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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 86 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 142 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(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) 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(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) 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(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) 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(2(x1)))) -> 3(2(1(x1))) 2(4(1(5(2(x1))))) -> 2(2(2(4(4(x1))))) 3(0(0(1(0(x1))))) -> 1(1(3(0(x1)))) 0(1(3(5(4(2(2(x1))))))) -> 3(5(1(3(1(2(x1)))))) 0(2(4(1(2(0(5(3(x1)))))))) -> 5(3(2(0(1(1(2(x1))))))) 2(1(2(3(0(3(5(0(x1)))))))) -> 2(1(2(0(4(5(3(0(x1)))))))) 3(4(0(3(2(1(2(2(x1)))))))) -> 1(2(5(0(2(0(2(x1))))))) 0(4(0(3(3(2(1(1(3(x1))))))))) -> 3(4(2(3(1(3(1(1(x1)))))))) 3(2(2(2(5(5(3(5(0(x1))))))))) -> 1(2(1(0(1(4(0(5(0(x1))))))))) 0(1(1(3(0(4(4(4(4(2(x1)))))))))) -> 3(0(5(4(1(1(3(5(x1)))))))) 4(0(2(0(2(0(2(0(5(4(x1)))))))))) -> 4(0(1(4(1(3(2(2(3(x1))))))))) 0(1(5(0(4(3(5(2(1(5(2(x1))))))))))) -> 5(4(2(0(1(2(4(4(5(1(2(2(x1)))))))))))) 1(1(2(4(0(2(2(1(5(5(0(x1))))))))))) -> 1(1(1(0(1(3(1(2(1(0(x1)))))))))) 5(1(3(5(1(1(2(1(2(0(2(2(x1)))))))))))) -> 3(5(1(0(3(4(3(5(2(0(4(3(x1)))))))))))) 5(2(4(2(4(0(5(1(1(0(4(3(x1)))))))))))) -> 1(2(0(5(5(0(3(2(2(1(3(x1))))))))))) 1(4(2(4(5(3(0(0(2(3(4(5(3(x1))))))))))))) -> 1(3(2(2(0(0(5(1(2(0(4(0(3(x1))))))))))))) 2(3(3(4(0(4(1(0(0(2(3(0(1(x1))))))))))))) -> 2(4(0(3(5(0(3(1(4(4(2(0(1(x1))))))))))))) 5(4(1(4(1(0(5(2(1(5(0(4(4(x1))))))))))))) -> 3(0(0(0(2(5(5(1(0(1(2(2(0(4(x1)))))))))))))) 0(3(0(1(1(0(4(5(3(5(1(0(3(2(x1)))))))))))))) -> 5(5(1(3(3(3(3(1(5(5(1(3(x1)))))))))))) 2(0(3(1(5(5(0(1(5(5(1(5(2(0(4(x1))))))))))))))) -> 2(1(3(4(1(4(3(4(1(2(4(1(1(4(4(x1))))))))))))))) 0(3(3(0(4(4(0(0(2(1(5(0(5(5(1(2(x1)))))))))))))))) -> 1(4(2(2(4(4(0(0(0(4(0(3(1(5(5(1(x1)))))))))))))))) 4(2(1(4(0(4(4(2(2(1(2(4(3(5(1(0(x1)))))))))))))))) -> 4(1(3(1(3(3(2(5(0(4(5(5(4(0(x1)))))))))))))) 2(2(1(4(4(4(5(4(0(2(2(5(3(2(3(3(2(x1))))))))))))))))) -> 2(2(4(4(2(4(0(2(3(1(1(1(1(4(1(1(x1)))))))))))))))) 2(1(2(0(1(5(1(0(3(2(0(1(1(1(4(4(0(2(x1)))))))))))))))))) -> 2(5(4(0(3(5(2(4(2(3(0(2(2(0(1(1(4(1(x1)))))))))))))))))) 4(2(0(5(2(5(5(5(5(5(5(3(3(3(0(0(2(5(x1)))))))))))))))))) -> 4(0(1(4(5(0(1(5(1(2(2(5(2(2(5(4(1(5(x1)))))))))))))))))) 0(3(1(0(0(3(5(1(2(1(2(2(0(3(3(0(5(4(2(x1))))))))))))))))))) -> 5(2(1(0(4(4(0(1(0(1(2(2(5(1(2(5(4(5(x1)))))))))))))))))) 0(1(3(4(4(5(3(0(2(0(3(5(2(1(0(1(5(2(0(1(x1)))))))))))))))))))) -> 1(5(3(2(4(4(4(1(2(3(0(3(2(4(1(0(0(0(2(1(x1)))))))))))))))))))) 0(5(1(5(2(1(1(5(3(2(3(1(3(4(1(2(3(5(4(3(x1)))))))))))))))))))) -> 2(1(4(0(1(0(2(0(3(3(4(0(3(5(4(3(2(5(4(1(3(x1))))))))))))))))))))) 2(2(1(5(0(4(3(4(5(3(3(3(3(1(5(3(1(4(0(1(x1)))))))))))))))))))) -> 2(5(0(5(4(2(1(2(5(5(4(4(1(1(4(5(3(1(3(1(x1)))))))))))))))))))) 3(0(1(3(5(1(3(4(2(5(4(0(5(0(1(5(5(5(1(0(2(x1))))))))))))))))))))) -> 3(0(1(5(5(0(3(2(4(4(0(2(4(5(3(3(2(5(1(1(1(x1))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_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)) 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 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] {(148,149,[0_1|0, 2_1|0, 3_1|0, 4_1|0, 1_1|0, 5_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]), (148,150,[0_1|1, 2_1|1, 3_1|1, 4_1|1, 1_1|1, 5_1|1]), (148,151,[3_1|2]), (148,153,[3_1|2]), (148,158,[1_1|2]), (148,177,[3_1|2]), (148,184,[5_1|2]), (148,195,[5_1|2]), (148,201,[3_1|2]), (148,208,[5_1|2]), (148,219,[1_1|2]), (148,234,[5_1|2]), (148,251,[2_1|2]), (148,271,[2_1|2]), (148,275,[2_1|2]), (148,282,[2_1|2]), (148,299,[2_1|2]), (148,311,[2_1|2]), (148,325,[2_1|2]), (148,340,[2_1|2]), (148,359,[1_1|2]), (148,362,[3_1|2]), (148,382,[1_1|2]), (148,388,[1_1|2]), (148,396,[4_1|2]), (148,404,[4_1|2]), (148,417,[4_1|2]), (148,434,[1_1|2]), (148,443,[1_1|2]), (148,455,[3_1|2]), (148,466,[1_1|2]), (148,476,[3_1|2]), (149,149,[cons_0_1|0, cons_2_1|0, cons_3_1|0, cons_4_1|0, cons_1_1|0, cons_5_1|0]), (150,149,[encArg_1|1]), (150,150,[0_1|1, 2_1|1, 3_1|1, 4_1|1, 1_1|1, 5_1|1]), (150,151,[3_1|2]), (150,153,[3_1|2]), (150,158,[1_1|2]), (150,177,[3_1|2]), (150,184,[5_1|2]), (150,195,[5_1|2]), (150,201,[3_1|2]), (150,208,[5_1|2]), (150,219,[1_1|2]), (150,234,[5_1|2]), (150,251,[2_1|2]), (150,271,[2_1|2]), (150,275,[2_1|2]), (150,282,[2_1|2]), (150,299,[2_1|2]), 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(398,399,[4_1|2]), (399,400,[1_1|2]), (400,401,[3_1|2]), (401,402,[2_1|2]), (402,403,[2_1|2]), (402,299,[2_1|2]), (403,150,[3_1|2]), (403,396,[3_1|2]), (403,404,[3_1|2]), (403,417,[3_1|2]), (403,185,[3_1|2]), (403,359,[1_1|2]), (403,362,[3_1|2]), (403,382,[1_1|2]), (403,388,[1_1|2]), (404,405,[1_1|2]), (405,406,[3_1|2]), (406,407,[1_1|2]), (407,408,[3_1|2]), (408,409,[3_1|2]), (409,410,[2_1|2]), (410,411,[5_1|2]), (411,412,[0_1|2]), (412,413,[4_1|2]), (413,414,[5_1|2]), (414,415,[5_1|2]), (415,416,[4_1|2]), (415,396,[4_1|2]), (416,150,[0_1|2]), (416,458,[0_1|2]), (416,151,[3_1|2]), (416,153,[3_1|2]), (416,158,[1_1|2]), (416,177,[3_1|2]), (416,184,[5_1|2]), (416,195,[5_1|2]), (416,201,[3_1|2]), (416,208,[5_1|2]), (416,219,[1_1|2]), (416,234,[5_1|2]), (416,251,[2_1|2]), (417,418,[0_1|2]), (418,419,[1_1|2]), (419,420,[4_1|2]), (420,421,[5_1|2]), (421,422,[0_1|2]), (422,423,[1_1|2]), (423,424,[5_1|2]), (424,425,[1_1|2]), (425,426,[2_1|2]), (426,427,[2_1|2]), (427,428,[5_1|2]), (428,429,[2_1|2]), (429,430,[2_1|2]), (430,431,[5_1|2]), (431,432,[4_1|2]), (432,433,[1_1|2]), (433,150,[5_1|2]), (433,184,[5_1|2]), (433,195,[5_1|2]), (433,208,[5_1|2]), (433,234,[5_1|2]), (433,283,[5_1|2]), (433,341,[5_1|2]), (433,455,[3_1|2]), (433,466,[1_1|2]), (433,476,[3_1|2]), (434,435,[1_1|2]), (435,436,[1_1|2]), (436,437,[0_1|2]), (437,438,[1_1|2]), (438,439,[3_1|2]), (439,440,[1_1|2]), (440,441,[2_1|2]), (441,442,[1_1|2]), (442,150,[0_1|2]), (442,151,[3_1|2]), (442,153,[3_1|2]), (442,158,[1_1|2]), (442,177,[3_1|2]), (442,184,[5_1|2]), (442,195,[5_1|2]), (442,201,[3_1|2]), (442,208,[5_1|2]), (442,219,[1_1|2]), (442,234,[5_1|2]), (442,251,[2_1|2]), (443,444,[3_1|2]), (444,445,[2_1|2]), (445,446,[2_1|2]), (446,447,[0_1|2]), (447,448,[0_1|2]), (448,449,[5_1|2]), (449,450,[1_1|2]), (450,451,[2_1|2]), (451,452,[0_1|2]), (451,201,[3_1|2]), (451,493,[3_1|3]), (452,453,[4_1|2]), (453,454,[0_1|2]), (453,208,[5_1|2]), (453,219,[1_1|2]), (453,234,[5_1|2]), (453,177,[3_1|2]), (454,150,[3_1|2]), (454,151,[3_1|2]), (454,153,[3_1|2]), (454,177,[3_1|2]), (454,201,[3_1|2]), (454,362,[3_1|2]), (454,455,[3_1|2]), (454,476,[3_1|2]), (454,196,[3_1|2]), (454,359,[1_1|2]), (454,382,[1_1|2]), (454,388,[1_1|2]), (455,456,[5_1|2]), (456,457,[1_1|2]), (457,458,[0_1|2]), (458,459,[3_1|2]), (459,460,[4_1|2]), (460,461,[3_1|2]), (461,462,[5_1|2]), (462,463,[2_1|2]), (463,464,[0_1|2]), (464,465,[4_1|2]), (465,150,[3_1|2]), (465,251,[3_1|2]), (465,271,[3_1|2]), (465,275,[3_1|2]), (465,282,[3_1|2]), (465,299,[3_1|2]), (465,311,[3_1|2]), (465,325,[3_1|2]), (465,340,[3_1|2]), (465,272,[3_1|2]), (465,326,[3_1|2]), (465,359,[1_1|2]), (465,362,[3_1|2]), (465,382,[1_1|2]), (465,388,[1_1|2]), (466,467,[2_1|2]), (467,468,[0_1|2]), (468,469,[5_1|2]), (469,470,[5_1|2]), (470,471,[0_1|2]), (471,472,[3_1|2]), (472,473,[2_1|2]), (473,474,[2_1|2]), (474,475,[1_1|2]), (475,150,[3_1|2]), (475,151,[3_1|2]), (475,153,[3_1|2]), (475,177,[3_1|2]), (475,201,[3_1|2]), (475,362,[3_1|2]), (475,455,[3_1|2]), (475,476,[3_1|2]), (475,359,[1_1|2]), (475,382,[1_1|2]), (475,388,[1_1|2]), (476,477,[0_1|2]), (477,478,[0_1|2]), (478,479,[0_1|2]), (479,480,[2_1|2]), (480,481,[5_1|2]), (481,482,[5_1|2]), (482,483,[1_1|2]), (483,484,[0_1|2]), (483,500,[3_1|3]), (484,485,[1_1|2]), (485,486,[2_1|2]), (486,487,[2_1|2]), (487,488,[0_1|2]), (487,201,[3_1|2]), (488,150,[4_1|2]), (488,396,[4_1|2]), (488,404,[4_1|2]), (488,417,[4_1|2]), (489,490,[2_1|3]), (490,245,[1_1|3]), (491,492,[2_1|3]), (492,272,[1_1|3]), (492,326,[1_1|3]), (493,494,[4_1|3]), (494,495,[2_1|3]), (495,496,[3_1|3]), (496,497,[1_1|3]), (497,498,[3_1|3]), (498,499,[1_1|3]), (499,444,[1_1|3]), (499,313,[1_1|3]), (500,501,[2_1|3]), (501,487,[1_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)