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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 77 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(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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: 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(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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: 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(1(2(2(x1))))) -> 2(1(3(2(x1)))) 0(1(2(0(3(0(x1)))))) -> 1(3(2(1(2(x1))))) 4(3(1(2(1(0(x1)))))) -> 4(3(5(1(0(0(x1)))))) 1(1(0(1(4(0(0(x1))))))) -> 1(0(3(4(1(0(x1)))))) 4(3(3(5(4(5(1(1(x1)))))))) -> 4(5(5(3(5(1(0(1(x1)))))))) 2(4(4(3(1(1(1(1(5(x1))))))))) -> 2(5(5(3(1(3(1(0(x1)))))))) 4(1(0(5(4(3(3(4(0(x1))))))))) -> 5(3(0(1(4(3(4(4(2(x1))))))))) 0(1(4(3(4(5(4(2(2(2(0(1(1(x1))))))))))))) -> 0(2(0(0(3(5(1(1(3(2(0(5(x1)))))))))))) 0(4(0(1(3(1(2(5(2(0(5(1(1(x1))))))))))))) -> 4(0(2(0(4(5(1(1(3(0(1(1(1(x1))))))))))))) 4(3(0(1(0(1(2(2(2(2(1(5(2(x1))))))))))))) -> 4(1(0(3(1(4(0(2(2(0(3(3(x1)))))))))))) 3(2(0(4(3(1(0(0(1(0(4(4(5(2(x1)))))))))))))) -> 3(2(4(1(4(5(3(2(0(1(0(4(2(x1))))))))))))) 5(1(0(2(5(3(5(0(3(1(3(1(2(1(5(x1))))))))))))))) -> 0(1(5(0(4(0(2(1(4(4(2(5(5(3(0(4(x1)))))))))))))))) 4(3(3(3(5(0(4(3(0(2(4(1(3(1(2(0(2(x1))))))))))))))))) -> 1(0(0(4(2(2(4(4(5(5(1(4(3(5(4(1(3(x1))))))))))))))))) 5(1(0(2(1(5(4(0(0(1(4(5(5(3(5(3(2(x1))))))))))))))))) -> 2(3(5(0(0(0(3(2(1(0(3(0(4(3(3(2(x1)))))))))))))))) 5(4(5(0(3(3(2(5(3(0(4(0(2(5(0(3(0(x1))))))))))))))))) -> 5(1(5(3(0(1(2(3(2(4(5(2(0(5(3(3(0(x1))))))))))))))))) 4(1(1(0(0(0(1(3(4(1(3(5(1(1(5(2(5(2(x1)))))))))))))))))) -> 4(2(5(4(2(3(0(0(5(3(5(3(5(5(2(2(5(2(x1)))))))))))))))))) 4(5(4(0(5(5(3(4(5(3(1(2(5(3(2(3(0(0(x1)))))))))))))))))) -> 4(5(3(2(1(5(5(4(4(3(2(3(5(5(0(0(5(0(x1)))))))))))))))))) 5(3(3(1(1(5(4(0(2(2(1(2(3(1(2(1(5(2(x1)))))))))))))))))) -> 1(5(4(2(2(2(2(4(2(4(5(1(2(4(3(1(3(2(x1)))))))))))))))))) 1(4(1(0(1(0(1(3(2(0(1(3(5(2(4(0(2(3(5(x1))))))))))))))))))) -> 4(4(3(5(4(2(4(0(0(4(4(4(0(3(0(1(0(0(0(x1))))))))))))))))))) 3(2(3(1(0(1(1(5(4(5(5(4(2(0(2(3(1(5(1(x1))))))))))))))))))) -> 3(2(0(1(2(3(0(4(4(2(4(4(3(0(0(5(4(4(3(x1))))))))))))))))))) 1(5(1(2(3(3(4(1(2(0(3(4(1(0(4(3(4(1(5(4(x1)))))))))))))))))))) -> 1(5(0(3(4(1(5(3(3(5(1(0(0(4(3(4(2(3(0(4(x1)))))))))))))))))))) 2(2(2(2(0(2(4(0(2(4(2(0(4(3(3(3(5(3(2(1(x1)))))))))))))))))))) -> 2(0(2(4(5(0(0(5(1(3(0(5(0(2(1(4(3(3(3(0(x1)))))))))))))))))))) 0(1(2(5(5(1(0(1(3(5(0(4(5(2(0(0(3(0(5(5(0(x1))))))))))))))))))))) -> 4(2(1(4(1(1(3(2(0(4(3(3(0(4(3(3(1(3(1(x1))))))))))))))))))) 4(3(0(1(4(1(4(0(2(4(3(0(0(0(2(4(5(4(2(1(1(x1))))))))))))))))))))) -> 2(3(4(0(5(5(4(3(4(3(5(4(1(4(2(5(4(1(4(1(x1)))))))))))))))))))) 5(0(4(2(2(4(1(3(3(1(4(2(2(4(5(0(3(0(4(3(4(x1))))))))))))))))))))) -> 5(0(1(5(0(5(5(5(3(0(0(0(1(2(0(3(1(1(1(2(4(x1))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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: 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 2. The certificate found is represented by the following graph. "[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] {(150,151,[0_1|0, 4_1|0, 1_1|0, 2_1|0, 3_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]), (150,152,[0_1|1, 4_1|1, 1_1|1, 2_1|1, 3_1|1, 5_1|1]), (150,153,[2_1|2]), (150,156,[1_1|2]), (150,160,[4_1|2]), (150,178,[0_1|2]), (150,189,[4_1|2]), (150,201,[4_1|2]), (150,206,[4_1|2]), (150,213,[1_1|2]), (150,229,[4_1|2]), (150,240,[2_1|2]), (150,259,[5_1|2]), (150,267,[4_1|2]), (150,284,[4_1|2]), (150,301,[1_1|2]), (150,306,[4_1|2]), (150,324,[1_1|2]), (150,343,[2_1|2]), (150,350,[2_1|2]), (150,369,[3_1|2]), (150,381,[3_1|2]), (150,399,[0_1|2]), (150,414,[2_1|2]), (150,429,[5_1|2]), (150,445,[1_1|2]), (150,462,[5_1|2]), (151,151,[cons_0_1|0, cons_4_1|0, cons_1_1|0, cons_2_1|0, cons_3_1|0, cons_5_1|0]), (152,151,[encArg_1|1]), (152,152,[0_1|1, 4_1|1, 1_1|1, 2_1|1, 3_1|1, 5_1|1]), (152,153,[2_1|2]), (152,156,[1_1|2]), (152,160,[4_1|2]), (152,178,[0_1|2]), (152,189,[4_1|2]), (152,201,[4_1|2]), (152,206,[4_1|2]), (152,213,[1_1|2]), (152,229,[4_1|2]), (152,240,[2_1|2]), (152,259,[5_1|2]), (152,267,[4_1|2]), (152,284,[4_1|2]), (152,301,[1_1|2]), (152,306,[4_1|2]), (152,324,[1_1|2]), (152,343,[2_1|2]), (152,350,[2_1|2]), (152,369,[3_1|2]), (152,381,[3_1|2]), (152,399,[0_1|2]), (152,414,[2_1|2]), (152,429,[5_1|2]), (152,445,[1_1|2]), (152,462,[5_1|2]), (153,154,[1_1|2]), (154,155,[3_1|2]), (154,369,[3_1|2]), (154,381,[3_1|2]), (155,152,[2_1|2]), (155,153,[2_1|2]), (155,240,[2_1|2]), (155,343,[2_1|2]), (155,350,[2_1|2]), (155,414,[2_1|2]), (156,157,[3_1|2]), (157,158,[2_1|2]), (158,159,[1_1|2]), (159,152,[2_1|2]), (159,178,[2_1|2]), (159,399,[2_1|2]), (159,343,[2_1|2]), (159,350,[2_1|2]), (160,161,[2_1|2]), (161,162,[1_1|2]), (162,163,[4_1|2]), (163,164,[1_1|2]), (164,165,[1_1|2]), (165,166,[3_1|2]), (166,167,[2_1|2]), (167,168,[0_1|2]), (168,169,[4_1|2]), (169,170,[3_1|2]), (170,171,[3_1|2]), (171,172,[0_1|2]), (172,173,[4_1|2]), (173,174,[3_1|2]), (174,175,[3_1|2]), (175,176,[1_1|2]), (176,177,[3_1|2]), (177,152,[1_1|2]), (177,178,[1_1|2]), (177,399,[1_1|2]), (177,463,[1_1|2]), (177,301,[1_1|2]), (177,306,[4_1|2]), (177,324,[1_1|2]), (178,179,[2_1|2]), (179,180,[0_1|2]), (180,181,[0_1|2]), (181,182,[3_1|2]), (182,183,[5_1|2]), (183,184,[1_1|2]), (184,185,[1_1|2]), (185,186,[3_1|2]), (186,187,[2_1|2]), (187,188,[0_1|2]), (188,152,[5_1|2]), (188,156,[5_1|2]), (188,213,[5_1|2]), (188,301,[5_1|2]), (188,324,[5_1|2]), (188,445,[5_1|2, 1_1|2]), (188,399,[0_1|2]), (188,414,[2_1|2]), (188,429,[5_1|2]), (188,462,[5_1|2]), (189,190,[0_1|2]), (190,191,[2_1|2]), (191,192,[0_1|2]), (192,193,[4_1|2]), (193,194,[5_1|2]), (194,195,[1_1|2]), (195,196,[1_1|2]), (196,197,[3_1|2]), (197,198,[0_1|2]), (198,199,[1_1|2]), (199,200,[1_1|2]), (199,301,[1_1|2]), (200,152,[1_1|2]), (200,156,[1_1|2]), (200,213,[1_1|2]), (200,301,[1_1|2]), (200,324,[1_1|2]), (200,445,[1_1|2]), (200,306,[4_1|2]), (201,202,[3_1|2]), (202,203,[5_1|2]), (203,204,[1_1|2]), (204,205,[0_1|2]), (205,152,[0_1|2]), (205,178,[0_1|2]), (205,399,[0_1|2]), (205,214,[0_1|2]), (205,302,[0_1|2]), (205,153,[2_1|2]), (205,156,[1_1|2]), (205,160,[4_1|2]), (205,189,[4_1|2]), (206,207,[5_1|2]), (207,208,[5_1|2]), (208,209,[3_1|2]), (209,210,[5_1|2]), (210,211,[1_1|2]), (211,212,[0_1|2]), (211,153,[2_1|2]), (211,156,[1_1|2]), (211,160,[4_1|2]), (211,178,[0_1|2]), (212,152,[1_1|2]), (212,156,[1_1|2]), (212,213,[1_1|2]), (212,301,[1_1|2]), (212,324,[1_1|2]), (212,445,[1_1|2]), (212,306,[4_1|2]), (213,214,[0_1|2]), (214,215,[0_1|2]), (215,216,[4_1|2]), (216,217,[2_1|2]), (217,218,[2_1|2]), (218,219,[4_1|2]), (219,220,[4_1|2]), (220,221,[5_1|2]), (221,222,[5_1|2]), (222,223,[1_1|2]), (223,224,[4_1|2]), (224,225,[3_1|2]), (225,226,[5_1|2]), (226,227,[4_1|2]), (227,228,[1_1|2]), (228,152,[3_1|2]), (228,153,[3_1|2]), (228,240,[3_1|2]), (228,343,[3_1|2]), (228,350,[3_1|2]), (228,414,[3_1|2]), (228,179,[3_1|2]), (228,352,[3_1|2]), (228,369,[3_1|2]), (228,381,[3_1|2]), (229,230,[1_1|2]), (230,231,[0_1|2]), (231,232,[3_1|2]), (232,233,[1_1|2]), (233,234,[4_1|2]), (234,235,[0_1|2]), (235,236,[2_1|2]), (236,237,[2_1|2]), (237,238,[0_1|2]), (238,239,[3_1|2]), (239,152,[3_1|2]), (239,153,[3_1|2]), (239,240,[3_1|2]), (239,343,[3_1|2]), (239,350,[3_1|2]), (239,414,[3_1|2]), (239,369,[3_1|2]), (239,381,[3_1|2]), (240,241,[3_1|2]), (241,242,[4_1|2]), (242,243,[0_1|2]), (243,244,[5_1|2]), (244,245,[5_1|2]), (245,246,[4_1|2]), (246,247,[3_1|2]), (247,248,[4_1|2]), (248,249,[3_1|2]), (249,250,[5_1|2]), (250,251,[4_1|2]), (251,252,[1_1|2]), (252,253,[4_1|2]), (253,254,[2_1|2]), (254,255,[5_1|2]), (255,256,[4_1|2]), (256,257,[1_1|2]), (256,306,[4_1|2]), (257,258,[4_1|2]), (257,259,[5_1|2]), (257,267,[4_1|2]), (258,152,[1_1|2]), (258,156,[1_1|2]), (258,213,[1_1|2]), (258,301,[1_1|2]), (258,324,[1_1|2]), (258,445,[1_1|2]), (258,306,[4_1|2]), (259,260,[3_1|2]), (260,261,[0_1|2]), (261,262,[1_1|2]), (262,263,[4_1|2]), (263,264,[3_1|2]), (264,265,[4_1|2]), (265,266,[4_1|2]), (266,152,[2_1|2]), (266,178,[2_1|2]), (266,399,[2_1|2]), (266,190,[2_1|2]), (266,343,[2_1|2]), (266,350,[2_1|2]), (267,268,[2_1|2]), (268,269,[5_1|2]), (269,270,[4_1|2]), (270,271,[2_1|2]), (271,272,[3_1|2]), (272,273,[0_1|2]), (273,274,[0_1|2]), (274,275,[5_1|2]), (275,276,[3_1|2]), (276,277,[5_1|2]), (277,278,[3_1|2]), (278,279,[5_1|2]), (279,280,[5_1|2]), (280,281,[2_1|2]), (281,282,[2_1|2]), (282,283,[5_1|2]), (283,152,[2_1|2]), (283,153,[2_1|2]), (283,240,[2_1|2]), (283,343,[2_1|2]), (283,350,[2_1|2]), (283,414,[2_1|2]), (284,285,[5_1|2]), (285,286,[3_1|2]), (286,287,[2_1|2]), (287,288,[1_1|2]), (288,289,[5_1|2]), (289,290,[5_1|2]), (290,291,[4_1|2]), (291,292,[4_1|2]), (292,293,[3_1|2]), (293,294,[2_1|2]), (294,295,[3_1|2]), (295,296,[5_1|2]), (296,297,[5_1|2]), (297,298,[0_1|2]), (298,299,[0_1|2]), (299,300,[5_1|2]), (299,462,[5_1|2]), (300,152,[0_1|2]), (300,178,[0_1|2]), (300,399,[0_1|2]), (300,153,[2_1|2]), (300,156,[1_1|2]), (300,160,[4_1|2]), (300,189,[4_1|2]), (301,302,[0_1|2]), (302,303,[3_1|2]), (303,304,[4_1|2]), (303,259,[5_1|2]), (304,305,[1_1|2]), (305,152,[0_1|2]), (305,178,[0_1|2]), (305,399,[0_1|2]), (305,153,[2_1|2]), (305,156,[1_1|2]), (305,160,[4_1|2]), (305,189,[4_1|2]), (306,307,[4_1|2]), (307,308,[3_1|2]), (308,309,[5_1|2]), (309,310,[4_1|2]), (310,311,[2_1|2]), (311,312,[4_1|2]), (312,313,[0_1|2]), (313,314,[0_1|2]), (314,315,[4_1|2]), (315,316,[4_1|2]), (316,317,[4_1|2]), (317,318,[0_1|2]), (318,319,[3_1|2]), (319,320,[0_1|2]), (320,321,[1_1|2]), (321,322,[0_1|2]), (322,323,[0_1|2]), (323,152,[0_1|2]), (323,259,[0_1|2]), (323,429,[0_1|2]), (323,462,[0_1|2]), (323,416,[0_1|2]), (323,153,[2_1|2]), (323,156,[1_1|2]), (323,160,[4_1|2]), (323,178,[0_1|2]), (323,189,[4_1|2]), (324,325,[5_1|2]), (325,326,[0_1|2]), (326,327,[3_1|2]), (327,328,[4_1|2]), (328,329,[1_1|2]), (329,330,[5_1|2]), (330,331,[3_1|2]), (331,332,[3_1|2]), (332,333,[5_1|2]), (333,334,[1_1|2]), (334,335,[0_1|2]), (335,336,[0_1|2]), (336,337,[4_1|2]), (337,338,[3_1|2]), (338,339,[4_1|2]), (339,340,[2_1|2]), (340,341,[3_1|2]), (341,342,[0_1|2]), (341,189,[4_1|2]), (342,152,[4_1|2]), (342,160,[4_1|2]), (342,189,[4_1|2]), (342,201,[4_1|2]), (342,206,[4_1|2]), (342,229,[4_1|2]), (342,267,[4_1|2]), (342,284,[4_1|2]), (342,306,[4_1|2]), (342,447,[4_1|2]), (342,213,[1_1|2]), (342,240,[2_1|2]), (342,259,[5_1|2]), (343,344,[5_1|2]), (344,345,[5_1|2]), (345,346,[3_1|2]), (346,347,[1_1|2]), (347,348,[3_1|2]), (348,349,[1_1|2]), (349,152,[0_1|2]), (349,259,[0_1|2]), (349,429,[0_1|2]), (349,462,[0_1|2]), (349,325,[0_1|2]), (349,446,[0_1|2]), (349,153,[2_1|2]), (349,156,[1_1|2]), (349,160,[4_1|2]), (349,178,[0_1|2]), (349,189,[4_1|2]), (350,351,[0_1|2]), (351,352,[2_1|2]), (352,353,[4_1|2]), (353,354,[5_1|2]), (354,355,[0_1|2]), (355,356,[0_1|2]), (356,357,[5_1|2]), (357,358,[1_1|2]), (358,359,[3_1|2]), (359,360,[0_1|2]), (360,361,[5_1|2]), (361,362,[0_1|2]), (362,363,[2_1|2]), (363,364,[1_1|2]), (364,365,[4_1|2]), (365,366,[3_1|2]), (366,367,[3_1|2]), (367,368,[3_1|2]), (368,152,[0_1|2]), (368,156,[0_1|2, 1_1|2]), (368,213,[0_1|2]), (368,301,[0_1|2]), (368,324,[0_1|2]), (368,445,[0_1|2]), (368,154,[0_1|2]), (368,153,[2_1|2]), (368,160,[4_1|2]), (368,178,[0_1|2]), (368,189,[4_1|2]), (369,370,[2_1|2]), (370,371,[4_1|2]), (371,372,[1_1|2]), (372,373,[4_1|2]), (373,374,[5_1|2]), (374,375,[3_1|2]), (375,376,[2_1|2]), (376,377,[0_1|2]), (377,378,[1_1|2]), (378,379,[0_1|2]), (379,380,[4_1|2]), (380,152,[2_1|2]), (380,153,[2_1|2]), (380,240,[2_1|2]), (380,343,[2_1|2]), (380,350,[2_1|2]), (380,414,[2_1|2]), (381,382,[2_1|2]), (382,383,[0_1|2]), (383,384,[1_1|2]), (384,385,[2_1|2]), (385,386,[3_1|2]), (386,387,[0_1|2]), (387,388,[4_1|2]), (388,389,[4_1|2]), (389,390,[2_1|2]), (390,391,[4_1|2]), (391,392,[4_1|2]), (392,393,[3_1|2]), (393,394,[0_1|2]), (394,395,[0_1|2]), (395,396,[5_1|2]), (396,397,[4_1|2]), (397,398,[4_1|2]), (397,201,[4_1|2]), (397,206,[4_1|2]), (397,213,[1_1|2]), (397,229,[4_1|2]), (397,240,[2_1|2]), (398,152,[3_1|2]), (398,156,[3_1|2]), (398,213,[3_1|2]), (398,301,[3_1|2]), (398,324,[3_1|2]), (398,445,[3_1|2]), (398,430,[3_1|2]), (398,369,[3_1|2]), (398,381,[3_1|2]), (399,400,[1_1|2]), (400,401,[5_1|2]), (401,402,[0_1|2]), (402,403,[4_1|2]), (403,404,[0_1|2]), (404,405,[2_1|2]), (405,406,[1_1|2]), (406,407,[4_1|2]), (407,408,[4_1|2]), (408,409,[2_1|2]), (409,410,[5_1|2]), (410,411,[5_1|2]), (411,412,[3_1|2]), (412,413,[0_1|2]), (412,189,[4_1|2]), (413,152,[4_1|2]), (413,259,[4_1|2, 5_1|2]), (413,429,[4_1|2]), (413,462,[4_1|2]), (413,325,[4_1|2]), (413,446,[4_1|2]), (413,201,[4_1|2]), (413,206,[4_1|2]), (413,213,[1_1|2]), (413,229,[4_1|2]), (413,240,[2_1|2]), (413,267,[4_1|2]), (413,284,[4_1|2]), (414,415,[3_1|2]), (415,416,[5_1|2]), (416,417,[0_1|2]), (417,418,[0_1|2]), (418,419,[0_1|2]), (419,420,[3_1|2]), (420,421,[2_1|2]), (421,422,[1_1|2]), (422,423,[0_1|2]), (423,424,[3_1|2]), (424,425,[0_1|2]), (425,426,[4_1|2]), (426,427,[3_1|2]), (427,428,[3_1|2]), (427,369,[3_1|2]), (427,381,[3_1|2]), (428,152,[2_1|2]), (428,153,[2_1|2]), (428,240,[2_1|2]), (428,343,[2_1|2]), (428,350,[2_1|2]), (428,414,[2_1|2]), (428,370,[2_1|2]), (428,382,[2_1|2]), (429,430,[1_1|2]), (430,431,[5_1|2]), (431,432,[3_1|2]), (432,433,[0_1|2]), (433,434,[1_1|2]), (434,435,[2_1|2]), (435,436,[3_1|2]), (436,437,[2_1|2]), (437,438,[4_1|2]), (438,439,[5_1|2]), (439,440,[2_1|2]), (440,441,[0_1|2]), (441,442,[5_1|2]), (442,443,[3_1|2]), (443,444,[3_1|2]), (444,152,[0_1|2]), (444,178,[0_1|2]), (444,399,[0_1|2]), (444,153,[2_1|2]), (444,156,[1_1|2]), (444,160,[4_1|2]), (444,189,[4_1|2]), (445,446,[5_1|2]), (446,447,[4_1|2]), (447,448,[2_1|2]), (448,449,[2_1|2]), (449,450,[2_1|2]), (450,451,[2_1|2]), (451,452,[4_1|2]), (452,453,[2_1|2]), (453,454,[4_1|2]), (454,455,[5_1|2]), (455,456,[1_1|2]), (456,457,[2_1|2]), (457,458,[4_1|2]), (458,459,[3_1|2]), (459,460,[1_1|2]), (460,461,[3_1|2]), (460,369,[3_1|2]), (460,381,[3_1|2]), (461,152,[2_1|2]), (461,153,[2_1|2]), (461,240,[2_1|2]), (461,343,[2_1|2]), (461,350,[2_1|2]), (461,414,[2_1|2]), (462,463,[0_1|2]), (463,464,[1_1|2]), (464,465,[5_1|2]), (465,466,[0_1|2]), (466,467,[5_1|2]), (467,468,[5_1|2]), (468,469,[5_1|2]), (469,470,[3_1|2]), (470,471,[0_1|2]), (471,472,[0_1|2]), (472,473,[0_1|2]), (473,474,[1_1|2]), (474,475,[2_1|2]), (475,476,[0_1|2]), (476,477,[3_1|2]), (477,478,[1_1|2]), (478,479,[1_1|2]), (479,480,[1_1|2]), (480,481,[2_1|2]), (480,343,[2_1|2]), (481,152,[4_1|2]), (481,160,[4_1|2]), (481,189,[4_1|2]), (481,201,[4_1|2]), (481,206,[4_1|2]), (481,229,[4_1|2]), (481,267,[4_1|2]), (481,284,[4_1|2]), (481,306,[4_1|2]), (481,213,[1_1|2]), (481,240,[2_1|2]), (481,259,[5_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)