1|2]), (463,266,[5_1|2]), (463,383,[5_1|2]), (463,446,[5_1|2]), (463,455,[5_1|2]), (463,464,[5_1|2]), (463,473,[5_1|2]), (463,482,[5_1|2]), (463,249,[5_1|2]), (463,357,[5_1|2]), (464,465,[5_1|2]), (465,466,[1_1|2]), (466,467,[1_1|2]), (467,468,[3_1|2]), (468,469,[5_1|2]), (469,470,[1_1|2]), (469,500,[4_1|2]), (470,471,[1_1|2]), (471,472,[5_1|2]), (471,473,[5_1|2]), (472,139,[2_1|2]), (472,374,[2_1|2]), (472,401,[2_1|2]), (472,410,[2_1|2]), (472,428,[2_1|2]), (472,437,[2_1|2]), (472,366,[2_1|2]), (472,383,[5_1|2]), (472,392,[0_1|2]), (472,419,[3_1|2]), (473,474,[0_1|2]), (474,475,[4_1|2]), (475,476,[3_1|2]), (476,477,[1_1|2]), (477,478,[0_1|2]), (478,479,[4_1|2]), (479,480,[5_1|2]), (480,481,[0_1|2]), (481,139,[5_1|2]), (481,491,[5_1|2]), (481,509,[5_1|2]), (481,447,[5_1|2]), (481,483,[5_1|2]), (481,446,[5_1|2]), (481,455,[5_1|2]), (481,464,[5_1|2]), (481,473,[5_1|2]), (481,482,[5_1|2]), (482,483,[1_1|2]), (483,484,[3_1|2]), (484,485,[5_1|2]), (485,486,[3_1|2]), (486,487,[3_1|2]), (487,488,[2_1|2]), (488,489,[0_1|2]), (489,490,[4_1|2]), (489,176,[3_1|2]), (489,185,[0_1|2]), (489,194,[4_1|2]), (489,203,[0_1|2]), (489,212,[0_1|2]), (489,581,[3_1|3]), (490,139,[2_1|2]), (490,374,[2_1|2]), (490,401,[2_1|2]), (490,410,[2_1|2]), (490,428,[2_1|2]), (490,437,[2_1|2]), (490,168,[2_1|2]), (490,383,[5_1|2]), (490,392,[0_1|2]), (490,419,[3_1|2]), (491,492,[2_1|2]), (492,493,[0_1|2]), (493,494,[5_1|2]), (494,495,[4_1|2]), (495,496,[2_1|2]), (496,497,[1_1|2]), (497,498,[1_1|2]), (498,499,[0_1|2]), (498,365,[3_1|2]), (499,139,[4_1|2]), (499,167,[4_1|2]), (499,194,[4_1|2]), (499,248,[4_1|2]), (499,356,[4_1|2]), (499,500,[4_1|2]), (499,176,[3_1|2]), (499,185,[0_1|2]), (499,203,[0_1|2]), (499,212,[0_1|2]), (499,221,[0_1|2]), (499,230,[3_1|2]), (499,239,[0_1|2]), (499,257,[5_1|2]), (499,266,[5_1|2]), (499,275,[0_1|2]), (499,284,[0_1|2]), (499,293,[0_1|2]), (499,527,[3_1|3]), (499,536,[0_1|3]), (500,501,[3_1|2]), (501,502,[1_1|2]), (502,503,[0_1|2]), (503,504,[2_1|2]), (504,505,[1_1|2]), (505,506,[1_1|2]), (506,507,[1_1|2]), (506,509,[1_1|2]), (507,508,[4_1|2]), (508,139,[0_1|2]), (508,158,[0_1|2]), (508,257,[0_1|2]), (508,266,[0_1|2]), (508,383,[0_1|2]), (508,446,[0_1|2]), (508,455,[0_1|2]), (508,464,[0_1|2]), (508,473,[0_1|2]), (508,482,[0_1|2]), (508,249,[0_1|2]), (508,357,[0_1|2]), (508,302,[0_1|2]), (508,311,[0_1|2]), (508,320,[3_1|2]), (508,329,[0_1|2]), (508,338,[0_1|2]), (508,347,[0_1|2]), (508,356,[4_1|2]), (508,365,[3_1|2]), (509,510,[0_1|2]), (510,511,[5_1|2]), (511,512,[3_1|2]), (512,513,[4_1|2]), (513,514,[0_1|2]), (514,515,[2_1|2]), (515,516,[3_1|2]), (516,517,[0_1|2]), (516,302,[0_1|2]), (516,311,[0_1|2]), (516,320,[3_1|2]), (516,329,[0_1|2]), (516,338,[0_1|2]), (517,139,[2_1|2]), (517,491,[2_1|2]), (517,509,[2_1|2]), (517,374,[2_1|2]), (517,383,[5_1|2]), (517,392,[0_1|2]), (517,401,[2_1|2]), (517,410,[2_1|2]), (517,419,[3_1|2]), (517,428,[2_1|2]), (517,437,[2_1|2]), (518,519,[5_1|3]), (519,520,[0_1|3]), (520,521,[3_1|3]), (521,522,[1_1|3]), (522,523,[0_1|3]), (523,524,[3_1|3]), (524,525,[0_1|3]), (525,526,[4_1|3]), (526,249,[0_1|3]), (526,357,[0_1|3]), (527,528,[1_1|3]), (528,529,[3_1|3]), (529,530,[0_1|3]), (530,531,[1_1|3]), (531,532,[0_1|3]), (532,533,[3_1|3]), (533,534,[0_1|3]), (534,535,[3_1|3]), (535,411,[0_1|3]), (535,493,[0_1|3]), (535,430,[0_1|3]), (536,537,[3_1|3]), (537,538,[0_1|3]), (538,539,[3_1|3]), (539,540,[5_1|3]), (540,541,[0_1|3]), (541,542,[5_1|3]), (542,543,[0_1|3]), (543,544,[5_1|3]), (544,447,[4_1|3]), (544,483,[4_1|3]), (545,546,[5_1|3]), (546,547,[0_1|3]), (547,548,[3_1|3]), (548,549,[1_1|3]), (549,550,[0_1|3]), (550,551,[3_1|3]), (551,552,[0_1|3]), (552,553,[4_1|3]), (553,158,[0_1|3]), (553,257,[0_1|3]), (553,266,[0_1|3]), (553,383,[0_1|3]), (553,446,[0_1|3]), (553,455,[0_1|3]), (553,464,[0_1|3]), (553,473,[0_1|3]), (553,482,[0_1|3]), (553,249,[0_1|3]), (554,555,[1_1|3]), (555,556,[2_1|3]), (556,557,[5_1|3]), (557,558,[5_1|3]), (558,559,[3_1|3]), (559,560,[4_1|3]), (560,561,[0_1|3]), (561,562,[0_1|3]), (562,429,[1_1|3]), (563,564,[5_1|3]), (564,565,[0_1|3]), (565,566,[0_1|3]), (566,567,[3_1|3]), (567,568,[3_1|3]), (568,569,[1_1|3]), (569,570,[5_1|3]), (570,571,[4_1|3]), (571,251,[3_1|3]), (572,573,[3_1|3]), (573,574,[0_1|3]), (574,575,[3_1|3]), (575,576,[5_1|3]), (576,577,[0_1|3]), (577,578,[5_1|3]), (578,579,[0_1|3]), (579,580,[5_1|3]), (580,491,[4_1|3]), (580,509,[4_1|3]), (580,447,[4_1|3]), (580,483,[4_1|3]), (580,527,[3_1|3]), (581,582,[1_1|3]), (582,583,[3_1|3]), (583,584,[0_1|3]), (584,585,[1_1|3]), (585,586,[0_1|3]), (586,587,[3_1|3]), (587,588,[0_1|3]), (588,589,[3_1|3]), (589,185,[0_1|3]), (589,203,[0_1|3]), (589,212,[0_1|3]), (589,221,[0_1|3]), (589,239,[0_1|3]), (589,275,[0_1|3]), (589,284,[0_1|3]), (589,293,[0_1|3]), (589,302,[0_1|3]), (589,311,[0_1|3]), (589,329,[0_1|3]), (589,338,[0_1|3]), (589,347,[0_1|3]), (589,392,[0_1|3]), (589,411,[0_1|3]), (589,430,[0_1|3]), (590,591,[1_1|3]), (591,592,[3_1|3]), (592,593,[0_1|3]), (593,594,[1_1|3]), (594,595,[0_1|3]), (595,596,[3_1|3]), (596,597,[0_1|3]), (597,598,[3_1|3]), (598,351,[0_1|3]), (599,600,[3_1|3]), (600,601,[3_1|3]), (601,602,[3_1|3]), (602,603,[0_1|3]), (603,604,[2_1|3]), (604,605,[3_1|3]), (605,606,[5_1|3]), (606,607,[0_1|3]), (607,456,[5_1|3]), (607,474,[5_1|3]), (607,252,[5_1|3]), (608,609,[5_1|3]), (609,610,[0_1|3]), (610,611,[3_1|3]), (611,612,[1_1|3]), (612,613,[0_1|3]), (613,614,[3_1|3]), (614,615,[0_1|3]), (615,616,[4_1|3]), (616,425,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)