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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 58 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 108 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(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(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_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_1(x_1)) -> 1(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 3. The certificate found is represented by the following graph. 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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] {(151,152,[0_1|0, 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(425,288,[4_1|2]), (425,501,[4_1|2]), (425,270,[4_1|2]), (425,519,[2_1|2]), (426,427,[3_1|2]), (427,428,[1_1|2]), (428,429,[3_1|2]), (429,430,[1_1|2]), (430,431,[4_1|2]), (431,432,[4_1|2]), (432,433,[2_1|2]), (433,434,[5_1|2]), (434,435,[4_1|2]), (435,436,[5_1|2]), (436,437,[4_1|2]), (437,438,[4_1|2]), (438,439,[1_1|2]), (438,484,[4_1|2]), (439,153,[0_1|2]), (439,158,[0_1|2]), (439,174,[0_1|2]), (439,216,[0_1|2]), (439,440,[0_1|2]), (439,458,[0_1|2]), (439,154,[2_1|2]), (439,169,[5_1|2]), (439,189,[1_1|2]), (439,196,[3_1|2]), (439,205,[2_1|2]), (439,231,[2_1|2]), (439,251,[2_1|2]), (440,441,[1_1|2]), (441,442,[2_1|2]), (442,443,[0_1|2]), (443,444,[5_1|2]), (444,445,[4_1|2]), (445,446,[3_1|2]), (446,447,[2_1|2]), (447,448,[0_1|2]), (448,449,[2_1|2]), (449,450,[3_1|2]), (450,451,[3_1|2]), (451,452,[3_1|2]), (452,453,[0_1|2]), (453,454,[1_1|2]), (454,455,[5_1|2]), (455,456,[5_1|2]), (456,457,[5_1|2]), (457,153,[1_1|2]), (457,196,[1_1|2]), (457,269,[1_1|2]), (457,312,[1_1|2]), (457,403,[1_1|2]), (457,155,[1_1|2]), (457,484,[4_1|2]), (457,501,[1_1|2]), (458,459,[3_1|2]), (459,460,[2_1|2]), (460,461,[5_1|2]), (461,462,[5_1|2]), (462,463,[2_1|2]), (463,464,[4_1|2]), (464,465,[4_1|2]), (465,466,[3_1|2]), (466,467,[5_1|2]), (467,468,[3_1|2]), (468,469,[4_1|2]), (469,470,[3_1|2]), (470,471,[4_1|2]), (471,153,[1_1|2]), (471,169,[1_1|2]), (471,298,[1_1|2]), (471,329,[1_1|2]), (471,344,[1_1|2]), (471,352,[1_1|2]), (471,362,[1_1|2]), (471,374,[1_1|2]), (471,426,[1_1|2]), (471,472,[1_1|2]), (471,190,[1_1|2]), (471,278,[1_1|2]), (471,502,[1_1|2]), (471,484,[4_1|2]), (471,501,[1_1|2]), (472,473,[3_1|2]), (473,474,[2_1|2]), (474,475,[3_1|2]), (475,476,[5_1|2]), (476,477,[1_1|2]), (477,478,[5_1|2]), (478,479,[3_1|2]), (479,480,[5_1|2]), (480,481,[3_1|2]), (481,482,[5_1|2]), (482,483,[1_1|2]), (483,153,[5_1|2]), (483,196,[5_1|2]), (483,269,[5_1|2]), (483,312,[5_1|2]), (483,403,[5_1|2]), (483,344,[5_1|2]), (483,352,[5_1|2]), (483,362,[5_1|2]), (484,485,[2_1|2]), (485,486,[2_1|2]), (486,487,[0_1|2]), (487,488,[0_1|2]), (488,489,[0_1|2]), (489,490,[0_1|2]), (490,491,[4_1|2]), (491,492,[1_1|2]), (492,493,[4_1|2]), (493,494,[0_1|2]), (494,495,[5_1|2]), (495,496,[3_1|2]), (496,497,[5_1|2]), (497,498,[0_1|2]), (498,499,[2_1|2]), (499,500,[4_1|2]), (499,519,[2_1|2]), (500,153,[3_1|2]), (500,196,[3_1|2]), (500,269,[3_1|2]), (500,312,[3_1|2]), (500,403,[3_1|2]), (500,277,[1_1|2]), (500,288,[1_1|2]), (500,298,[5_1|2]), (500,329,[5_1|2]), (501,502,[5_1|2]), (502,503,[3_1|2]), (503,504,[2_1|2]), (504,505,[5_1|2]), (505,506,[0_1|2]), (506,507,[5_1|2]), (507,508,[2_1|2]), (508,509,[3_1|2]), (509,510,[0_1|2]), (510,511,[0_1|2]), (511,512,[3_1|2]), (512,513,[1_1|2]), (513,514,[5_1|2]), (514,515,[3_1|2]), (515,516,[4_1|2]), (516,517,[4_1|2]), (517,518,[3_1|2]), (517,329,[5_1|2]), (518,153,[0_1|2]), (518,158,[0_1|2]), (518,174,[0_1|2]), (518,216,[0_1|2]), (518,440,[0_1|2]), (518,458,[0_1|2]), (518,154,[2_1|2]), (518,169,[5_1|2]), (518,189,[1_1|2]), (518,196,[3_1|2]), (518,205,[2_1|2]), (518,231,[2_1|2]), (518,251,[2_1|2]), (519,520,[5_1|2]), (520,521,[5_1|2]), (521,522,[1_1|2]), (522,523,[1_1|2]), (523,524,[1_1|2]), (524,525,[4_1|2]), (525,526,[5_1|2]), (526,527,[3_1|2]), (527,528,[2_1|2]), (528,529,[3_1|2]), (529,530,[3_1|2]), (530,531,[0_1|2]), (531,532,[4_1|2]), (532,533,[2_1|2]), (533,534,[3_1|2]), (534,535,[1_1|2]), (535,536,[5_1|2]), (536,537,[4_1|2]), (537,153,[5_1|2]), (537,196,[5_1|2]), (537,269,[5_1|2]), (537,312,[5_1|2]), (537,403,[5_1|2]), (537,197,[5_1|2]), (537,156,[5_1|2]), (537,344,[5_1|2]), (537,352,[5_1|2]), (537,362,[5_1|2]), (538,539,[3_1|3]), (539,540,[3_1|3]), (540,541,[2_1|3]), (541,206,[2_1|3]), (542,543,[3_1|3]), (543,544,[3_1|3]), (544,545,[2_1|3]), (545,290,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)