WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 48 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 124 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(1(2(2(x1)))) -> 0(1(1(x1))) 1(3(0(2(4(3(2(x1))))))) -> 0(2(4(5(3(4(x1)))))) 4(5(3(0(4(0(3(x1))))))) -> 3(5(1(2(2(5(0(x1))))))) 5(2(0(3(3(2(5(2(x1)))))))) -> 3(4(1(3(4(2(2(1(x1)))))))) 1(5(1(1(4(5(4(2(0(x1))))))))) -> 5(1(5(0(2(2(1(0(x1)))))))) 2(5(4(4(0(0(2(1(2(x1))))))))) -> 2(2(0(4(4(3(0(2(x1)))))))) 1(5(1(5(1(0(4(0(3(1(x1)))))))))) -> 1(3(0(0(0(1(4(2(4(1(1(x1))))))))))) 4(3(2(1(2(0(1(0(3(3(x1)))))))))) -> 0(1(5(5(0(2(0(3(3(x1))))))))) 1(4(2(0(2(3(1(3(4(0(4(x1))))))))))) -> 0(4(2(5(2(0(3(2(5(4(1(x1))))))))))) 3(2(4(5(3(3(4(1(5(4(0(3(x1)))))))))))) -> 3(0(4(1(5(1(3(3(3(4(3(x1))))))))))) 4(0(3(3(3(1(4(2(2(1(0(2(x1)))))))))))) -> 4(0(4(2(2(2(2(4(4(2(1(0(1(x1))))))))))))) 1(0(2(3(3(2(5(2(1(3(3(1(4(x1))))))))))))) -> 0(3(3(4(0(0(0(2(3(3(3(0(x1)))))))))))) 1(2(5(3(5(2(4(1(0(2(4(3(2(x1))))))))))))) -> 5(3(1(4(3(2(2(0(0(5(3(4(x1)))))))))))) 2(2(0(3(2(2(3(1(1(1(1(0(4(x1))))))))))))) -> 2(2(3(0(2(4(5(0(3(3(2(1(1(x1))))))))))))) 3(0(1(0(0(5(3(5(2(0(1(0(2(x1))))))))))))) -> 2(3(2(5(5(2(1(2(2(0(4(4(5(2(x1)))))))))))))) 4(2(4(3(3(0(1(0(3(3(2(1(0(x1))))))))))))) -> 4(5(4(0(3(1(4(2(5(1(5(2(0(x1))))))))))))) 3(1(4(3(5(5(1(2(5(1(3(2(2(0(4(x1))))))))))))))) -> 3(5(5(5(2(3(4(1(3(4(2(0(0(2(2(x1))))))))))))))) 2(3(5(4(0(1(3(4(4(4(2(5(4(4(1(2(x1)))))))))))))))) -> 4(2(0(0(0(0(5(2(5(1(3(3(2(1(x1)))))))))))))) 1(2(3(0(5(4(4(0(1(2(0(5(0(3(0(4(4(x1))))))))))))))))) -> 1(4(0(1(0(2(4(4(5(3(0(2(1(0(1(2(x1)))))))))))))))) 1(3(2(4(1(1(5(5(1(2(0(2(0(2(2(4(1(x1))))))))))))))))) -> 1(0(2(4(3(5(2(1(1(1(0(1(5(4(0(1(x1)))))))))))))))) 1(3(5(2(3(5(1(2(0(5(0(4(2(4(3(3(0(x1))))))))))))))))) -> 5(5(2(0(0(0(5(5(2(1(4(4(3(0(1(0(x1)))))))))))))))) 1(4(1(3(0(5(4(1(2(3(0(3(5(4(4(1(4(2(x1)))))))))))))))))) -> 1(3(1(3(2(2(0(0(5(1(4(1(5(5(1(5(1(x1))))))))))))))))) 3(1(0(2(1(3(5(4(3(5(5(2(4(5(3(0(0(2(x1)))))))))))))))))) -> 2(2(2(0(2(3(1(4(5(3(5(4(4(4(0(4(0(5(x1)))))))))))))))))) 3(4(1(2(3(1(3(5(0(4(0(1(1(5(5(2(2(3(x1)))))))))))))))))) -> 0(3(5(2(3(2(0(0(0(0(4(0(0(5(3(2(3(x1))))))))))))))))) 5(2(4(3(2(3(1(5(5(1(3(0(5(1(5(1(4(4(x1)))))))))))))))))) -> 5(5(5(4(4(4(2(5(0(0(3(4(2(1(5(4(5(2(x1)))))))))))))))))) 3(4(3(1(0(0(5(5(2(0(2(3(0(1(4(0(1(2(2(x1))))))))))))))))))) -> 0(5(2(2(1(0(3(1(4(5(3(2(2(1(5(4(4(2(1(x1))))))))))))))))))) 4(2(3(0(0(3(5(1(0(4(0(0(3(5(2(2(5(2(2(x1))))))))))))))))))) -> 5(5(4(5(3(2(2(0(2(2(5(2(0(5(4(4(4(0(1(x1))))))))))))))))))) 4(2(5(1(0(4(4(2(2(5(5(3(4(3(2(3(4(1(4(x1))))))))))))))))))) -> 4(0(4(2(3(3(2(2(1(4(4(0(3(0(0(5(1(1(x1)))))))))))))))))) 2(4(4(0(4(2(1(1(0(4(3(2(2(0(3(4(4(5(2(4(4(x1))))))))))))))))))))) -> 2(4(1(4(3(0(3(5(4(5(4(2(1(3(0(5(2(4(0(3(2(x1))))))))))))))))))))) 5(2(0(5(3(4(0(2(3(5(0(2(3(2(1(5(0(1(5(0(3(x1))))))))))))))))))))) -> 5(5(3(2(1(4(1(3(5(0(1(5(0(5(1(4(2(3(0(0(3(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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_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(1(2(2(x1)))) -> 0(1(1(x1))) 1(3(0(2(4(3(2(x1))))))) -> 0(2(4(5(3(4(x1)))))) 4(5(3(0(4(0(3(x1))))))) -> 3(5(1(2(2(5(0(x1))))))) 5(2(0(3(3(2(5(2(x1)))))))) -> 3(4(1(3(4(2(2(1(x1)))))))) 1(5(1(1(4(5(4(2(0(x1))))))))) -> 5(1(5(0(2(2(1(0(x1)))))))) 2(5(4(4(0(0(2(1(2(x1))))))))) -> 2(2(0(4(4(3(0(2(x1)))))))) 1(5(1(5(1(0(4(0(3(1(x1)))))))))) -> 1(3(0(0(0(1(4(2(4(1(1(x1))))))))))) 4(3(2(1(2(0(1(0(3(3(x1)))))))))) -> 0(1(5(5(0(2(0(3(3(x1))))))))) 1(4(2(0(2(3(1(3(4(0(4(x1))))))))))) -> 0(4(2(5(2(0(3(2(5(4(1(x1))))))))))) 3(2(4(5(3(3(4(1(5(4(0(3(x1)))))))))))) -> 3(0(4(1(5(1(3(3(3(4(3(x1))))))))))) 4(0(3(3(3(1(4(2(2(1(0(2(x1)))))))))))) -> 4(0(4(2(2(2(2(4(4(2(1(0(1(x1))))))))))))) 1(0(2(3(3(2(5(2(1(3(3(1(4(x1))))))))))))) -> 0(3(3(4(0(0(0(2(3(3(3(0(x1)))))))))))) 1(2(5(3(5(2(4(1(0(2(4(3(2(x1))))))))))))) -> 5(3(1(4(3(2(2(0(0(5(3(4(x1)))))))))))) 2(2(0(3(2(2(3(1(1(1(1(0(4(x1))))))))))))) -> 2(2(3(0(2(4(5(0(3(3(2(1(1(x1))))))))))))) 3(0(1(0(0(5(3(5(2(0(1(0(2(x1))))))))))))) -> 2(3(2(5(5(2(1(2(2(0(4(4(5(2(x1)))))))))))))) 4(2(4(3(3(0(1(0(3(3(2(1(0(x1))))))))))))) -> 4(5(4(0(3(1(4(2(5(1(5(2(0(x1))))))))))))) 3(1(4(3(5(5(1(2(5(1(3(2(2(0(4(x1))))))))))))))) -> 3(5(5(5(2(3(4(1(3(4(2(0(0(2(2(x1))))))))))))))) 2(3(5(4(0(1(3(4(4(4(2(5(4(4(1(2(x1)))))))))))))))) -> 4(2(0(0(0(0(5(2(5(1(3(3(2(1(x1)))))))))))))) 1(2(3(0(5(4(4(0(1(2(0(5(0(3(0(4(4(x1))))))))))))))))) -> 1(4(0(1(0(2(4(4(5(3(0(2(1(0(1(2(x1)))))))))))))))) 1(3(2(4(1(1(5(5(1(2(0(2(0(2(2(4(1(x1))))))))))))))))) -> 1(0(2(4(3(5(2(1(1(1(0(1(5(4(0(1(x1)))))))))))))))) 1(3(5(2(3(5(1(2(0(5(0(4(2(4(3(3(0(x1))))))))))))))))) -> 5(5(2(0(0(0(5(5(2(1(4(4(3(0(1(0(x1)))))))))))))))) 1(4(1(3(0(5(4(1(2(3(0(3(5(4(4(1(4(2(x1)))))))))))))))))) -> 1(3(1(3(2(2(0(0(5(1(4(1(5(5(1(5(1(x1))))))))))))))))) 3(1(0(2(1(3(5(4(3(5(5(2(4(5(3(0(0(2(x1)))))))))))))))))) -> 2(2(2(0(2(3(1(4(5(3(5(4(4(4(0(4(0(5(x1)))))))))))))))))) 3(4(1(2(3(1(3(5(0(4(0(1(1(5(5(2(2(3(x1)))))))))))))))))) -> 0(3(5(2(3(2(0(0(0(0(4(0(0(5(3(2(3(x1))))))))))))))))) 5(2(4(3(2(3(1(5(5(1(3(0(5(1(5(1(4(4(x1)))))))))))))))))) -> 5(5(5(4(4(4(2(5(0(0(3(4(2(1(5(4(5(2(x1)))))))))))))))))) 3(4(3(1(0(0(5(5(2(0(2(3(0(1(4(0(1(2(2(x1))))))))))))))))))) -> 0(5(2(2(1(0(3(1(4(5(3(2(2(1(5(4(4(2(1(x1))))))))))))))))))) 4(2(3(0(0(3(5(1(0(4(0(0(3(5(2(2(5(2(2(x1))))))))))))))))))) -> 5(5(4(5(3(2(2(0(2(2(5(2(0(5(4(4(4(0(1(x1))))))))))))))))))) 4(2(5(1(0(4(4(2(2(5(5(3(4(3(2(3(4(1(4(x1))))))))))))))))))) -> 4(0(4(2(3(3(2(2(1(4(4(0(3(0(0(5(1(1(x1)))))))))))))))))) 2(4(4(0(4(2(1(1(0(4(3(2(2(0(3(4(4(5(2(4(4(x1))))))))))))))))))))) -> 2(4(1(4(3(0(3(5(4(5(4(2(1(3(0(5(2(4(0(3(2(x1))))))))))))))))))))) 5(2(0(5(3(4(0(2(3(5(0(2(3(2(1(5(0(1(5(0(3(x1))))))))))))))))))))) -> 5(5(3(2(1(4(1(3(5(0(1(5(0(5(1(4(2(3(0(0(3(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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_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(1(2(2(x1)))) -> 0(1(1(x1))) 1(3(0(2(4(3(2(x1))))))) -> 0(2(4(5(3(4(x1)))))) 4(5(3(0(4(0(3(x1))))))) -> 3(5(1(2(2(5(0(x1))))))) 5(2(0(3(3(2(5(2(x1)))))))) -> 3(4(1(3(4(2(2(1(x1)))))))) 1(5(1(1(4(5(4(2(0(x1))))))))) -> 5(1(5(0(2(2(1(0(x1)))))))) 2(5(4(4(0(0(2(1(2(x1))))))))) -> 2(2(0(4(4(3(0(2(x1)))))))) 1(5(1(5(1(0(4(0(3(1(x1)))))))))) -> 1(3(0(0(0(1(4(2(4(1(1(x1))))))))))) 4(3(2(1(2(0(1(0(3(3(x1)))))))))) -> 0(1(5(5(0(2(0(3(3(x1))))))))) 1(4(2(0(2(3(1(3(4(0(4(x1))))))))))) -> 0(4(2(5(2(0(3(2(5(4(1(x1))))))))))) 3(2(4(5(3(3(4(1(5(4(0(3(x1)))))))))))) -> 3(0(4(1(5(1(3(3(3(4(3(x1))))))))))) 4(0(3(3(3(1(4(2(2(1(0(2(x1)))))))))))) -> 4(0(4(2(2(2(2(4(4(2(1(0(1(x1))))))))))))) 1(0(2(3(3(2(5(2(1(3(3(1(4(x1))))))))))))) -> 0(3(3(4(0(0(0(2(3(3(3(0(x1)))))))))))) 1(2(5(3(5(2(4(1(0(2(4(3(2(x1))))))))))))) -> 5(3(1(4(3(2(2(0(0(5(3(4(x1)))))))))))) 2(2(0(3(2(2(3(1(1(1(1(0(4(x1))))))))))))) -> 2(2(3(0(2(4(5(0(3(3(2(1(1(x1))))))))))))) 3(0(1(0(0(5(3(5(2(0(1(0(2(x1))))))))))))) -> 2(3(2(5(5(2(1(2(2(0(4(4(5(2(x1)))))))))))))) 4(2(4(3(3(0(1(0(3(3(2(1(0(x1))))))))))))) -> 4(5(4(0(3(1(4(2(5(1(5(2(0(x1))))))))))))) 3(1(4(3(5(5(1(2(5(1(3(2(2(0(4(x1))))))))))))))) -> 3(5(5(5(2(3(4(1(3(4(2(0(0(2(2(x1))))))))))))))) 2(3(5(4(0(1(3(4(4(4(2(5(4(4(1(2(x1)))))))))))))))) -> 4(2(0(0(0(0(5(2(5(1(3(3(2(1(x1)))))))))))))) 1(2(3(0(5(4(4(0(1(2(0(5(0(3(0(4(4(x1))))))))))))))))) -> 1(4(0(1(0(2(4(4(5(3(0(2(1(0(1(2(x1)))))))))))))))) 1(3(2(4(1(1(5(5(1(2(0(2(0(2(2(4(1(x1))))))))))))))))) -> 1(0(2(4(3(5(2(1(1(1(0(1(5(4(0(1(x1)))))))))))))))) 1(3(5(2(3(5(1(2(0(5(0(4(2(4(3(3(0(x1))))))))))))))))) -> 5(5(2(0(0(0(5(5(2(1(4(4(3(0(1(0(x1)))))))))))))))) 1(4(1(3(0(5(4(1(2(3(0(3(5(4(4(1(4(2(x1)))))))))))))))))) -> 1(3(1(3(2(2(0(0(5(1(4(1(5(5(1(5(1(x1))))))))))))))))) 3(1(0(2(1(3(5(4(3(5(5(2(4(5(3(0(0(2(x1)))))))))))))))))) -> 2(2(2(0(2(3(1(4(5(3(5(4(4(4(0(4(0(5(x1)))))))))))))))))) 3(4(1(2(3(1(3(5(0(4(0(1(1(5(5(2(2(3(x1)))))))))))))))))) -> 0(3(5(2(3(2(0(0(0(0(4(0(0(5(3(2(3(x1))))))))))))))))) 5(2(4(3(2(3(1(5(5(1(3(0(5(1(5(1(4(4(x1)))))))))))))))))) -> 5(5(5(4(4(4(2(5(0(0(3(4(2(1(5(4(5(2(x1)))))))))))))))))) 3(4(3(1(0(0(5(5(2(0(2(3(0(1(4(0(1(2(2(x1))))))))))))))))))) -> 0(5(2(2(1(0(3(1(4(5(3(2(2(1(5(4(4(2(1(x1))))))))))))))))))) 4(2(3(0(0(3(5(1(0(4(0(0(3(5(2(2(5(2(2(x1))))))))))))))))))) -> 5(5(4(5(3(2(2(0(2(2(5(2(0(5(4(4(4(0(1(x1))))))))))))))))))) 4(2(5(1(0(4(4(2(2(5(5(3(4(3(2(3(4(1(4(x1))))))))))))))))))) -> 4(0(4(2(3(3(2(2(1(4(4(0(3(0(0(5(1(1(x1)))))))))))))))))) 2(4(4(0(4(2(1(1(0(4(3(2(2(0(3(4(4(5(2(4(4(x1))))))))))))))))))))) -> 2(4(1(4(3(0(3(5(4(5(4(2(1(3(0(5(2(4(0(3(2(x1))))))))))))))))))))) 5(2(0(5(3(4(0(2(3(5(0(2(3(2(1(5(0(1(5(0(3(x1))))))))))))))))))))) -> 5(5(3(2(1(4(1(3(5(0(1(5(0(5(1(4(2(3(0(0(3(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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_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(1(2(2(x1)))) -> 0(1(1(x1))) 1(3(0(2(4(3(2(x1))))))) -> 0(2(4(5(3(4(x1)))))) 4(5(3(0(4(0(3(x1))))))) -> 3(5(1(2(2(5(0(x1))))))) 5(2(0(3(3(2(5(2(x1)))))))) -> 3(4(1(3(4(2(2(1(x1)))))))) 1(5(1(1(4(5(4(2(0(x1))))))))) -> 5(1(5(0(2(2(1(0(x1)))))))) 2(5(4(4(0(0(2(1(2(x1))))))))) -> 2(2(0(4(4(3(0(2(x1)))))))) 1(5(1(5(1(0(4(0(3(1(x1)))))))))) -> 1(3(0(0(0(1(4(2(4(1(1(x1))))))))))) 4(3(2(1(2(0(1(0(3(3(x1)))))))))) -> 0(1(5(5(0(2(0(3(3(x1))))))))) 1(4(2(0(2(3(1(3(4(0(4(x1))))))))))) -> 0(4(2(5(2(0(3(2(5(4(1(x1))))))))))) 3(2(4(5(3(3(4(1(5(4(0(3(x1)))))))))))) -> 3(0(4(1(5(1(3(3(3(4(3(x1))))))))))) 4(0(3(3(3(1(4(2(2(1(0(2(x1)))))))))))) -> 4(0(4(2(2(2(2(4(4(2(1(0(1(x1))))))))))))) 1(0(2(3(3(2(5(2(1(3(3(1(4(x1))))))))))))) -> 0(3(3(4(0(0(0(2(3(3(3(0(x1)))))))))))) 1(2(5(3(5(2(4(1(0(2(4(3(2(x1))))))))))))) -> 5(3(1(4(3(2(2(0(0(5(3(4(x1)))))))))))) 2(2(0(3(2(2(3(1(1(1(1(0(4(x1))))))))))))) -> 2(2(3(0(2(4(5(0(3(3(2(1(1(x1))))))))))))) 3(0(1(0(0(5(3(5(2(0(1(0(2(x1))))))))))))) -> 2(3(2(5(5(2(1(2(2(0(4(4(5(2(x1)))))))))))))) 4(2(4(3(3(0(1(0(3(3(2(1(0(x1))))))))))))) -> 4(5(4(0(3(1(4(2(5(1(5(2(0(x1))))))))))))) 3(1(4(3(5(5(1(2(5(1(3(2(2(0(4(x1))))))))))))))) -> 3(5(5(5(2(3(4(1(3(4(2(0(0(2(2(x1))))))))))))))) 2(3(5(4(0(1(3(4(4(4(2(5(4(4(1(2(x1)))))))))))))))) -> 4(2(0(0(0(0(5(2(5(1(3(3(2(1(x1)))))))))))))) 1(2(3(0(5(4(4(0(1(2(0(5(0(3(0(4(4(x1))))))))))))))))) -> 1(4(0(1(0(2(4(4(5(3(0(2(1(0(1(2(x1)))))))))))))))) 1(3(2(4(1(1(5(5(1(2(0(2(0(2(2(4(1(x1))))))))))))))))) -> 1(0(2(4(3(5(2(1(1(1(0(1(5(4(0(1(x1)))))))))))))))) 1(3(5(2(3(5(1(2(0(5(0(4(2(4(3(3(0(x1))))))))))))))))) -> 5(5(2(0(0(0(5(5(2(1(4(4(3(0(1(0(x1)))))))))))))))) 1(4(1(3(0(5(4(1(2(3(0(3(5(4(4(1(4(2(x1)))))))))))))))))) -> 1(3(1(3(2(2(0(0(5(1(4(1(5(5(1(5(1(x1))))))))))))))))) 3(1(0(2(1(3(5(4(3(5(5(2(4(5(3(0(0(2(x1)))))))))))))))))) -> 2(2(2(0(2(3(1(4(5(3(5(4(4(4(0(4(0(5(x1)))))))))))))))))) 3(4(1(2(3(1(3(5(0(4(0(1(1(5(5(2(2(3(x1)))))))))))))))))) -> 0(3(5(2(3(2(0(0(0(0(4(0(0(5(3(2(3(x1))))))))))))))))) 5(2(4(3(2(3(1(5(5(1(3(0(5(1(5(1(4(4(x1)))))))))))))))))) -> 5(5(5(4(4(4(2(5(0(0(3(4(2(1(5(4(5(2(x1)))))))))))))))))) 3(4(3(1(0(0(5(5(2(0(2(3(0(1(4(0(1(2(2(x1))))))))))))))))))) -> 0(5(2(2(1(0(3(1(4(5(3(2(2(1(5(4(4(2(1(x1))))))))))))))))))) 4(2(3(0(0(3(5(1(0(4(0(0(3(5(2(2(5(2(2(x1))))))))))))))))))) -> 5(5(4(5(3(2(2(0(2(2(5(2(0(5(4(4(4(0(1(x1))))))))))))))))))) 4(2(5(1(0(4(4(2(2(5(5(3(4(3(2(3(4(1(4(x1))))))))))))))))))) -> 4(0(4(2(3(3(2(2(1(4(4(0(3(0(0(5(1(1(x1)))))))))))))))))) 2(4(4(0(4(2(1(1(0(4(3(2(2(0(3(4(4(5(2(4(4(x1))))))))))))))))))))) -> 2(4(1(4(3(0(3(5(4(5(4(2(1(3(0(5(2(4(0(3(2(x1))))))))))))))))))))) 5(2(0(5(3(4(0(2(3(5(0(2(3(2(1(5(0(1(5(0(3(x1))))))))))))))))))))) -> 5(5(3(2(1(4(1(3(5(0(1(5(0(5(1(4(2(3(0(0(3(x1))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(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_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] {(151,152,[0_1|0, 1_1|0, 4_1|0, 5_1|0, 2_1|0, 3_1|0, encArg_1|0, encode_0_1|0, 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(419,176,[5_1|2]), (419,191,[5_1|2]), (419,198,[1_1|2]), (419,208,[0_1|2]), (419,218,[1_1|2]), (419,234,[0_1|2]), (419,245,[5_1|2]), (419,256,[1_1|2]), (420,421,[4_1|2]), (421,422,[1_1|2]), (422,423,[4_1|2]), (423,424,[3_1|2]), (424,425,[0_1|2]), (425,426,[3_1|2]), (426,427,[5_1|2]), (427,428,[4_1|2]), (428,429,[5_1|2]), (429,430,[4_1|2]), (430,431,[2_1|2]), (431,432,[1_1|2]), (432,433,[3_1|2]), (433,434,[0_1|2]), (434,435,[5_1|2]), (435,436,[2_1|2]), (436,437,[4_1|2]), (437,438,[0_1|2]), (438,439,[3_1|2]), (438,440,[3_1|2]), (439,153,[2_1|2]), (439,285,[2_1|2]), (439,297,[2_1|2]), (439,327,[2_1|2]), (439,407,[2_1|2, 4_1|2]), (439,388,[2_1|2]), (439,395,[2_1|2]), (439,420,[2_1|2]), (440,441,[0_1|2]), (441,442,[4_1|2]), (442,443,[1_1|2]), (443,444,[5_1|2]), (444,445,[1_1|2]), (445,446,[3_1|2]), (446,447,[3_1|2]), (447,448,[3_1|2]), (447,510,[0_1|2]), (448,449,[4_1|2]), (448,277,[0_1|2]), (449,153,[3_1|2]), (449,271,[3_1|2]), (449,344,[3_1|2]), (449,440,[3_1|2]), (449,463,[3_1|2]), (449,235,[3_1|2]), (449,495,[3_1|2]), (449,450,[2_1|2]), (449,477,[2_1|2]), (449,494,[0_1|2]), (449,510,[0_1|2]), (450,451,[3_1|2]), (451,452,[2_1|2]), (452,453,[5_1|2]), (453,454,[5_1|2]), (454,455,[2_1|2]), (455,456,[1_1|2]), (456,457,[2_1|2]), (457,458,[2_1|2]), (458,459,[0_1|2]), (459,460,[4_1|2]), (460,461,[4_1|2]), (461,462,[5_1|2]), (461,344,[3_1|2]), (461,351,[5_1|2]), (461,371,[5_1|2]), (462,153,[2_1|2]), (462,388,[2_1|2]), (462,395,[2_1|2]), (462,420,[2_1|2]), (462,450,[2_1|2]), (462,477,[2_1|2]), (462,157,[2_1|2]), (462,163,[2_1|2]), (462,407,[4_1|2]), (463,464,[5_1|2]), (464,465,[5_1|2]), (465,466,[5_1|2]), (466,467,[2_1|2]), (467,468,[3_1|2]), (468,469,[4_1|2]), (469,470,[1_1|2]), (470,471,[3_1|2]), (471,472,[4_1|2]), (472,473,[2_1|2]), (473,474,[0_1|2]), (474,475,[0_1|2]), (475,476,[2_1|2]), (475,395,[2_1|2]), (476,153,[2_1|2]), (476,285,[2_1|2]), (476,297,[2_1|2]), (476,327,[2_1|2]), (476,407,[2_1|2, 4_1|2]), (476,209,[2_1|2]), (476,391,[2_1|2]), (476,388,[2_1|2]), (476,395,[2_1|2]), (476,420,[2_1|2]), (477,478,[2_1|2]), (478,479,[2_1|2]), (479,480,[0_1|2]), (480,481,[2_1|2]), (481,482,[3_1|2]), (482,483,[1_1|2]), (483,484,[4_1|2]), (484,485,[5_1|2]), (485,486,[3_1|2]), (486,487,[5_1|2]), (487,488,[4_1|2]), (488,489,[4_1|2]), (489,490,[4_1|2]), (490,491,[0_1|2]), (491,492,[4_1|2]), (492,493,[0_1|2]), (493,153,[5_1|2]), (493,388,[5_1|2]), (493,395,[5_1|2]), (493,420,[5_1|2]), (493,450,[5_1|2]), (493,477,[5_1|2]), (493,157,[5_1|2]), (493,344,[3_1|2]), (493,351,[5_1|2]), (493,371,[5_1|2]), (494,495,[3_1|2]), (495,496,[5_1|2]), (496,497,[2_1|2]), (497,498,[3_1|2]), (498,499,[2_1|2]), (499,500,[0_1|2]), (500,501,[0_1|2]), (501,502,[0_1|2]), (502,503,[0_1|2]), (503,504,[4_1|2]), (504,505,[0_1|2]), (505,506,[0_1|2]), (506,507,[5_1|2]), (507,508,[3_1|2]), (508,509,[2_1|2]), (508,407,[4_1|2]), (509,153,[3_1|2]), (509,271,[3_1|2]), (509,344,[3_1|2]), (509,440,[3_1|2]), (509,463,[3_1|2]), (509,451,[3_1|2]), (509,397,[3_1|2]), (509,450,[2_1|2]), (509,477,[2_1|2]), (509,494,[0_1|2]), (509,510,[0_1|2]), (510,511,[5_1|2]), (511,512,[2_1|2]), (512,513,[2_1|2]), (513,514,[1_1|2]), (514,515,[0_1|2]), (515,516,[3_1|2]), (516,517,[1_1|2]), (517,518,[4_1|2]), (518,519,[5_1|2]), (519,520,[3_1|2]), (520,521,[2_1|2]), (521,522,[2_1|2]), (522,523,[1_1|2]), (523,524,[5_1|2]), (524,525,[4_1|2]), (525,526,[4_1|2]), (526,527,[2_1|2]), (527,153,[1_1|2]), (527,388,[1_1|2]), (527,395,[1_1|2]), (527,420,[1_1|2]), (527,450,[1_1|2]), (527,477,[1_1|2]), (527,389,[1_1|2]), (527,396,[1_1|2]), (527,478,[1_1|2]), (527,156,[0_1|2]), (527,161,[1_1|2]), (527,176,[5_1|2]), (527,191,[5_1|2]), (527,198,[1_1|2]), (527,208,[0_1|2]), (527,218,[1_1|2]), (527,234,[0_1|2]), (527,245,[5_1|2]), (527,256,[1_1|2]), (528,529,[1_1|3]), (529,389,[1_1|3]), (529,396,[1_1|3]), (529,478,[1_1|3]), (529,479,[1_1|3]), (530,531,[1_1|3]), (531,388,[1_1|3]), (531,395,[1_1|3]), (531,420,[1_1|3]), (531,450,[1_1|3]), (531,477,[1_1|3]), (531,408,[1_1|3]), (531,389,[1_1|3]), (531,396,[1_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)