/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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, 143 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(0(x1))) -> 0(4(2(2(3(2(3(3(5(5(x1)))))))))) 0(0(5(x1))) -> 1(2(3(4(1(5(0(3(3(4(x1)))))))))) 0(0(0(2(x1)))) -> 0(2(2(1(0(4(2(3(4(2(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(2(2(2(3(4(3(0(3(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(3(3(5(5(4(0(2(4(x1)))))))))) 5(4(2(0(x1)))) -> 5(3(4(4(0(3(5(2(3(2(x1)))))))))) 0(0(0(4(0(x1))))) -> 1(4(3(0(5(5(5(5(1(2(x1)))))))))) 0(1(3(0(0(x1))))) -> 0(5(1(0(2(3(5(2(3(0(x1)))))))))) 4(0(1(3(4(x1))))) -> 4(2(3(2(4(4(1(4(4(4(x1)))))))))) 4(0(2(0(2(x1))))) -> 5(3(1(4(3(2(1(1(2(3(x1)))))))))) 4(0(2(5(0(x1))))) -> 3(1(2(3(4(4(5(1(0(3(x1)))))))))) 5(0(0(2(5(x1))))) -> 0(1(0(3(3(4(5(3(0(5(x1)))))))))) 5(2(0(0(0(x1))))) -> 5(5(2(3(3(4(2(5(0(5(x1)))))))))) 1(0(0(5(0(0(x1)))))) -> 2(1(5(5(5(5(0(1(0(4(x1)))))))))) 1(3(5(0(5(0(x1)))))) -> 2(4(2(3(2(3(5(3(3(0(x1)))))))))) 1(5(4(0(2(4(x1)))))) -> 1(0(1(2(3(1(1(0(5(5(x1)))))))))) 2(0(0(0(0(3(x1)))))) -> 1(0(5(4(3(3(1(3(5(1(x1)))))))))) 2(0(0(5(1(0(x1)))))) -> 2(5(3(5(2(2(3(5(2(3(x1)))))))))) 2(2(1(3(1(4(x1)))))) -> 1(1(5(3(2(2(2(1(4(3(x1)))))))))) 3(0(2(0(1(3(x1)))))) -> 3(0(5(5(1(5(3(3(4(3(x1)))))))))) 4(1(2(1(4(1(x1)))))) -> 3(5(5(2(1(4(4(4(5(1(x1)))))))))) 5(0(1(3(3(2(x1)))))) -> 0(3(3(5(5(5(1(5(1(0(x1)))))))))) 5(1(3(0(2(0(x1)))))) -> 2(4(5(5(1(0(1(5(3(4(x1)))))))))) 0(0(0(1(4(0(4(x1))))))) -> 2(4(4(2(1(2(1(2(4(5(x1)))))))))) 0(2(0(2(1(4(2(x1))))))) -> 2(4(4(2(4(1(1(3(4(2(x1)))))))))) 0(3(0(0(2(2(4(x1))))))) -> 0(1(0(5(3(5(5(4(3(0(x1)))))))))) 0(3(5(0(0(2(0(x1))))))) -> 0(2(3(1(5(4(3(0(4(4(x1)))))))))) 1(0(1(3(0(0(2(x1))))))) -> 1(0(1(0(3(5(0(1(1(4(x1)))))))))) 1(0(2(0(0(2(0(x1))))))) -> 1(2(2(1(3(1(1(0(4(4(x1)))))))))) 2(0(0(2(1(3(5(x1))))))) -> 2(5(4(2(2(1(0(0(3(4(x1)))))))))) 2(0(0(4(0(0(3(x1))))))) -> 1(1(1(1(0(0(5(4(2(3(x1)))))))))) 2(1(4(0(0(0(0(x1))))))) -> 2(1(4(0(2(1(1(3(1(4(x1)))))))))) 2(4(0(5(3(2(1(x1))))))) -> 2(0(3(0(4(2(3(1(2(1(x1)))))))))) 2(5(2(0(4(5(0(x1))))))) -> 1(2(2(2(2(5(0(2(1(0(x1)))))))))) 3(0(2(2(0(1(3(x1))))))) -> 3(1(4(5(1(1(4(2(3(4(x1)))))))))) 3(4(0(2(4(0(0(x1))))))) -> 3(2(4(5(3(1(1(0(3(0(x1)))))))))) 4(0(2(4(0(1(3(x1))))))) -> 4(3(1(1(2(2(1(3(0(3(x1)))))))))) 4(4(0(5(4(0(4(x1))))))) -> 4(4(5(1(1(0(3(4(2(4(x1)))))))))) 5(0(2(5(0(2(0(x1))))))) -> 3(2(3(4(2(5(0(2(4(4(x1)))))))))) 5(2(1(4(1(4(2(x1))))))) -> 1(3(2(4(3(1(2(0(3(0(x1)))))))))) 5(4(1(1(0(5(0(x1))))))) -> 4(3(4(5(5(5(1(4(0(3(x1)))))))))) 5(4(1(1(2(0(4(x1))))))) -> 4(3(5(5(4(4(4(2(2(4(x1)))))))))) 5(4(2(0(0(1(3(x1))))))) -> 2(3(4(2(4(3(1(1(2(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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_1(x_1) -> 1(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(0(x1))) -> 0(4(2(2(3(2(3(3(5(5(x1)))))))))) 0(0(5(x1))) -> 1(2(3(4(1(5(0(3(3(4(x1)))))))))) 0(0(0(2(x1)))) -> 0(2(2(1(0(4(2(3(4(2(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(2(2(2(3(4(3(0(3(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(3(3(5(5(4(0(2(4(x1)))))))))) 5(4(2(0(x1)))) -> 5(3(4(4(0(3(5(2(3(2(x1)))))))))) 0(0(0(4(0(x1))))) -> 1(4(3(0(5(5(5(5(1(2(x1)))))))))) 0(1(3(0(0(x1))))) -> 0(5(1(0(2(3(5(2(3(0(x1)))))))))) 4(0(1(3(4(x1))))) -> 4(2(3(2(4(4(1(4(4(4(x1)))))))))) 4(0(2(0(2(x1))))) -> 5(3(1(4(3(2(1(1(2(3(x1)))))))))) 4(0(2(5(0(x1))))) -> 3(1(2(3(4(4(5(1(0(3(x1)))))))))) 5(0(0(2(5(x1))))) -> 0(1(0(3(3(4(5(3(0(5(x1)))))))))) 5(2(0(0(0(x1))))) -> 5(5(2(3(3(4(2(5(0(5(x1)))))))))) 1(0(0(5(0(0(x1)))))) -> 2(1(5(5(5(5(0(1(0(4(x1)))))))))) 1(3(5(0(5(0(x1)))))) -> 2(4(2(3(2(3(5(3(3(0(x1)))))))))) 1(5(4(0(2(4(x1)))))) -> 1(0(1(2(3(1(1(0(5(5(x1)))))))))) 2(0(0(0(0(3(x1)))))) -> 1(0(5(4(3(3(1(3(5(1(x1)))))))))) 2(0(0(5(1(0(x1)))))) -> 2(5(3(5(2(2(3(5(2(3(x1)))))))))) 2(2(1(3(1(4(x1)))))) -> 1(1(5(3(2(2(2(1(4(3(x1)))))))))) 3(0(2(0(1(3(x1)))))) -> 3(0(5(5(1(5(3(3(4(3(x1)))))))))) 4(1(2(1(4(1(x1)))))) -> 3(5(5(2(1(4(4(4(5(1(x1)))))))))) 5(0(1(3(3(2(x1)))))) -> 0(3(3(5(5(5(1(5(1(0(x1)))))))))) 5(1(3(0(2(0(x1)))))) -> 2(4(5(5(1(0(1(5(3(4(x1)))))))))) 0(0(0(1(4(0(4(x1))))))) -> 2(4(4(2(1(2(1(2(4(5(x1)))))))))) 0(2(0(2(1(4(2(x1))))))) -> 2(4(4(2(4(1(1(3(4(2(x1)))))))))) 0(3(0(0(2(2(4(x1))))))) -> 0(1(0(5(3(5(5(4(3(0(x1)))))))))) 0(3(5(0(0(2(0(x1))))))) -> 0(2(3(1(5(4(3(0(4(4(x1)))))))))) 1(0(1(3(0(0(2(x1))))))) -> 1(0(1(0(3(5(0(1(1(4(x1)))))))))) 1(0(2(0(0(2(0(x1))))))) -> 1(2(2(1(3(1(1(0(4(4(x1)))))))))) 2(0(0(2(1(3(5(x1))))))) -> 2(5(4(2(2(1(0(0(3(4(x1)))))))))) 2(0(0(4(0(0(3(x1))))))) -> 1(1(1(1(0(0(5(4(2(3(x1)))))))))) 2(1(4(0(0(0(0(x1))))))) -> 2(1(4(0(2(1(1(3(1(4(x1)))))))))) 2(4(0(5(3(2(1(x1))))))) -> 2(0(3(0(4(2(3(1(2(1(x1)))))))))) 2(5(2(0(4(5(0(x1))))))) -> 1(2(2(2(2(5(0(2(1(0(x1)))))))))) 3(0(2(2(0(1(3(x1))))))) -> 3(1(4(5(1(1(4(2(3(4(x1)))))))))) 3(4(0(2(4(0(0(x1))))))) -> 3(2(4(5(3(1(1(0(3(0(x1)))))))))) 4(0(2(4(0(1(3(x1))))))) -> 4(3(1(1(2(2(1(3(0(3(x1)))))))))) 4(4(0(5(4(0(4(x1))))))) -> 4(4(5(1(1(0(3(4(2(4(x1)))))))))) 5(0(2(5(0(2(0(x1))))))) -> 3(2(3(4(2(5(0(2(4(4(x1)))))))))) 5(2(1(4(1(4(2(x1))))))) -> 1(3(2(4(3(1(2(0(3(0(x1)))))))))) 5(4(1(1(0(5(0(x1))))))) -> 4(3(4(5(5(5(1(4(0(3(x1)))))))))) 5(4(1(1(2(0(4(x1))))))) -> 4(3(5(5(4(4(4(2(2(4(x1)))))))))) 5(4(2(0(0(1(3(x1))))))) -> 2(3(4(2(4(3(1(1(2(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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_1(x_1) -> 1(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(0(x1))) -> 0(4(2(2(3(2(3(3(5(5(x1)))))))))) 0(0(5(x1))) -> 1(2(3(4(1(5(0(3(3(4(x1)))))))))) 0(0(0(2(x1)))) -> 0(2(2(1(0(4(2(3(4(2(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(2(2(2(3(4(3(0(3(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(3(3(5(5(4(0(2(4(x1)))))))))) 5(4(2(0(x1)))) -> 5(3(4(4(0(3(5(2(3(2(x1)))))))))) 0(0(0(4(0(x1))))) -> 1(4(3(0(5(5(5(5(1(2(x1)))))))))) 0(1(3(0(0(x1))))) -> 0(5(1(0(2(3(5(2(3(0(x1)))))))))) 4(0(1(3(4(x1))))) -> 4(2(3(2(4(4(1(4(4(4(x1)))))))))) 4(0(2(0(2(x1))))) -> 5(3(1(4(3(2(1(1(2(3(x1)))))))))) 4(0(2(5(0(x1))))) -> 3(1(2(3(4(4(5(1(0(3(x1)))))))))) 5(0(0(2(5(x1))))) -> 0(1(0(3(3(4(5(3(0(5(x1)))))))))) 5(2(0(0(0(x1))))) -> 5(5(2(3(3(4(2(5(0(5(x1)))))))))) 1(0(0(5(0(0(x1)))))) -> 2(1(5(5(5(5(0(1(0(4(x1)))))))))) 1(3(5(0(5(0(x1)))))) -> 2(4(2(3(2(3(5(3(3(0(x1)))))))))) 1(5(4(0(2(4(x1)))))) -> 1(0(1(2(3(1(1(0(5(5(x1)))))))))) 2(0(0(0(0(3(x1)))))) -> 1(0(5(4(3(3(1(3(5(1(x1)))))))))) 2(0(0(5(1(0(x1)))))) -> 2(5(3(5(2(2(3(5(2(3(x1)))))))))) 2(2(1(3(1(4(x1)))))) -> 1(1(5(3(2(2(2(1(4(3(x1)))))))))) 3(0(2(0(1(3(x1)))))) -> 3(0(5(5(1(5(3(3(4(3(x1)))))))))) 4(1(2(1(4(1(x1)))))) -> 3(5(5(2(1(4(4(4(5(1(x1)))))))))) 5(0(1(3(3(2(x1)))))) -> 0(3(3(5(5(5(1(5(1(0(x1)))))))))) 5(1(3(0(2(0(x1)))))) -> 2(4(5(5(1(0(1(5(3(4(x1)))))))))) 0(0(0(1(4(0(4(x1))))))) -> 2(4(4(2(1(2(1(2(4(5(x1)))))))))) 0(2(0(2(1(4(2(x1))))))) -> 2(4(4(2(4(1(1(3(4(2(x1)))))))))) 0(3(0(0(2(2(4(x1))))))) -> 0(1(0(5(3(5(5(4(3(0(x1)))))))))) 0(3(5(0(0(2(0(x1))))))) -> 0(2(3(1(5(4(3(0(4(4(x1)))))))))) 1(0(1(3(0(0(2(x1))))))) -> 1(0(1(0(3(5(0(1(1(4(x1)))))))))) 1(0(2(0(0(2(0(x1))))))) -> 1(2(2(1(3(1(1(0(4(4(x1)))))))))) 2(0(0(2(1(3(5(x1))))))) -> 2(5(4(2(2(1(0(0(3(4(x1)))))))))) 2(0(0(4(0(0(3(x1))))))) -> 1(1(1(1(0(0(5(4(2(3(x1)))))))))) 2(1(4(0(0(0(0(x1))))))) -> 2(1(4(0(2(1(1(3(1(4(x1)))))))))) 2(4(0(5(3(2(1(x1))))))) -> 2(0(3(0(4(2(3(1(2(1(x1)))))))))) 2(5(2(0(4(5(0(x1))))))) -> 1(2(2(2(2(5(0(2(1(0(x1)))))))))) 3(0(2(2(0(1(3(x1))))))) -> 3(1(4(5(1(1(4(2(3(4(x1)))))))))) 3(4(0(2(4(0(0(x1))))))) -> 3(2(4(5(3(1(1(0(3(0(x1)))))))))) 4(0(2(4(0(1(3(x1))))))) -> 4(3(1(1(2(2(1(3(0(3(x1)))))))))) 4(4(0(5(4(0(4(x1))))))) -> 4(4(5(1(1(0(3(4(2(4(x1)))))))))) 5(0(2(5(0(2(0(x1))))))) -> 3(2(3(4(2(5(0(2(4(4(x1)))))))))) 5(2(1(4(1(4(2(x1))))))) -> 1(3(2(4(3(1(2(0(3(0(x1)))))))))) 5(4(1(1(0(5(0(x1))))))) -> 4(3(4(5(5(5(1(4(0(3(x1)))))))))) 5(4(1(1(2(0(4(x1))))))) -> 4(3(5(5(4(4(4(2(2(4(x1)))))))))) 5(4(2(0(0(1(3(x1))))))) -> 2(3(4(2(4(3(1(1(2(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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_1(x_1) -> 1(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(0(x1))) -> 0(4(2(2(3(2(3(3(5(5(x1)))))))))) 0(0(5(x1))) -> 1(2(3(4(1(5(0(3(3(4(x1)))))))))) 0(0(0(2(x1)))) -> 0(2(2(1(0(4(2(3(4(2(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(2(2(2(3(4(3(0(3(x1)))))))))) 1(3(0(0(x1)))) -> 1(2(3(3(5(5(4(0(2(4(x1)))))))))) 5(4(2(0(x1)))) -> 5(3(4(4(0(3(5(2(3(2(x1)))))))))) 0(0(0(4(0(x1))))) -> 1(4(3(0(5(5(5(5(1(2(x1)))))))))) 0(1(3(0(0(x1))))) -> 0(5(1(0(2(3(5(2(3(0(x1)))))))))) 4(0(1(3(4(x1))))) -> 4(2(3(2(4(4(1(4(4(4(x1)))))))))) 4(0(2(0(2(x1))))) -> 5(3(1(4(3(2(1(1(2(3(x1)))))))))) 4(0(2(5(0(x1))))) -> 3(1(2(3(4(4(5(1(0(3(x1)))))))))) 5(0(0(2(5(x1))))) -> 0(1(0(3(3(4(5(3(0(5(x1)))))))))) 5(2(0(0(0(x1))))) -> 5(5(2(3(3(4(2(5(0(5(x1)))))))))) 1(0(0(5(0(0(x1)))))) -> 2(1(5(5(5(5(0(1(0(4(x1)))))))))) 1(3(5(0(5(0(x1)))))) -> 2(4(2(3(2(3(5(3(3(0(x1)))))))))) 1(5(4(0(2(4(x1)))))) -> 1(0(1(2(3(1(1(0(5(5(x1)))))))))) 2(0(0(0(0(3(x1)))))) -> 1(0(5(4(3(3(1(3(5(1(x1)))))))))) 2(0(0(5(1(0(x1)))))) -> 2(5(3(5(2(2(3(5(2(3(x1)))))))))) 2(2(1(3(1(4(x1)))))) -> 1(1(5(3(2(2(2(1(4(3(x1)))))))))) 3(0(2(0(1(3(x1)))))) -> 3(0(5(5(1(5(3(3(4(3(x1)))))))))) 4(1(2(1(4(1(x1)))))) -> 3(5(5(2(1(4(4(4(5(1(x1)))))))))) 5(0(1(3(3(2(x1)))))) -> 0(3(3(5(5(5(1(5(1(0(x1)))))))))) 5(1(3(0(2(0(x1)))))) -> 2(4(5(5(1(0(1(5(3(4(x1)))))))))) 0(0(0(1(4(0(4(x1))))))) -> 2(4(4(2(1(2(1(2(4(5(x1)))))))))) 0(2(0(2(1(4(2(x1))))))) -> 2(4(4(2(4(1(1(3(4(2(x1)))))))))) 0(3(0(0(2(2(4(x1))))))) -> 0(1(0(5(3(5(5(4(3(0(x1)))))))))) 0(3(5(0(0(2(0(x1))))))) -> 0(2(3(1(5(4(3(0(4(4(x1)))))))))) 1(0(1(3(0(0(2(x1))))))) -> 1(0(1(0(3(5(0(1(1(4(x1)))))))))) 1(0(2(0(0(2(0(x1))))))) -> 1(2(2(1(3(1(1(0(4(4(x1)))))))))) 2(0(0(2(1(3(5(x1))))))) -> 2(5(4(2(2(1(0(0(3(4(x1)))))))))) 2(0(0(4(0(0(3(x1))))))) -> 1(1(1(1(0(0(5(4(2(3(x1)))))))))) 2(1(4(0(0(0(0(x1))))))) -> 2(1(4(0(2(1(1(3(1(4(x1)))))))))) 2(4(0(5(3(2(1(x1))))))) -> 2(0(3(0(4(2(3(1(2(1(x1)))))))))) 2(5(2(0(4(5(0(x1))))))) -> 1(2(2(2(2(5(0(2(1(0(x1)))))))))) 3(0(2(2(0(1(3(x1))))))) -> 3(1(4(5(1(1(4(2(3(4(x1)))))))))) 3(4(0(2(4(0(0(x1))))))) -> 3(2(4(5(3(1(1(0(3(0(x1)))))))))) 4(0(2(4(0(1(3(x1))))))) -> 4(3(1(1(2(2(1(3(0(3(x1)))))))))) 4(4(0(5(4(0(4(x1))))))) -> 4(4(5(1(1(0(3(4(2(4(x1)))))))))) 5(0(2(5(0(2(0(x1))))))) -> 3(2(3(4(2(5(0(2(4(4(x1)))))))))) 5(2(1(4(1(4(2(x1))))))) -> 1(3(2(4(3(1(2(0(3(0(x1)))))))))) 5(4(1(1(0(5(0(x1))))))) -> 4(3(4(5(5(5(1(4(0(3(x1)))))))))) 5(4(1(1(2(0(4(x1))))))) -> 4(3(5(5(4(4(4(2(2(4(x1)))))))))) 5(4(2(0(0(1(3(x1))))))) -> 2(3(4(2(4(3(1(1(2(4(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_4(x_1) -> 4(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_1(x_1) -> 1(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. "[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] {(148,149,[0_1|0, 1_1|0, 5_1|0, 4_1|0, 2_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_4_1|0, encode_2_1|0, encode_3_1|0, encode_5_1|0, encode_1_1|0]), (148,150,[0_1|1, 1_1|1, 5_1|1, 4_1|1, 2_1|1, 3_1|1]), (148,151,[0_1|2]), (148,160,[0_1|2]), (148,169,[1_1|2]), (148,178,[2_1|2]), (148,187,[1_1|2]), (148,196,[0_1|2]), (148,205,[2_1|2]), (148,214,[0_1|2]), (148,223,[0_1|2]), (148,232,[1_1|2]), (148,241,[1_1|2]), (148,250,[2_1|2]), (148,259,[2_1|2]), (148,268,[1_1|2]), (148,277,[1_1|2]), (148,286,[1_1|2]), (148,295,[5_1|2]), (148,304,[2_1|2]), (148,313,[4_1|2]), (148,322,[4_1|2]), (148,331,[0_1|2]), (148,340,[0_1|2]), (148,349,[3_1|2]), (148,358,[5_1|2]), (148,367,[1_1|2]), (148,376,[2_1|2]), (148,385,[4_1|2]), (148,394,[5_1|2]), (148,403,[3_1|2]), (148,412,[4_1|2]), (148,421,[3_1|2]), (148,430,[4_1|2]), (148,439,[1_1|2]), (148,448,[2_1|2]), (148,457,[2_1|2]), (148,466,[1_1|2]), (148,475,[1_1|2]), (148,484,[2_1|2]), (148,493,[2_1|2]), 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(366,295,[5_1|2]), (366,304,[2_1|2]), (366,313,[4_1|2]), (366,322,[4_1|2]), (366,349,[3_1|2]), (366,358,[5_1|2]), (366,367,[1_1|2]), (366,376,[2_1|2]), (367,368,[3_1|2]), (368,369,[2_1|2]), (369,370,[4_1|2]), (370,371,[3_1|2]), (371,372,[1_1|2]), (372,373,[2_1|2]), (373,374,[0_1|2]), (373,214,[0_1|2]), (374,375,[3_1|2]), (374,511,[3_1|2]), (374,520,[3_1|2]), (375,150,[0_1|2]), (375,178,[0_1|2, 2_1|2]), (375,205,[0_1|2, 2_1|2]), (375,250,[0_1|2]), (375,259,[0_1|2]), (375,304,[0_1|2]), (375,376,[0_1|2]), (375,448,[0_1|2]), (375,457,[0_1|2]), (375,484,[0_1|2]), (375,493,[0_1|2]), (375,386,[0_1|2]), (375,151,[0_1|2]), (375,160,[0_1|2]), (375,169,[1_1|2]), (375,187,[1_1|2]), (375,196,[0_1|2]), (375,214,[0_1|2]), (375,223,[0_1|2]), (375,538,[1_1|3]), (376,377,[4_1|2]), (377,378,[5_1|2]), (378,379,[5_1|2]), (379,380,[1_1|2]), (380,381,[0_1|2]), (381,382,[1_1|2]), (382,383,[5_1|2]), (383,384,[3_1|2]), (383,529,[3_1|2]), (384,150,[4_1|2]), (384,151,[4_1|2]), (384,160,[4_1|2]), (384,196,[4_1|2]), (384,214,[4_1|2]), (384,223,[4_1|2]), (384,331,[4_1|2]), (384,340,[4_1|2]), (384,494,[4_1|2]), (384,385,[4_1|2]), (384,394,[5_1|2]), (384,403,[3_1|2]), (384,412,[4_1|2]), (384,421,[3_1|2]), (384,430,[4_1|2]), (385,386,[2_1|2]), (386,387,[3_1|2]), (387,388,[2_1|2]), (388,389,[4_1|2]), (389,390,[4_1|2]), (390,391,[1_1|2]), (391,392,[4_1|2]), (392,393,[4_1|2]), (392,430,[4_1|2]), (393,150,[4_1|2]), (393,313,[4_1|2]), (393,322,[4_1|2]), (393,385,[4_1|2]), (393,412,[4_1|2]), (393,430,[4_1|2]), (393,394,[5_1|2]), (393,403,[3_1|2]), (393,421,[3_1|2]), (394,395,[3_1|2]), (395,396,[1_1|2]), (396,397,[4_1|2]), (397,398,[3_1|2]), (398,399,[2_1|2]), (399,400,[1_1|2]), (400,401,[1_1|2]), (401,402,[2_1|2]), (402,150,[3_1|2]), (402,178,[3_1|2]), (402,205,[3_1|2]), (402,250,[3_1|2]), (402,259,[3_1|2]), (402,304,[3_1|2]), (402,376,[3_1|2]), (402,448,[3_1|2]), (402,457,[3_1|2]), (402,484,[3_1|2]), (402,493,[3_1|2]), (402,161,[3_1|2]), (402,224,[3_1|2]), (402,511,[3_1|2]), (402,520,[3_1|2]), (402,529,[3_1|2]), (403,404,[1_1|2]), (404,405,[2_1|2]), (405,406,[3_1|2]), (406,407,[4_1|2]), (407,408,[4_1|2]), (408,409,[5_1|2]), (409,410,[1_1|2]), (410,411,[0_1|2]), (410,214,[0_1|2]), (410,223,[0_1|2]), (411,150,[3_1|2]), (411,151,[3_1|2]), (411,160,[3_1|2]), (411,196,[3_1|2]), (411,214,[3_1|2]), (411,223,[3_1|2]), (411,331,[3_1|2]), (411,340,[3_1|2]), (411,511,[3_1|2]), (411,520,[3_1|2]), (411,529,[3_1|2]), (412,413,[3_1|2]), (413,414,[1_1|2]), (414,415,[1_1|2]), (415,416,[2_1|2]), (416,417,[2_1|2]), (417,418,[1_1|2]), (418,419,[3_1|2]), (419,420,[0_1|2]), (419,214,[0_1|2]), (419,223,[0_1|2]), (420,150,[3_1|2]), (420,349,[3_1|2]), (420,403,[3_1|2]), (420,421,[3_1|2]), (420,511,[3_1|2]), (420,520,[3_1|2]), (420,529,[3_1|2]), (420,368,[3_1|2]), (421,422,[5_1|2]), (422,423,[5_1|2]), (423,424,[2_1|2]), (424,425,[1_1|2]), (425,426,[4_1|2]), (426,427,[4_1|2]), (427,428,[4_1|2]), (428,429,[5_1|2]), (428,376,[2_1|2]), (429,150,[1_1|2]), (429,169,[1_1|2]), (429,187,[1_1|2]), (429,232,[1_1|2]), (429,241,[1_1|2]), (429,268,[1_1|2]), (429,277,[1_1|2]), (429,286,[1_1|2]), (429,367,[1_1|2]), (429,439,[1_1|2]), (429,466,[1_1|2]), (429,475,[1_1|2]), (429,502,[1_1|2]), (429,250,[2_1|2]), (429,259,[2_1|2]), (430,431,[4_1|2]), (431,432,[5_1|2]), (432,433,[1_1|2]), (433,434,[1_1|2]), (434,435,[0_1|2]), (435,436,[3_1|2]), (436,437,[4_1|2]), (437,438,[2_1|2]), (437,493,[2_1|2]), (438,150,[4_1|2]), (438,313,[4_1|2]), (438,322,[4_1|2]), (438,385,[4_1|2]), (438,412,[4_1|2]), (438,430,[4_1|2]), (438,152,[4_1|2]), (438,394,[5_1|2]), (438,403,[3_1|2]), (438,421,[3_1|2]), (439,440,[0_1|2]), (440,441,[5_1|2]), (441,442,[4_1|2]), (442,443,[3_1|2]), (443,444,[3_1|2]), (444,445,[1_1|2]), (445,446,[3_1|2]), (446,447,[5_1|2]), (446,376,[2_1|2]), (447,150,[1_1|2]), (447,349,[1_1|2]), (447,403,[1_1|2]), (447,421,[1_1|2]), (447,511,[1_1|2]), (447,520,[1_1|2]), (447,529,[1_1|2]), (447,341,[1_1|2]), (447,232,[1_1|2]), (447,241,[1_1|2]), (447,250,[2_1|2]), (447,259,[2_1|2]), (447,268,[1_1|2]), (447,277,[1_1|2]), (447,286,[1_1|2]), (448,449,[5_1|2]), (449,450,[3_1|2]), (450,451,[5_1|2]), (451,452,[2_1|2]), (452,453,[2_1|2]), (453,454,[3_1|2]), (454,455,[5_1|2]), (455,456,[2_1|2]), (456,150,[3_1|2]), (456,151,[3_1|2]), (456,160,[3_1|2]), (456,196,[3_1|2]), (456,214,[3_1|2]), (456,223,[3_1|2]), (456,331,[3_1|2]), (456,340,[3_1|2]), (456,269,[3_1|2]), (456,287,[3_1|2]), (456,440,[3_1|2]), (456,199,[3_1|2]), (456,511,[3_1|2]), (456,520,[3_1|2]), (456,529,[3_1|2]), (457,458,[5_1|2]), (458,459,[4_1|2]), (459,460,[2_1|2]), (460,461,[2_1|2]), (461,462,[1_1|2]), (462,463,[0_1|2]), (463,464,[0_1|2]), (464,465,[3_1|2]), (464,529,[3_1|2]), (465,150,[4_1|2]), (465,295,[4_1|2]), (465,358,[4_1|2]), (465,394,[4_1|2, 5_1|2]), (465,422,[4_1|2]), (465,385,[4_1|2]), (465,403,[3_1|2]), (465,412,[4_1|2]), (465,421,[3_1|2]), (465,430,[4_1|2]), (466,467,[1_1|2]), (467,468,[1_1|2]), (468,469,[1_1|2]), (469,470,[0_1|2]), (469,547,[1_1|3]), (470,471,[0_1|2]), (471,472,[5_1|2]), (472,473,[4_1|2]), (473,474,[2_1|2]), (474,150,[3_1|2]), (474,349,[3_1|2]), (474,403,[3_1|2]), (474,421,[3_1|2]), (474,511,[3_1|2]), (474,520,[3_1|2]), (474,529,[3_1|2]), (474,341,[3_1|2]), (475,476,[1_1|2]), (476,477,[5_1|2]), (477,478,[3_1|2]), (478,479,[2_1|2]), (479,480,[2_1|2]), (480,481,[2_1|2]), (481,482,[1_1|2]), (482,483,[4_1|2]), (483,150,[3_1|2]), (483,313,[3_1|2]), (483,322,[3_1|2]), (483,385,[3_1|2]), (483,412,[3_1|2]), (483,430,[3_1|2]), (483,170,[3_1|2]), (483,522,[3_1|2]), (483,511,[3_1|2]), (483,520,[3_1|2]), (483,529,[3_1|2]), (484,485,[1_1|2]), (485,486,[4_1|2]), (486,487,[0_1|2]), (487,488,[2_1|2]), (488,489,[1_1|2]), (489,490,[1_1|2]), (490,491,[3_1|2]), (491,492,[1_1|2]), (492,150,[4_1|2]), (492,151,[4_1|2]), (492,160,[4_1|2]), (492,196,[4_1|2]), (492,214,[4_1|2]), (492,223,[4_1|2]), (492,331,[4_1|2]), (492,340,[4_1|2]), (492,385,[4_1|2]), (492,394,[5_1|2]), (492,403,[3_1|2]), (492,412,[4_1|2]), (492,421,[3_1|2]), (492,430,[4_1|2]), (493,494,[0_1|2]), (494,495,[3_1|2]), (495,496,[0_1|2]), (496,497,[4_1|2]), (497,498,[2_1|2]), (498,499,[3_1|2]), (499,500,[1_1|2]), (500,501,[2_1|2]), (500,484,[2_1|2]), (501,150,[1_1|2]), (501,169,[1_1|2]), (501,187,[1_1|2]), (501,232,[1_1|2]), (501,241,[1_1|2]), (501,268,[1_1|2]), (501,277,[1_1|2]), (501,286,[1_1|2]), (501,367,[1_1|2]), (501,439,[1_1|2]), (501,466,[1_1|2]), (501,475,[1_1|2]), (501,502,[1_1|2]), (501,260,[1_1|2]), (501,485,[1_1|2]), (501,250,[2_1|2]), (501,259,[2_1|2]), (502,503,[2_1|2]), (503,504,[2_1|2]), (504,505,[2_1|2]), (505,506,[2_1|2]), (506,507,[5_1|2]), (507,508,[0_1|2]), (508,509,[2_1|2]), (509,510,[1_1|2]), (509,259,[2_1|2]), (509,268,[1_1|2]), (509,277,[1_1|2]), (510,150,[0_1|2]), (510,151,[0_1|2]), (510,160,[0_1|2]), (510,196,[0_1|2]), (510,214,[0_1|2]), (510,223,[0_1|2]), (510,331,[0_1|2]), (510,340,[0_1|2]), (510,169,[1_1|2]), (510,178,[2_1|2]), (510,187,[1_1|2]), (510,205,[2_1|2]), (510,538,[1_1|3]), (511,512,[0_1|2]), (512,513,[5_1|2]), (513,514,[5_1|2]), (514,515,[1_1|2]), (515,516,[5_1|2]), (516,517,[3_1|2]), (517,518,[3_1|2]), (518,519,[4_1|2]), (519,150,[3_1|2]), (519,349,[3_1|2]), (519,403,[3_1|2]), (519,421,[3_1|2]), (519,511,[3_1|2]), (519,520,[3_1|2]), (519,529,[3_1|2]), (519,368,[3_1|2]), (520,521,[1_1|2]), (521,522,[4_1|2]), (522,523,[5_1|2]), (523,524,[1_1|2]), (524,525,[1_1|2]), (525,526,[4_1|2]), (526,527,[2_1|2]), (527,528,[3_1|2]), (527,529,[3_1|2]), (528,150,[4_1|2]), (528,349,[4_1|2]), (528,403,[4_1|2, 3_1|2]), (528,421,[4_1|2, 3_1|2]), (528,511,[4_1|2]), (528,520,[4_1|2]), (528,529,[4_1|2]), (528,368,[4_1|2]), (528,385,[4_1|2]), (528,394,[5_1|2]), (528,412,[4_1|2]), (528,430,[4_1|2]), (529,530,[2_1|2]), (530,531,[4_1|2]), (531,532,[5_1|2]), (532,533,[3_1|2]), (533,534,[1_1|2]), (534,535,[1_1|2]), (535,536,[0_1|2]), (535,214,[0_1|2]), (536,537,[3_1|2]), (536,511,[3_1|2]), (536,520,[3_1|2]), (537,150,[0_1|2]), (537,151,[0_1|2]), (537,160,[0_1|2]), (537,196,[0_1|2]), (537,214,[0_1|2]), (537,223,[0_1|2]), (537,331,[0_1|2]), (537,340,[0_1|2]), (537,169,[1_1|2]), (537,178,[2_1|2]), (537,187,[1_1|2]), (537,205,[2_1|2]), (537,538,[1_1|3]), (538,539,[2_1|3]), (539,540,[3_1|3]), (540,541,[4_1|3]), (541,542,[1_1|3]), (542,543,[5_1|3]), (543,544,[0_1|3]), (544,545,[3_1|3]), (545,546,[3_1|3]), (546,197,[4_1|3]), (547,548,[2_1|3]), (548,549,[3_1|3]), (549,550,[4_1|3]), (550,551,[1_1|3]), (551,552,[5_1|3]), (552,553,[0_1|3]), (553,554,[3_1|3]), (554,555,[3_1|3]), (555,472,[4_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)