WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 46 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 75 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 0(1(3(x1))) 1(0(3(2(3(0(x1)))))) -> 3(2(4(2(0(x1))))) 5(5(1(4(4(5(x1)))))) -> 5(0(2(1(2(2(x1)))))) 0(5(0(4(0(2(3(x1))))))) -> 4(4(4(1(3(2(3(x1))))))) 4(0(0(0(4(5(1(x1))))))) -> 4(5(1(0(1(1(x1)))))) 1(2(3(4(3(1(4(3(x1)))))))) -> 5(2(3(1(2(0(5(1(x1)))))))) 1(3(1(4(1(1(2(5(x1)))))))) -> 1(3(1(1(2(0(2(5(x1)))))))) 3(0(2(3(3(1(0(1(2(x1))))))))) -> 1(1(2(0(2(2(0(2(3(x1))))))))) 0(1(1(3(3(3(0(0(4(3(x1)))))))))) -> 4(1(0(0(1(5(5(1(0(1(x1)))))))))) 4(2(4(4(5(2(0(5(0(4(x1)))))))))) -> 4(1(2(0(3(3(5(3(4(x1))))))))) 0(3(3(5(4(4(2(3(4(2(5(x1))))))))))) -> 0(3(1(4(1(3(2(4(2(1(2(x1))))))))))) 0(5(4(5(0(5(2(4(2(5(5(x1))))))))))) -> 2(5(4(4(0(0(4(5(2(4(5(x1))))))))))) 2(2(4(2(4(2(4(5(0(4(3(x1))))))))))) -> 2(0(1(3(2(5(5(5(1(1(4(4(x1)))))))))))) 0(2(4(0(0(5(1(5(3(3(4(1(x1)))))))))))) -> 2(3(1(0(1(4(1(5(2(4(1(x1))))))))))) 1(3(2(2(0(5(2(0(4(3(3(0(x1)))))))))))) -> 1(3(5(4(1(1(1(5(1(4(0(x1))))))))))) 1(5(3(5(2(2(4(2(5(5(3(5(x1)))))))))))) -> 4(0(3(5(0(5(4(0(3(4(0(3(x1)))))))))))) 0(0(0(0(3(5(3(3(2(2(0(4(5(2(x1)))))))))))))) -> 5(2(3(5(1(3(1(5(1(4(0(4(1(x1))))))))))))) 3(1(2(2(1(5(0(4(3(2(5(0(4(4(x1)))))))))))))) -> 3(1(4(1(1(3(4(1(3(4(4(4(x1)))))))))))) 5(2(4(3(4(5(1(2(0(1(5(1(1(5(5(x1))))))))))))))) -> 5(4(2(2(2(2(5(1(5(0(4(0(1(3(x1)))))))))))))) 0(3(4(5(0(2(2(2(4(2(2(5(0(3(2(5(x1)))))))))))))))) -> 5(2(3(0(1(5(4(4(5(4(0(5(4(5(2(0(x1)))))))))))))))) 4(5(3(2(1(5(2(0(4(5(5(3(3(0(2(5(x1)))))))))))))))) -> 4(2(0(3(1(0(5(1(0(2(1(3(4(5(x1)))))))))))))) 0(2(2(3(3(2(3(5(4(4(3(3(1(3(2(0(0(x1))))))))))))))))) -> 4(4(3(1(2(4(3(3(1(5(0(0(4(5(3(0(3(x1))))))))))))))))) 0(4(4(4(4(3(2(1(3(2(4(4(4(1(5(4(3(x1))))))))))))))))) -> 2(5(0(2(5(4(5(1(2(0(5(0(3(4(5(x1))))))))))))))) 1(2(1(2(2(5(4(1(4(4(2(1(0(1(1(3(5(x1))))))))))))))))) -> 1(1(0(2(0(1(4(5(3(5(5(3(0(0(1(5(x1)))))))))))))))) 0(5(1(4(0(0(1(4(3(5(4(1(0(4(1(1(0(1(5(x1))))))))))))))))))) -> 2(4(2(1(4(0(4(0(5(0(0(5(0(4(1(5(5(1(x1)))))))))))))))))) 3(0(5(4(5(5(4(5(0(3(3(1(4(1(3(2(2(5(0(x1))))))))))))))))))) -> 4(5(4(5(2(5(1(4(4(3(1(2(3(4(4(2(1(2(0(x1))))))))))))))))))) 4(2(4(0(5(0(2(2(4(0(2(3(5(2(5(4(1(0(0(x1))))))))))))))))))) -> 4(2(5(3(0(1(0(4(2(0(4(4(1(1(5(0(0(0(3(x1))))))))))))))))))) 2(2(5(3(2(3(5(1(3(5(1(5(5(4(2(4(0(3(5(0(x1)))))))))))))))))))) -> 1(2(4(2(3(4(1(1(2(0(3(0(1(4(1(3(0(5(3(0(x1)))))))))))))))))))) 4(1(5(3(0(1(4(0(3(4(0(0(5(2(5(1(5(1(2(3(x1)))))))))))))))))))) -> 4(2(5(1(1(5(2(4(3(3(1(1(3(0(0(3(4(5(x1)))))))))))))))))) 0(0(4(4(4(3(2(3(1(0(0(2(5(2(5(0(0(1(5(2(2(x1))))))))))))))))))))) -> 1(3(2(3(1(4(1(0(4(1(2(1(2(5(4(1(2(0(2(2(x1)))))))))))))))))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 0(1(3(x1))) 1(0(3(2(3(0(x1)))))) -> 3(2(4(2(0(x1))))) 5(5(1(4(4(5(x1)))))) -> 5(0(2(1(2(2(x1)))))) 0(5(0(4(0(2(3(x1))))))) -> 4(4(4(1(3(2(3(x1))))))) 4(0(0(0(4(5(1(x1))))))) -> 4(5(1(0(1(1(x1)))))) 1(2(3(4(3(1(4(3(x1)))))))) -> 5(2(3(1(2(0(5(1(x1)))))))) 1(3(1(4(1(1(2(5(x1)))))))) -> 1(3(1(1(2(0(2(5(x1)))))))) 3(0(2(3(3(1(0(1(2(x1))))))))) -> 1(1(2(0(2(2(0(2(3(x1))))))))) 0(1(1(3(3(3(0(0(4(3(x1)))))))))) -> 4(1(0(0(1(5(5(1(0(1(x1)))))))))) 4(2(4(4(5(2(0(5(0(4(x1)))))))))) -> 4(1(2(0(3(3(5(3(4(x1))))))))) 0(3(3(5(4(4(2(3(4(2(5(x1))))))))))) -> 0(3(1(4(1(3(2(4(2(1(2(x1))))))))))) 0(5(4(5(0(5(2(4(2(5(5(x1))))))))))) -> 2(5(4(4(0(0(4(5(2(4(5(x1))))))))))) 2(2(4(2(4(2(4(5(0(4(3(x1))))))))))) -> 2(0(1(3(2(5(5(5(1(1(4(4(x1)))))))))))) 0(2(4(0(0(5(1(5(3(3(4(1(x1)))))))))))) -> 2(3(1(0(1(4(1(5(2(4(1(x1))))))))))) 1(3(2(2(0(5(2(0(4(3(3(0(x1)))))))))))) -> 1(3(5(4(1(1(1(5(1(4(0(x1))))))))))) 1(5(3(5(2(2(4(2(5(5(3(5(x1)))))))))))) -> 4(0(3(5(0(5(4(0(3(4(0(3(x1)))))))))))) 0(0(0(0(3(5(3(3(2(2(0(4(5(2(x1)))))))))))))) -> 5(2(3(5(1(3(1(5(1(4(0(4(1(x1))))))))))))) 3(1(2(2(1(5(0(4(3(2(5(0(4(4(x1)))))))))))))) -> 3(1(4(1(1(3(4(1(3(4(4(4(x1)))))))))))) 5(2(4(3(4(5(1(2(0(1(5(1(1(5(5(x1))))))))))))))) -> 5(4(2(2(2(2(5(1(5(0(4(0(1(3(x1)))))))))))))) 0(3(4(5(0(2(2(2(4(2(2(5(0(3(2(5(x1)))))))))))))))) -> 5(2(3(0(1(5(4(4(5(4(0(5(4(5(2(0(x1)))))))))))))))) 4(5(3(2(1(5(2(0(4(5(5(3(3(0(2(5(x1)))))))))))))))) -> 4(2(0(3(1(0(5(1(0(2(1(3(4(5(x1)))))))))))))) 0(2(2(3(3(2(3(5(4(4(3(3(1(3(2(0(0(x1))))))))))))))))) -> 4(4(3(1(2(4(3(3(1(5(0(0(4(5(3(0(3(x1))))))))))))))))) 0(4(4(4(4(3(2(1(3(2(4(4(4(1(5(4(3(x1))))))))))))))))) -> 2(5(0(2(5(4(5(1(2(0(5(0(3(4(5(x1))))))))))))))) 1(2(1(2(2(5(4(1(4(4(2(1(0(1(1(3(5(x1))))))))))))))))) -> 1(1(0(2(0(1(4(5(3(5(5(3(0(0(1(5(x1)))))))))))))))) 0(5(1(4(0(0(1(4(3(5(4(1(0(4(1(1(0(1(5(x1))))))))))))))))))) -> 2(4(2(1(4(0(4(0(5(0(0(5(0(4(1(5(5(1(x1)))))))))))))))))) 3(0(5(4(5(5(4(5(0(3(3(1(4(1(3(2(2(5(0(x1))))))))))))))))))) -> 4(5(4(5(2(5(1(4(4(3(1(2(3(4(4(2(1(2(0(x1))))))))))))))))))) 4(2(4(0(5(0(2(2(4(0(2(3(5(2(5(4(1(0(0(x1))))))))))))))))))) -> 4(2(5(3(0(1(0(4(2(0(4(4(1(1(5(0(0(0(3(x1))))))))))))))))))) 2(2(5(3(2(3(5(1(3(5(1(5(5(4(2(4(0(3(5(0(x1)))))))))))))))))))) -> 1(2(4(2(3(4(1(1(2(0(3(0(1(4(1(3(0(5(3(0(x1)))))))))))))))))))) 4(1(5(3(0(1(4(0(3(4(0(0(5(2(5(1(5(1(2(3(x1)))))))))))))))))))) -> 4(2(5(1(1(5(2(4(3(3(1(1(3(0(0(3(4(5(x1)))))))))))))))))) 0(0(4(4(4(3(2(3(1(0(0(2(5(2(5(0(0(1(5(2(2(x1))))))))))))))))))))) -> 1(3(2(3(1(4(1(0(4(1(2(1(2(5(4(1(2(0(2(2(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 0(1(3(x1))) 1(0(3(2(3(0(x1)))))) -> 3(2(4(2(0(x1))))) 5(5(1(4(4(5(x1)))))) -> 5(0(2(1(2(2(x1)))))) 0(5(0(4(0(2(3(x1))))))) -> 4(4(4(1(3(2(3(x1))))))) 4(0(0(0(4(5(1(x1))))))) -> 4(5(1(0(1(1(x1)))))) 1(2(3(4(3(1(4(3(x1)))))))) -> 5(2(3(1(2(0(5(1(x1)))))))) 1(3(1(4(1(1(2(5(x1)))))))) -> 1(3(1(1(2(0(2(5(x1)))))))) 3(0(2(3(3(1(0(1(2(x1))))))))) -> 1(1(2(0(2(2(0(2(3(x1))))))))) 0(1(1(3(3(3(0(0(4(3(x1)))))))))) -> 4(1(0(0(1(5(5(1(0(1(x1)))))))))) 4(2(4(4(5(2(0(5(0(4(x1)))))))))) -> 4(1(2(0(3(3(5(3(4(x1))))))))) 0(3(3(5(4(4(2(3(4(2(5(x1))))))))))) -> 0(3(1(4(1(3(2(4(2(1(2(x1))))))))))) 0(5(4(5(0(5(2(4(2(5(5(x1))))))))))) -> 2(5(4(4(0(0(4(5(2(4(5(x1))))))))))) 2(2(4(2(4(2(4(5(0(4(3(x1))))))))))) -> 2(0(1(3(2(5(5(5(1(1(4(4(x1)))))))))))) 0(2(4(0(0(5(1(5(3(3(4(1(x1)))))))))))) -> 2(3(1(0(1(4(1(5(2(4(1(x1))))))))))) 1(3(2(2(0(5(2(0(4(3(3(0(x1)))))))))))) -> 1(3(5(4(1(1(1(5(1(4(0(x1))))))))))) 1(5(3(5(2(2(4(2(5(5(3(5(x1)))))))))))) -> 4(0(3(5(0(5(4(0(3(4(0(3(x1)))))))))))) 0(0(0(0(3(5(3(3(2(2(0(4(5(2(x1)))))))))))))) -> 5(2(3(5(1(3(1(5(1(4(0(4(1(x1))))))))))))) 3(1(2(2(1(5(0(4(3(2(5(0(4(4(x1)))))))))))))) -> 3(1(4(1(1(3(4(1(3(4(4(4(x1)))))))))))) 5(2(4(3(4(5(1(2(0(1(5(1(1(5(5(x1))))))))))))))) -> 5(4(2(2(2(2(5(1(5(0(4(0(1(3(x1)))))))))))))) 0(3(4(5(0(2(2(2(4(2(2(5(0(3(2(5(x1)))))))))))))))) -> 5(2(3(0(1(5(4(4(5(4(0(5(4(5(2(0(x1)))))))))))))))) 4(5(3(2(1(5(2(0(4(5(5(3(3(0(2(5(x1)))))))))))))))) -> 4(2(0(3(1(0(5(1(0(2(1(3(4(5(x1)))))))))))))) 0(2(2(3(3(2(3(5(4(4(3(3(1(3(2(0(0(x1))))))))))))))))) -> 4(4(3(1(2(4(3(3(1(5(0(0(4(5(3(0(3(x1))))))))))))))))) 0(4(4(4(4(3(2(1(3(2(4(4(4(1(5(4(3(x1))))))))))))))))) -> 2(5(0(2(5(4(5(1(2(0(5(0(3(4(5(x1))))))))))))))) 1(2(1(2(2(5(4(1(4(4(2(1(0(1(1(3(5(x1))))))))))))))))) -> 1(1(0(2(0(1(4(5(3(5(5(3(0(0(1(5(x1)))))))))))))))) 0(5(1(4(0(0(1(4(3(5(4(1(0(4(1(1(0(1(5(x1))))))))))))))))))) -> 2(4(2(1(4(0(4(0(5(0(0(5(0(4(1(5(5(1(x1)))))))))))))))))) 3(0(5(4(5(5(4(5(0(3(3(1(4(1(3(2(2(5(0(x1))))))))))))))))))) -> 4(5(4(5(2(5(1(4(4(3(1(2(3(4(4(2(1(2(0(x1))))))))))))))))))) 4(2(4(0(5(0(2(2(4(0(2(3(5(2(5(4(1(0(0(x1))))))))))))))))))) -> 4(2(5(3(0(1(0(4(2(0(4(4(1(1(5(0(0(0(3(x1))))))))))))))))))) 2(2(5(3(2(3(5(1(3(5(1(5(5(4(2(4(0(3(5(0(x1)))))))))))))))))))) -> 1(2(4(2(3(4(1(1(2(0(3(0(1(4(1(3(0(5(3(0(x1)))))))))))))))))))) 4(1(5(3(0(1(4(0(3(4(0(0(5(2(5(1(5(1(2(3(x1)))))))))))))))))))) -> 4(2(5(1(1(5(2(4(3(3(1(1(3(0(0(3(4(5(x1)))))))))))))))))) 0(0(4(4(4(3(2(3(1(0(0(2(5(2(5(0(0(1(5(2(2(x1))))))))))))))))))))) -> 1(3(2(3(1(4(1(0(4(1(2(1(2(5(4(1(2(0(2(2(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(x1)))) -> 0(1(3(x1))) 1(0(3(2(3(0(x1)))))) -> 3(2(4(2(0(x1))))) 5(5(1(4(4(5(x1)))))) -> 5(0(2(1(2(2(x1)))))) 0(5(0(4(0(2(3(x1))))))) -> 4(4(4(1(3(2(3(x1))))))) 4(0(0(0(4(5(1(x1))))))) -> 4(5(1(0(1(1(x1)))))) 1(2(3(4(3(1(4(3(x1)))))))) -> 5(2(3(1(2(0(5(1(x1)))))))) 1(3(1(4(1(1(2(5(x1)))))))) -> 1(3(1(1(2(0(2(5(x1)))))))) 3(0(2(3(3(1(0(1(2(x1))))))))) -> 1(1(2(0(2(2(0(2(3(x1))))))))) 0(1(1(3(3(3(0(0(4(3(x1)))))))))) -> 4(1(0(0(1(5(5(1(0(1(x1)))))))))) 4(2(4(4(5(2(0(5(0(4(x1)))))))))) -> 4(1(2(0(3(3(5(3(4(x1))))))))) 0(3(3(5(4(4(2(3(4(2(5(x1))))))))))) -> 0(3(1(4(1(3(2(4(2(1(2(x1))))))))))) 0(5(4(5(0(5(2(4(2(5(5(x1))))))))))) -> 2(5(4(4(0(0(4(5(2(4(5(x1))))))))))) 2(2(4(2(4(2(4(5(0(4(3(x1))))))))))) -> 2(0(1(3(2(5(5(5(1(1(4(4(x1)))))))))))) 0(2(4(0(0(5(1(5(3(3(4(1(x1)))))))))))) -> 2(3(1(0(1(4(1(5(2(4(1(x1))))))))))) 1(3(2(2(0(5(2(0(4(3(3(0(x1)))))))))))) -> 1(3(5(4(1(1(1(5(1(4(0(x1))))))))))) 1(5(3(5(2(2(4(2(5(5(3(5(x1)))))))))))) -> 4(0(3(5(0(5(4(0(3(4(0(3(x1)))))))))))) 0(0(0(0(3(5(3(3(2(2(0(4(5(2(x1)))))))))))))) -> 5(2(3(5(1(3(1(5(1(4(0(4(1(x1))))))))))))) 3(1(2(2(1(5(0(4(3(2(5(0(4(4(x1)))))))))))))) -> 3(1(4(1(1(3(4(1(3(4(4(4(x1)))))))))))) 5(2(4(3(4(5(1(2(0(1(5(1(1(5(5(x1))))))))))))))) -> 5(4(2(2(2(2(5(1(5(0(4(0(1(3(x1)))))))))))))) 0(3(4(5(0(2(2(2(4(2(2(5(0(3(2(5(x1)))))))))))))))) -> 5(2(3(0(1(5(4(4(5(4(0(5(4(5(2(0(x1)))))))))))))))) 4(5(3(2(1(5(2(0(4(5(5(3(3(0(2(5(x1)))))))))))))))) -> 4(2(0(3(1(0(5(1(0(2(1(3(4(5(x1)))))))))))))) 0(2(2(3(3(2(3(5(4(4(3(3(1(3(2(0(0(x1))))))))))))))))) -> 4(4(3(1(2(4(3(3(1(5(0(0(4(5(3(0(3(x1))))))))))))))))) 0(4(4(4(4(3(2(1(3(2(4(4(4(1(5(4(3(x1))))))))))))))))) -> 2(5(0(2(5(4(5(1(2(0(5(0(3(4(5(x1))))))))))))))) 1(2(1(2(2(5(4(1(4(4(2(1(0(1(1(3(5(x1))))))))))))))))) -> 1(1(0(2(0(1(4(5(3(5(5(3(0(0(1(5(x1)))))))))))))))) 0(5(1(4(0(0(1(4(3(5(4(1(0(4(1(1(0(1(5(x1))))))))))))))))))) -> 2(4(2(1(4(0(4(0(5(0(0(5(0(4(1(5(5(1(x1)))))))))))))))))) 3(0(5(4(5(5(4(5(0(3(3(1(4(1(3(2(2(5(0(x1))))))))))))))))))) -> 4(5(4(5(2(5(1(4(4(3(1(2(3(4(4(2(1(2(0(x1))))))))))))))))))) 4(2(4(0(5(0(2(2(4(0(2(3(5(2(5(4(1(0(0(x1))))))))))))))))))) -> 4(2(5(3(0(1(0(4(2(0(4(4(1(1(5(0(0(0(3(x1))))))))))))))))))) 2(2(5(3(2(3(5(1(3(5(1(5(5(4(2(4(0(3(5(0(x1)))))))))))))))))))) -> 1(2(4(2(3(4(1(1(2(0(3(0(1(4(1(3(0(5(3(0(x1)))))))))))))))))))) 4(1(5(3(0(1(4(0(3(4(0(0(5(2(5(1(5(1(2(3(x1)))))))))))))))))))) -> 4(2(5(1(1(5(2(4(3(3(1(1(3(0(0(3(4(5(x1)))))))))))))))))) 0(0(4(4(4(3(2(3(1(0(0(2(5(2(5(0(0(1(5(2(2(x1))))))))))))))))))))) -> 1(3(2(3(1(4(1(0(4(1(2(1(2(5(4(1(2(0(2(2(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_2(x_1)) -> 2(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: FULL ---------------------------------------- (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 2. The certificate found is represented by the following graph. "[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] {(112,113,[0_1|0, 1_1|0, 5_1|0, 4_1|0, 3_1|0, 2_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]), (112,114,[0_1|1, 1_1|1, 5_1|1, 4_1|1, 3_1|1, 2_1|1]), (112,115,[0_1|2]), (112,117,[4_1|2]), (112,126,[4_1|2]), (112,132,[2_1|2]), (112,142,[2_1|2]), (112,159,[0_1|2]), (112,169,[5_1|2]), (112,184,[2_1|2]), (112,194,[4_1|2]), (112,210,[5_1|2]), (112,222,[1_1|2]), (112,241,[2_1|2]), (112,255,[3_1|2]), (112,259,[5_1|2]), (112,266,[1_1|2]), (112,281,[1_1|2]), (112,288,[1_1|2]), (112,298,[4_1|2]), (112,309,[5_1|2]), (112,314,[5_1|2]), (112,327,[4_1|2]), (112,332,[4_1|2]), (112,340,[4_1|2]), (112,358,[4_1|2]), (112,371,[4_1|2]), (112,388,[1_1|2]), (112,396,[4_1|2]), (112,414,[3_1|2]), (112,425,[2_1|2]), (112,436,[1_1|2]), (113,113,[cons_0_1|0, cons_1_1|0, cons_5_1|0, cons_4_1|0, cons_3_1|0, cons_2_1|0]), (114,113,[encArg_1|1]), (114,114,[0_1|1, 1_1|1, 5_1|1, 4_1|1, 3_1|1, 2_1|1]), (114,115,[0_1|2]), (114,117,[4_1|2]), (114,126,[4_1|2]), (114,132,[2_1|2]), (114,142,[2_1|2]), (114,159,[0_1|2]), (114,169,[5_1|2]), (114,184,[2_1|2]), (114,194,[4_1|2]), (114,210,[5_1|2]), (114,222,[1_1|2]), (114,241,[2_1|2]), (114,255,[3_1|2]), (114,259,[5_1|2]), (114,266,[1_1|2]), (114,281,[1_1|2]), (114,288,[1_1|2]), (114,298,[4_1|2]), 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(385,386,[3_1|2]), (386,387,[4_1|2]), (386,358,[4_1|2]), (387,114,[5_1|2]), (387,255,[5_1|2]), (387,414,[5_1|2]), (387,185,[5_1|2]), (387,309,[5_1|2]), (387,314,[5_1|2]), (388,389,[1_1|2]), (389,390,[2_1|2]), (390,391,[0_1|2]), (391,392,[2_1|2]), (392,393,[2_1|2]), (393,394,[0_1|2]), (394,395,[2_1|2]), (395,114,[3_1|2]), (395,132,[3_1|2]), (395,142,[3_1|2]), (395,184,[3_1|2]), (395,241,[3_1|2]), (395,425,[3_1|2]), (395,437,[3_1|2]), (395,388,[1_1|2]), (395,396,[4_1|2]), (395,414,[3_1|2]), (396,397,[5_1|2]), (397,398,[4_1|2]), (398,399,[5_1|2]), (399,400,[2_1|2]), (400,401,[5_1|2]), (401,402,[1_1|2]), (402,403,[4_1|2]), (403,404,[4_1|2]), (404,405,[3_1|2]), (405,406,[1_1|2]), (406,407,[2_1|2]), (407,408,[3_1|2]), (408,409,[4_1|2]), (409,410,[4_1|2]), (410,411,[2_1|2]), (411,412,[1_1|2]), (412,413,[2_1|2]), (413,114,[0_1|2]), (413,115,[0_1|2]), (413,159,[0_1|2]), (413,310,[0_1|2]), (413,243,[0_1|2]), (413,117,[4_1|2]), (413,126,[4_1|2]), (413,132,[2_1|2]), (413,142,[2_1|2]), (413,169,[5_1|2]), (413,184,[2_1|2]), (413,194,[4_1|2]), (413,210,[5_1|2]), (413,222,[1_1|2]), (413,241,[2_1|2]), (414,415,[1_1|2]), (415,416,[4_1|2]), (416,417,[1_1|2]), (417,418,[1_1|2]), (418,419,[3_1|2]), (419,420,[4_1|2]), (420,421,[1_1|2]), (421,422,[3_1|2]), (422,423,[4_1|2]), (423,424,[4_1|2]), (424,114,[4_1|2]), (424,117,[4_1|2]), (424,126,[4_1|2]), (424,194,[4_1|2]), (424,298,[4_1|2]), (424,327,[4_1|2]), (424,332,[4_1|2]), (424,340,[4_1|2]), (424,358,[4_1|2]), (424,371,[4_1|2]), (424,396,[4_1|2]), (424,127,[4_1|2]), (424,195,[4_1|2]), (425,426,[0_1|2]), (426,427,[1_1|2]), (427,428,[3_1|2]), (428,429,[2_1|2]), (429,430,[5_1|2]), (430,431,[5_1|2]), (431,432,[5_1|2]), (432,433,[1_1|2]), (433,434,[1_1|2]), (434,435,[4_1|2]), (435,114,[4_1|2]), (435,255,[4_1|2]), (435,414,[4_1|2]), (435,327,[4_1|2]), (435,332,[4_1|2]), (435,340,[4_1|2]), (435,358,[4_1|2]), (435,371,[4_1|2]), (436,437,[2_1|2]), (437,438,[4_1|2]), (438,439,[2_1|2]), (439,440,[3_1|2]), (440,441,[4_1|2]), (441,442,[1_1|2]), (442,443,[1_1|2]), (443,444,[2_1|2]), (444,445,[0_1|2]), (445,446,[3_1|2]), (446,447,[0_1|2]), (447,448,[1_1|2]), (448,449,[4_1|2]), (449,450,[1_1|2]), (450,451,[3_1|2]), (451,452,[0_1|2]), (452,453,[5_1|2]), (453,454,[3_1|2]), (453,388,[1_1|2]), (453,396,[4_1|2]), (454,114,[0_1|2]), (454,115,[0_1|2]), (454,159,[0_1|2]), (454,310,[0_1|2]), (454,302,[0_1|2]), (454,117,[4_1|2]), (454,126,[4_1|2]), (454,132,[2_1|2]), (454,142,[2_1|2]), (454,169,[5_1|2]), (454,184,[2_1|2]), (454,194,[4_1|2]), (454,210,[5_1|2]), (454,222,[1_1|2]), (454,241,[2_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)