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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 43 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 180 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(x1))) -> 2(3(3(3(0(2(2(4(4(2(x1)))))))))) 2(5(0(x1))) -> 4(2(2(3(4(3(0(2(3(0(x1)))))))))) 0(1(0(3(x1)))) -> 5(2(2(2(0(0(3(2(0(3(x1)))))))))) 0(1(3(3(x1)))) -> 4(5(0(2(2(2(0(2(3(3(x1)))))))))) 0(1(2(4(5(x1))))) -> 4(2(2(0(3(1(3(3(4(2(x1)))))))))) 0(1(5(5(1(x1))))) -> 1(2(2(4(4(2(0(3(2(5(x1)))))))))) 1(2(5(1(5(x1))))) -> 2(3(5(0(2(2(3(4(4(4(x1)))))))))) 2(0(5(1(1(x1))))) -> 2(2(0(3(3(4(3(0(4(1(x1)))))))))) 2(0(5(4(0(x1))))) -> 2(0(0(0(0(4(0(4(5(0(x1)))))))))) 4(0(1(3(0(x1))))) -> 4(2(4(0(2(3(4(2(3(0(x1)))))))))) 5(0(0(5(2(x1))))) -> 0(5(2(0(3(0(4(0(2(3(x1)))))))))) 5(0(1(1(5(x1))))) -> 5(5(2(0(3(1(4(4(1(3(x1)))))))))) 0(0(1(5(4(1(x1)))))) -> 0(3(1(4(3(3(3(0(0(1(x1)))))))))) 0(1(1(1(0(4(x1)))))) -> 2(1(5(4(3(0(3(5(5(5(x1)))))))))) 0(1(2(4(1(5(x1)))))) -> 4(3(5(2(3(2(5(5(1(3(x1)))))))))) 0(1(5(0(1(5(x1)))))) -> 0(0(3(0(1(4(0(5(2(3(x1)))))))))) 0(5(0(1(4(5(x1)))))) -> 3(4(5(0(3(1(3(5(4(5(x1)))))))))) 1(2(0(0(5(5(x1)))))) -> 4(0(4(0(3(4(1(5(4(5(x1)))))))))) 1(2(0(2(4(1(x1)))))) -> 1(4(3(1(2(3(4(4(2(5(x1)))))))))) 2(2(5(0(0(5(x1)))))) -> 2(3(0(5(2(2(2(3(4(1(x1)))))))))) 2(5(2(5(0(4(x1)))))) -> 5(3(5(3(0(4(2(4(0(4(x1)))))))))) 3(2(1(5(1(3(x1)))))) -> 0(2(4(4(0(0(4(3(0(3(x1)))))))))) 0(0(0(1(2(1(5(x1))))))) -> 4(0(0(5(3(1(4(1(4(5(x1)))))))))) 0(0(0(5(1(4(4(x1))))))) -> 4(2(3(0(4(4(1(1(3(0(x1)))))))))) 0(1(0(0(5(0(5(x1))))))) -> 4(1(4(2(4(2(4(2(1(2(x1)))))))))) 0(1(0(2(5(0(5(x1))))))) -> 0(0(4(3(2(5(3(5(4(4(x1)))))))))) 1(0(1(5(1(1(0(x1))))))) -> 5(2(2(1(2(4(4(3(1(0(x1)))))))))) 1(0(5(0(5(0(5(x1))))))) -> 1(2(2(3(0(3(3(3(1(2(x1)))))))))) 1(1(0(1(1(2(4(x1))))))) -> 3(5(2(5(0(0(3(4(4(4(x1)))))))))) 1(2(0(5(0(1(3(x1))))))) -> 5(5(1(3(1(2(1(4(2(3(x1)))))))))) 1(3(2(0(1(2(2(x1))))))) -> 3(3(4(4(0(1(2(3(2(2(x1)))))))))) 1(4(0(0(5(0(5(x1))))))) -> 4(4(4(5(5(4(3(0(5(2(x1)))))))))) 1(4(0(5(1(0(4(x1))))))) -> 1(4(3(5(4(5(1(3(3(0(x1)))))))))) 1(5(0(0(1(4(4(x1))))))) -> 4(5(4(4(3(3(4(3(4(5(x1)))))))))) 2(0(1(0(2(4(1(x1))))))) -> 3(0(4(5(1(0(3(4(2(5(x1)))))))))) 2(0(1(1(5(1(3(x1))))))) -> 3(0(4(2(1(0(3(3(5(5(x1)))))))))) 2(0(1(3(2(4(3(x1))))))) -> 4(0(0(4(4(3(5(0(5(2(x1)))))))))) 2(1(2(0(0(5(5(x1))))))) -> 4(3(0(1(2(2(3(1(2(5(x1)))))))))) 2(2(5(5(0(0(5(x1))))))) -> 5(3(4(3(1(2(0(3(0(1(x1)))))))))) 2(5(0(5(2(4(4(x1))))))) -> 4(4(0(4(2(3(1(0(0(3(x1)))))))))) 2(5(1(1(2(4(2(x1))))))) -> 0(3(1(2(0(3(2(4(2(2(x1)))))))))) 2(5(4(2(1(0(2(x1))))))) -> 2(4(0(3(3(5(1(4(0(2(x1)))))))))) 3(5(1(1(0(1(5(x1))))))) -> 0(3(4(1(4(0(2(5(3(1(x1)))))))))) 4(1(4(0(4(1(1(x1))))))) -> 0(0(3(0(2(1(4(5(2(1(x1)))))))))) 5(0(0(1(3(5(0(x1))))))) -> 1(3(4(0(3(1(2(3(1(0(x1)))))))))) 5(1(1(0(0(1(4(x1))))))) -> 5(1(1(4(2(2(4(4(2(2(x1)))))))))) 5(1(5(0(5(1(5(x1))))))) -> 0(3(5(2(4(0(4(1(3(4(x1)))))))))) 5(3(2(0(1(0(0(x1))))))) -> 2(3(2(3(2(5(1(4(2(2(x1)))))))))) 5(4(2(5(0(4(5(x1))))))) -> 5(2(3(5(3(0(4(2(3(1(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_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)) encArg(cons_5(x_1)) -> 5(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(x1))) -> 2(3(3(3(0(2(2(4(4(2(x1)))))))))) 2(5(0(x1))) -> 4(2(2(3(4(3(0(2(3(0(x1)))))))))) 0(1(0(3(x1)))) -> 5(2(2(2(0(0(3(2(0(3(x1)))))))))) 0(1(3(3(x1)))) -> 4(5(0(2(2(2(0(2(3(3(x1)))))))))) 0(1(2(4(5(x1))))) -> 4(2(2(0(3(1(3(3(4(2(x1)))))))))) 0(1(5(5(1(x1))))) -> 1(2(2(4(4(2(0(3(2(5(x1)))))))))) 1(2(5(1(5(x1))))) -> 2(3(5(0(2(2(3(4(4(4(x1)))))))))) 2(0(5(1(1(x1))))) -> 2(2(0(3(3(4(3(0(4(1(x1)))))))))) 2(0(5(4(0(x1))))) -> 2(0(0(0(0(4(0(4(5(0(x1)))))))))) 4(0(1(3(0(x1))))) -> 4(2(4(0(2(3(4(2(3(0(x1)))))))))) 5(0(0(5(2(x1))))) -> 0(5(2(0(3(0(4(0(2(3(x1)))))))))) 5(0(1(1(5(x1))))) -> 5(5(2(0(3(1(4(4(1(3(x1)))))))))) 0(0(1(5(4(1(x1)))))) -> 0(3(1(4(3(3(3(0(0(1(x1)))))))))) 0(1(1(1(0(4(x1)))))) -> 2(1(5(4(3(0(3(5(5(5(x1)))))))))) 0(1(2(4(1(5(x1)))))) -> 4(3(5(2(3(2(5(5(1(3(x1)))))))))) 0(1(5(0(1(5(x1)))))) -> 0(0(3(0(1(4(0(5(2(3(x1)))))))))) 0(5(0(1(4(5(x1)))))) -> 3(4(5(0(3(1(3(5(4(5(x1)))))))))) 1(2(0(0(5(5(x1)))))) -> 4(0(4(0(3(4(1(5(4(5(x1)))))))))) 1(2(0(2(4(1(x1)))))) -> 1(4(3(1(2(3(4(4(2(5(x1)))))))))) 2(2(5(0(0(5(x1)))))) -> 2(3(0(5(2(2(2(3(4(1(x1)))))))))) 2(5(2(5(0(4(x1)))))) -> 5(3(5(3(0(4(2(4(0(4(x1)))))))))) 3(2(1(5(1(3(x1)))))) -> 0(2(4(4(0(0(4(3(0(3(x1)))))))))) 0(0(0(1(2(1(5(x1))))))) -> 4(0(0(5(3(1(4(1(4(5(x1)))))))))) 0(0(0(5(1(4(4(x1))))))) -> 4(2(3(0(4(4(1(1(3(0(x1)))))))))) 0(1(0(0(5(0(5(x1))))))) -> 4(1(4(2(4(2(4(2(1(2(x1)))))))))) 0(1(0(2(5(0(5(x1))))))) -> 0(0(4(3(2(5(3(5(4(4(x1)))))))))) 1(0(1(5(1(1(0(x1))))))) -> 5(2(2(1(2(4(4(3(1(0(x1)))))))))) 1(0(5(0(5(0(5(x1))))))) -> 1(2(2(3(0(3(3(3(1(2(x1)))))))))) 1(1(0(1(1(2(4(x1))))))) -> 3(5(2(5(0(0(3(4(4(4(x1)))))))))) 1(2(0(5(0(1(3(x1))))))) -> 5(5(1(3(1(2(1(4(2(3(x1)))))))))) 1(3(2(0(1(2(2(x1))))))) -> 3(3(4(4(0(1(2(3(2(2(x1)))))))))) 1(4(0(0(5(0(5(x1))))))) -> 4(4(4(5(5(4(3(0(5(2(x1)))))))))) 1(4(0(5(1(0(4(x1))))))) -> 1(4(3(5(4(5(1(3(3(0(x1)))))))))) 1(5(0(0(1(4(4(x1))))))) -> 4(5(4(4(3(3(4(3(4(5(x1)))))))))) 2(0(1(0(2(4(1(x1))))))) -> 3(0(4(5(1(0(3(4(2(5(x1)))))))))) 2(0(1(1(5(1(3(x1))))))) -> 3(0(4(2(1(0(3(3(5(5(x1)))))))))) 2(0(1(3(2(4(3(x1))))))) -> 4(0(0(4(4(3(5(0(5(2(x1)))))))))) 2(1(2(0(0(5(5(x1))))))) -> 4(3(0(1(2(2(3(1(2(5(x1)))))))))) 2(2(5(5(0(0(5(x1))))))) -> 5(3(4(3(1(2(0(3(0(1(x1)))))))))) 2(5(0(5(2(4(4(x1))))))) -> 4(4(0(4(2(3(1(0(0(3(x1)))))))))) 2(5(1(1(2(4(2(x1))))))) -> 0(3(1(2(0(3(2(4(2(2(x1)))))))))) 2(5(4(2(1(0(2(x1))))))) -> 2(4(0(3(3(5(1(4(0(2(x1)))))))))) 3(5(1(1(0(1(5(x1))))))) -> 0(3(4(1(4(0(2(5(3(1(x1)))))))))) 4(1(4(0(4(1(1(x1))))))) -> 0(0(3(0(2(1(4(5(2(1(x1)))))))))) 5(0(0(1(3(5(0(x1))))))) -> 1(3(4(0(3(1(2(3(1(0(x1)))))))))) 5(1(1(0(0(1(4(x1))))))) -> 5(1(1(4(2(2(4(4(2(2(x1)))))))))) 5(1(5(0(5(1(5(x1))))))) -> 0(3(5(2(4(0(4(1(3(4(x1)))))))))) 5(3(2(0(1(0(0(x1))))))) -> 2(3(2(3(2(5(1(4(2(2(x1)))))))))) 5(4(2(5(0(4(5(x1))))))) -> 5(2(3(5(3(0(4(2(3(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_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_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(x1))) -> 2(3(3(3(0(2(2(4(4(2(x1)))))))))) 2(5(0(x1))) -> 4(2(2(3(4(3(0(2(3(0(x1)))))))))) 0(1(0(3(x1)))) -> 5(2(2(2(0(0(3(2(0(3(x1)))))))))) 0(1(3(3(x1)))) -> 4(5(0(2(2(2(0(2(3(3(x1)))))))))) 0(1(2(4(5(x1))))) -> 4(2(2(0(3(1(3(3(4(2(x1)))))))))) 0(1(5(5(1(x1))))) -> 1(2(2(4(4(2(0(3(2(5(x1)))))))))) 1(2(5(1(5(x1))))) -> 2(3(5(0(2(2(3(4(4(4(x1)))))))))) 2(0(5(1(1(x1))))) -> 2(2(0(3(3(4(3(0(4(1(x1)))))))))) 2(0(5(4(0(x1))))) -> 2(0(0(0(0(4(0(4(5(0(x1)))))))))) 4(0(1(3(0(x1))))) -> 4(2(4(0(2(3(4(2(3(0(x1)))))))))) 5(0(0(5(2(x1))))) -> 0(5(2(0(3(0(4(0(2(3(x1)))))))))) 5(0(1(1(5(x1))))) -> 5(5(2(0(3(1(4(4(1(3(x1)))))))))) 0(0(1(5(4(1(x1)))))) -> 0(3(1(4(3(3(3(0(0(1(x1)))))))))) 0(1(1(1(0(4(x1)))))) -> 2(1(5(4(3(0(3(5(5(5(x1)))))))))) 0(1(2(4(1(5(x1)))))) -> 4(3(5(2(3(2(5(5(1(3(x1)))))))))) 0(1(5(0(1(5(x1)))))) -> 0(0(3(0(1(4(0(5(2(3(x1)))))))))) 0(5(0(1(4(5(x1)))))) -> 3(4(5(0(3(1(3(5(4(5(x1)))))))))) 1(2(0(0(5(5(x1)))))) -> 4(0(4(0(3(4(1(5(4(5(x1)))))))))) 1(2(0(2(4(1(x1)))))) -> 1(4(3(1(2(3(4(4(2(5(x1)))))))))) 2(2(5(0(0(5(x1)))))) -> 2(3(0(5(2(2(2(3(4(1(x1)))))))))) 2(5(2(5(0(4(x1)))))) -> 5(3(5(3(0(4(2(4(0(4(x1)))))))))) 3(2(1(5(1(3(x1)))))) -> 0(2(4(4(0(0(4(3(0(3(x1)))))))))) 0(0(0(1(2(1(5(x1))))))) -> 4(0(0(5(3(1(4(1(4(5(x1)))))))))) 0(0(0(5(1(4(4(x1))))))) -> 4(2(3(0(4(4(1(1(3(0(x1)))))))))) 0(1(0(0(5(0(5(x1))))))) -> 4(1(4(2(4(2(4(2(1(2(x1)))))))))) 0(1(0(2(5(0(5(x1))))))) -> 0(0(4(3(2(5(3(5(4(4(x1)))))))))) 1(0(1(5(1(1(0(x1))))))) -> 5(2(2(1(2(4(4(3(1(0(x1)))))))))) 1(0(5(0(5(0(5(x1))))))) -> 1(2(2(3(0(3(3(3(1(2(x1)))))))))) 1(1(0(1(1(2(4(x1))))))) -> 3(5(2(5(0(0(3(4(4(4(x1)))))))))) 1(2(0(5(0(1(3(x1))))))) -> 5(5(1(3(1(2(1(4(2(3(x1)))))))))) 1(3(2(0(1(2(2(x1))))))) -> 3(3(4(4(0(1(2(3(2(2(x1)))))))))) 1(4(0(0(5(0(5(x1))))))) -> 4(4(4(5(5(4(3(0(5(2(x1)))))))))) 1(4(0(5(1(0(4(x1))))))) -> 1(4(3(5(4(5(1(3(3(0(x1)))))))))) 1(5(0(0(1(4(4(x1))))))) -> 4(5(4(4(3(3(4(3(4(5(x1)))))))))) 2(0(1(0(2(4(1(x1))))))) -> 3(0(4(5(1(0(3(4(2(5(x1)))))))))) 2(0(1(1(5(1(3(x1))))))) -> 3(0(4(2(1(0(3(3(5(5(x1)))))))))) 2(0(1(3(2(4(3(x1))))))) -> 4(0(0(4(4(3(5(0(5(2(x1)))))))))) 2(1(2(0(0(5(5(x1))))))) -> 4(3(0(1(2(2(3(1(2(5(x1)))))))))) 2(2(5(5(0(0(5(x1))))))) -> 5(3(4(3(1(2(0(3(0(1(x1)))))))))) 2(5(0(5(2(4(4(x1))))))) -> 4(4(0(4(2(3(1(0(0(3(x1)))))))))) 2(5(1(1(2(4(2(x1))))))) -> 0(3(1(2(0(3(2(4(2(2(x1)))))))))) 2(5(4(2(1(0(2(x1))))))) -> 2(4(0(3(3(5(1(4(0(2(x1)))))))))) 3(5(1(1(0(1(5(x1))))))) -> 0(3(4(1(4(0(2(5(3(1(x1)))))))))) 4(1(4(0(4(1(1(x1))))))) -> 0(0(3(0(2(1(4(5(2(1(x1)))))))))) 5(0(0(1(3(5(0(x1))))))) -> 1(3(4(0(3(1(2(3(1(0(x1)))))))))) 5(1(1(0(0(1(4(x1))))))) -> 5(1(1(4(2(2(4(4(2(2(x1)))))))))) 5(1(5(0(5(1(5(x1))))))) -> 0(3(5(2(4(0(4(1(3(4(x1)))))))))) 5(3(2(0(1(0(0(x1))))))) -> 2(3(2(3(2(5(1(4(2(2(x1)))))))))) 5(4(2(5(0(4(5(x1))))))) -> 5(2(3(5(3(0(4(2(3(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_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_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(x1))) -> 2(3(3(3(0(2(2(4(4(2(x1)))))))))) 2(5(0(x1))) -> 4(2(2(3(4(3(0(2(3(0(x1)))))))))) 0(1(0(3(x1)))) -> 5(2(2(2(0(0(3(2(0(3(x1)))))))))) 0(1(3(3(x1)))) -> 4(5(0(2(2(2(0(2(3(3(x1)))))))))) 0(1(2(4(5(x1))))) -> 4(2(2(0(3(1(3(3(4(2(x1)))))))))) 0(1(5(5(1(x1))))) -> 1(2(2(4(4(2(0(3(2(5(x1)))))))))) 1(2(5(1(5(x1))))) -> 2(3(5(0(2(2(3(4(4(4(x1)))))))))) 2(0(5(1(1(x1))))) -> 2(2(0(3(3(4(3(0(4(1(x1)))))))))) 2(0(5(4(0(x1))))) -> 2(0(0(0(0(4(0(4(5(0(x1)))))))))) 4(0(1(3(0(x1))))) -> 4(2(4(0(2(3(4(2(3(0(x1)))))))))) 5(0(0(5(2(x1))))) -> 0(5(2(0(3(0(4(0(2(3(x1)))))))))) 5(0(1(1(5(x1))))) -> 5(5(2(0(3(1(4(4(1(3(x1)))))))))) 0(0(1(5(4(1(x1)))))) -> 0(3(1(4(3(3(3(0(0(1(x1)))))))))) 0(1(1(1(0(4(x1)))))) -> 2(1(5(4(3(0(3(5(5(5(x1)))))))))) 0(1(2(4(1(5(x1)))))) -> 4(3(5(2(3(2(5(5(1(3(x1)))))))))) 0(1(5(0(1(5(x1)))))) -> 0(0(3(0(1(4(0(5(2(3(x1)))))))))) 0(5(0(1(4(5(x1)))))) -> 3(4(5(0(3(1(3(5(4(5(x1)))))))))) 1(2(0(0(5(5(x1)))))) -> 4(0(4(0(3(4(1(5(4(5(x1)))))))))) 1(2(0(2(4(1(x1)))))) -> 1(4(3(1(2(3(4(4(2(5(x1)))))))))) 2(2(5(0(0(5(x1)))))) -> 2(3(0(5(2(2(2(3(4(1(x1)))))))))) 2(5(2(5(0(4(x1)))))) -> 5(3(5(3(0(4(2(4(0(4(x1)))))))))) 3(2(1(5(1(3(x1)))))) -> 0(2(4(4(0(0(4(3(0(3(x1)))))))))) 0(0(0(1(2(1(5(x1))))))) -> 4(0(0(5(3(1(4(1(4(5(x1)))))))))) 0(0(0(5(1(4(4(x1))))))) -> 4(2(3(0(4(4(1(1(3(0(x1)))))))))) 0(1(0(0(5(0(5(x1))))))) -> 4(1(4(2(4(2(4(2(1(2(x1)))))))))) 0(1(0(2(5(0(5(x1))))))) -> 0(0(4(3(2(5(3(5(4(4(x1)))))))))) 1(0(1(5(1(1(0(x1))))))) -> 5(2(2(1(2(4(4(3(1(0(x1)))))))))) 1(0(5(0(5(0(5(x1))))))) -> 1(2(2(3(0(3(3(3(1(2(x1)))))))))) 1(1(0(1(1(2(4(x1))))))) -> 3(5(2(5(0(0(3(4(4(4(x1)))))))))) 1(2(0(5(0(1(3(x1))))))) -> 5(5(1(3(1(2(1(4(2(3(x1)))))))))) 1(3(2(0(1(2(2(x1))))))) -> 3(3(4(4(0(1(2(3(2(2(x1)))))))))) 1(4(0(0(5(0(5(x1))))))) -> 4(4(4(5(5(4(3(0(5(2(x1)))))))))) 1(4(0(5(1(0(4(x1))))))) -> 1(4(3(5(4(5(1(3(3(0(x1)))))))))) 1(5(0(0(1(4(4(x1))))))) -> 4(5(4(4(3(3(4(3(4(5(x1)))))))))) 2(0(1(0(2(4(1(x1))))))) -> 3(0(4(5(1(0(3(4(2(5(x1)))))))))) 2(0(1(1(5(1(3(x1))))))) -> 3(0(4(2(1(0(3(3(5(5(x1)))))))))) 2(0(1(3(2(4(3(x1))))))) -> 4(0(0(4(4(3(5(0(5(2(x1)))))))))) 2(1(2(0(0(5(5(x1))))))) -> 4(3(0(1(2(2(3(1(2(5(x1)))))))))) 2(2(5(5(0(0(5(x1))))))) -> 5(3(4(3(1(2(0(3(0(1(x1)))))))))) 2(5(0(5(2(4(4(x1))))))) -> 4(4(0(4(2(3(1(0(0(3(x1)))))))))) 2(5(1(1(2(4(2(x1))))))) -> 0(3(1(2(0(3(2(4(2(2(x1)))))))))) 2(5(4(2(1(0(2(x1))))))) -> 2(4(0(3(3(5(1(4(0(2(x1)))))))))) 3(5(1(1(0(1(5(x1))))))) -> 0(3(4(1(4(0(2(5(3(1(x1)))))))))) 4(1(4(0(4(1(1(x1))))))) -> 0(0(3(0(2(1(4(5(2(1(x1)))))))))) 5(0(0(1(3(5(0(x1))))))) -> 1(3(4(0(3(1(2(3(1(0(x1)))))))))) 5(1(1(0(0(1(4(x1))))))) -> 5(1(1(4(2(2(4(4(2(2(x1)))))))))) 5(1(5(0(5(1(5(x1))))))) -> 0(3(5(2(4(0(4(1(3(4(x1)))))))))) 5(3(2(0(1(0(0(x1))))))) -> 2(3(2(3(2(5(1(4(2(2(x1)))))))))) 5(4(2(5(0(4(5(x1))))))) -> 5(2(3(5(3(0(4(2(3(1(x1)))))))))) encArg(cons_0(x_1)) -> 0(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)) encArg(cons_5(x_1)) -> 5(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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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, 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, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638] {(96,97,[0_1|0, 2_1|0, 1_1|0, 4_1|0, 5_1|0, 3_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]), (96,98,[0_1|1, 2_1|1, 1_1|1, 4_1|1, 5_1|1, 3_1|1]), (96,99,[2_1|2]), (96,108,[4_1|2]), (96,117,[4_1|2]), (96,126,[5_1|2]), 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(458,405,[3_1|2]), (458,414,[4_1|2]), (458,432,[4_1|2]), (459,460,[5_1|2]), (460,461,[2_1|2]), (461,462,[0_1|2]), (462,463,[3_1|2]), (463,464,[0_1|2]), (464,465,[4_1|2]), (465,466,[0_1|2]), (466,467,[2_1|2]), (467,98,[3_1|2]), (467,99,[3_1|2]), (467,180,[3_1|2]), (467,261,[3_1|2]), (467,270,[3_1|2]), (467,279,[3_1|2]), (467,315,[3_1|2]), (467,342,[3_1|2]), (467,504,[3_1|2]), (467,127,[3_1|2]), (467,379,[3_1|2]), (467,514,[3_1|2]), (467,461,[3_1|2]), (467,522,[0_1|2]), (467,531,[0_1|2]), (468,469,[3_1|2]), (469,470,[4_1|2]), (470,471,[0_1|2]), (471,472,[3_1|2]), (472,473,[1_1|2]), (473,474,[2_1|2]), (474,475,[3_1|2]), (475,476,[1_1|2]), (475,378,[5_1|2]), (475,387,[1_1|2]), (476,98,[0_1|2]), (476,144,[0_1|2]), (476,171,[0_1|2]), (476,189,[0_1|2]), (476,252,[0_1|2]), (476,450,[0_1|2]), (476,459,[0_1|2]), (476,495,[0_1|2]), (476,522,[0_1|2]), (476,531,[0_1|2]), (476,99,[2_1|2]), (476,108,[4_1|2]), (476,117,[4_1|2]), (476,126,[5_1|2]), (476,135,[4_1|2]), (476,153,[4_1|2]), (476,162,[1_1|2]), (476,180,[2_1|2]), (476,198,[4_1|2]), (476,207,[4_1|2]), (476,216,[3_1|2]), (476,585,[2_1|3]), (477,478,[5_1|2]), (478,479,[2_1|2]), (479,480,[0_1|2]), (480,481,[3_1|2]), (481,482,[1_1|2]), (482,483,[4_1|2]), (483,484,[4_1|2]), (484,485,[1_1|2]), (484,405,[3_1|2]), (485,98,[3_1|2]), (485,126,[3_1|2]), (485,243,[3_1|2]), (485,324,[3_1|2]), (485,369,[3_1|2]), (485,378,[3_1|2]), (485,477,[3_1|2]), (485,486,[3_1|2]), (485,513,[3_1|2]), (485,522,[0_1|2]), (485,531,[0_1|2]), (486,487,[1_1|2]), (487,488,[1_1|2]), (488,489,[4_1|2]), (489,490,[2_1|2]), (490,491,[2_1|2]), (491,492,[4_1|2]), (492,493,[4_1|2]), (493,494,[2_1|2]), (493,315,[2_1|2]), (493,324,[5_1|2]), (494,98,[2_1|2]), (494,108,[2_1|2]), (494,117,[2_1|2]), (494,135,[2_1|2]), (494,153,[2_1|2]), (494,198,[2_1|2]), (494,207,[2_1|2]), (494,225,[2_1|2, 4_1|2]), (494,234,[2_1|2, 4_1|2]), (494,306,[2_1|2, 4_1|2]), (494,333,[2_1|2, 4_1|2]), (494,351,[2_1|2]), (494,414,[2_1|2]), (494,432,[2_1|2]), (494,441,[2_1|2]), (494,361,[2_1|2]), (494,424,[2_1|2]), (494,243,[5_1|2]), (494,252,[0_1|2]), (494,261,[2_1|2]), (494,270,[2_1|2]), (494,279,[2_1|2]), (494,288,[3_1|2]), (494,297,[3_1|2]), (494,315,[2_1|2]), (494,324,[5_1|2]), (494,630,[4_1|3]), (495,496,[3_1|2]), (496,497,[5_1|2]), (497,498,[2_1|2]), (498,499,[4_1|2]), (499,500,[0_1|2]), (500,501,[4_1|2]), (501,502,[1_1|2]), (502,503,[3_1|2]), (503,98,[4_1|2]), (503,126,[4_1|2]), (503,243,[4_1|2]), (503,324,[4_1|2]), (503,369,[4_1|2]), (503,378,[4_1|2]), (503,477,[4_1|2]), (503,486,[4_1|2]), (503,513,[4_1|2]), (503,441,[4_1|2]), (503,450,[0_1|2]), (504,505,[3_1|2]), (505,506,[2_1|2]), (506,507,[3_1|2]), (507,508,[2_1|2]), (508,509,[5_1|2]), (509,510,[1_1|2]), (510,511,[4_1|2]), (511,512,[2_1|2]), (511,315,[2_1|2]), (511,324,[5_1|2]), (512,98,[2_1|2]), (512,144,[2_1|2]), (512,171,[2_1|2]), (512,189,[2_1|2]), (512,252,[2_1|2, 0_1|2]), (512,450,[2_1|2]), (512,459,[2_1|2]), (512,495,[2_1|2]), (512,522,[2_1|2]), (512,531,[2_1|2]), (512,145,[2_1|2]), (512,172,[2_1|2]), (512,451,[2_1|2]), (512,225,[4_1|2]), (512,234,[4_1|2]), (512,243,[5_1|2]), (512,261,[2_1|2]), (512,270,[2_1|2]), (512,279,[2_1|2]), (512,288,[3_1|2]), (512,297,[3_1|2]), (512,306,[4_1|2]), (512,315,[2_1|2]), (512,324,[5_1|2]), (512,333,[4_1|2]), (513,514,[2_1|2]), (514,515,[3_1|2]), (515,516,[5_1|2]), (516,517,[3_1|2]), (517,518,[0_1|2]), (518,519,[4_1|2]), (519,520,[2_1|2]), (520,521,[3_1|2]), (521,98,[1_1|2]), (521,126,[1_1|2]), (521,243,[1_1|2]), (521,324,[1_1|2]), (521,369,[1_1|2, 5_1|2]), (521,378,[1_1|2, 5_1|2]), (521,477,[1_1|2]), (521,486,[1_1|2]), (521,513,[1_1|2]), (521,154,[1_1|2]), (521,433,[1_1|2]), (521,342,[2_1|2]), (521,351,[4_1|2]), (521,360,[1_1|2]), (521,387,[1_1|2]), (521,396,[3_1|2]), (521,405,[3_1|2]), (521,414,[4_1|2]), (521,423,[1_1|2]), (521,432,[4_1|2]), (522,523,[2_1|2]), (523,524,[4_1|2]), (524,525,[4_1|2]), (525,526,[0_1|2]), (526,527,[0_1|2]), (527,528,[4_1|2]), (528,529,[3_1|2]), (529,530,[0_1|2]), (530,98,[3_1|2]), (530,216,[3_1|2]), (530,288,[3_1|2]), (530,297,[3_1|2]), (530,396,[3_1|2]), (530,405,[3_1|2]), (530,469,[3_1|2]), (530,522,[0_1|2]), (530,531,[0_1|2]), (531,532,[3_1|2]), (532,533,[4_1|2]), (533,534,[1_1|2]), (534,535,[4_1|2]), (535,536,[0_1|2]), (536,537,[2_1|2]), (537,538,[5_1|2]), (538,539,[3_1|2]), (539,98,[1_1|2]), (539,126,[1_1|2]), (539,243,[1_1|2]), (539,324,[1_1|2]), (539,369,[1_1|2, 5_1|2]), (539,378,[1_1|2, 5_1|2]), (539,477,[1_1|2]), (539,486,[1_1|2]), (539,513,[1_1|2]), (539,342,[2_1|2]), (539,351,[4_1|2]), (539,360,[1_1|2]), (539,387,[1_1|2]), (539,396,[3_1|2]), (539,405,[3_1|2]), (539,414,[4_1|2]), (539,423,[1_1|2]), (539,432,[4_1|2]), (540,541,[2_1|3]), (541,542,[2_1|3]), (542,543,[3_1|3]), (543,544,[4_1|3]), (544,545,[3_1|3]), (545,546,[0_1|3]), (546,547,[2_1|3]), (547,548,[3_1|3]), (548,144,[0_1|3]), (548,171,[0_1|3]), (548,189,[0_1|3]), (548,252,[0_1|3]), (548,450,[0_1|3]), (548,459,[0_1|3]), (548,495,[0_1|3]), (548,522,[0_1|3]), (548,531,[0_1|3]), (549,550,[3_1|3]), (550,551,[3_1|3]), (551,552,[3_1|3]), (552,553,[0_1|3]), (553,554,[2_1|3]), (554,555,[2_1|3]), (555,556,[4_1|3]), (556,557,[4_1|3]), (557,99,[2_1|3]), (557,180,[2_1|3]), (557,261,[2_1|3]), (557,270,[2_1|3]), (557,279,[2_1|3]), (557,315,[2_1|3]), (557,342,[2_1|3]), (557,504,[2_1|3]), (557,163,[2_1|3]), (557,388,[2_1|3]), (557,127,[2_1|3]), (557,379,[2_1|3]), (557,514,[2_1|3]), (557,461,[2_1|3]), (558,559,[2_1|3]), (559,560,[2_1|3]), (560,561,[2_1|3]), (561,562,[0_1|3]), (562,563,[0_1|3]), (563,564,[3_1|3]), (564,565,[2_1|3]), (565,566,[0_1|3]), (566,190,[3_1|3]), (566,253,[3_1|3]), (566,496,[3_1|3]), (566,532,[3_1|3]), (567,568,[2_1|3]), (568,569,[2_1|3]), (569,570,[4_1|3]), (570,571,[4_1|3]), (571,572,[2_1|3]), (572,573,[0_1|3]), (573,574,[3_1|3]), (574,575,[2_1|3]), (575,371,[5_1|3]), (576,577,[5_1|3]), (577,578,[0_1|3]), (578,579,[2_1|3]), (579,580,[2_1|3]), (580,581,[2_1|3]), (581,582,[0_1|3]), (582,583,[2_1|3]), (583,584,[3_1|3]), (584,406,[3_1|3]), (585,586,[3_1|3]), (586,587,[3_1|3]), (587,588,[3_1|3]), (588,589,[0_1|3]), (589,590,[2_1|3]), (590,591,[2_1|3]), (591,592,[4_1|3]), (592,593,[4_1|3]), (593,163,[2_1|3]), (593,388,[2_1|3]), (593,568,[2_1|3]), (594,595,[5_1|3]), (595,596,[2_1|3]), (596,597,[0_1|3]), (597,598,[3_1|3]), (598,599,[0_1|3]), (599,600,[4_1|3]), (600,601,[0_1|3]), (601,602,[2_1|3]), (602,461,[3_1|3]), (603,604,[3_1|3]), (604,605,[3_1|3]), (605,606,[3_1|3]), (606,607,[0_1|3]), (607,608,[2_1|3]), (608,609,[2_1|3]), (609,610,[4_1|3]), (610,611,[4_1|3]), (611,337,[2_1|3]), (612,613,[2_1|3]), (613,614,[2_1|3]), (614,615,[3_1|3]), (615,616,[4_1|3]), (616,617,[3_1|3]), (617,618,[0_1|3]), (618,619,[2_1|3]), (619,620,[3_1|3]), (620,400,[0_1|3]), (621,622,[3_1|3]), (622,623,[3_1|3]), (623,624,[3_1|3]), (624,625,[0_1|3]), (625,626,[2_1|3]), (626,627,[2_1|3]), (627,628,[4_1|3]), (628,629,[4_1|3]), (629,411,[2_1|3]), (630,631,[2_1|3]), (631,632,[2_1|3]), (632,633,[3_1|3]), (633,634,[4_1|3]), (634,635,[3_1|3]), (635,636,[0_1|3]), (636,637,[2_1|3]), (637,638,[3_1|3]), (638,155,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)