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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 79 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 187 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))) -> 0(0(3(4(0(3(0(3(0(2(x1)))))))))) 0(5(1(4(4(x1))))) -> 0(5(1(5(0(0(1(0(3(2(x1)))))))))) 1(2(5(5(2(x1))))) -> 1(0(5(0(3(5(4(0(3(2(x1)))))))))) 2(5(5(1(3(x1))))) -> 2(5(0(1(1(5(5(1(5(5(x1)))))))))) 3(5(2(5(2(x1))))) -> 3(5(5(3(0(0(3(0(4(3(x1)))))))))) 4(3(1(2(5(x1))))) -> 4(3(0(2(5(0(3(4(3(1(x1)))))))))) 4(5(1(1(2(x1))))) -> 1(0(1(5(5(0(5(2(0(3(x1)))))))))) 5(1(1(0(2(x1))))) -> 5(5(3(1(1(0(0(4(2(2(x1)))))))))) 0(1(2(4(2(4(x1)))))) -> 0(5(3(4(1(0(0(3(2(0(x1)))))))))) 0(1(2(5(1(3(x1)))))) -> 0(4(0(3(0(2(1(5(5(0(x1)))))))))) 0(5(2(0(1(1(x1)))))) -> 0(5(5(5(1(0(3(0(1(1(x1)))))))))) 1(4(1(0(1(2(x1)))))) -> 2(5(2(0(2(0(0(0(3(3(x1)))))))))) 2(1(1(4(1(3(x1)))))) -> 4(1(1(5(5(1(5(2(1(5(x1)))))))))) 2(3(1(2(1(2(x1)))))) -> 1(3(3(3(0(2(0(0(3(4(x1)))))))))) 2(3(5(1(1(2(x1)))))) -> 2(3(4(3(2(2(0(4(0(3(x1)))))))))) 2(3(5(1(1(3(x1)))))) -> 1(0(3(4(0(4(0(1(0(1(x1)))))))))) 2(3(5(1(2(4(x1)))))) -> 1(2(1(5(5(0(3(3(0(0(x1)))))))))) 3(2(5(2(1(2(x1)))))) -> 4(3(1(0(0(4(2(0(4(3(x1)))))))))) 3(3(1(2(5(3(x1)))))) -> 0(4(2(2(2(2(3(3(1(3(x1)))))))))) 4(5(1(2(0(5(x1)))))) -> 1(3(4(0(3(5(5(3(0(3(x1)))))))))) 5(2(4(4(1(2(x1)))))) -> 5(2(3(3(0(0(3(2(2(3(x1)))))))))) 0(4(1(1(4(2(3(x1))))))) -> 0(1(0(0(0(1(2(0(4(3(x1)))))))))) 0(5(2(5(2(3(1(x1))))))) -> 0(0(1(0(2(5(5(1(2(1(x1)))))))))) 1(0(1(2(5(1(2(x1))))))) -> 1(5(2(3(5(1(1(0(0(3(x1)))))))))) 1(1(4(1(2(1(4(x1))))))) -> 1(0(2(3(0(5(0(4(3(0(x1)))))))))) 1(1(4(2(1(4(1(x1))))))) -> 1(3(0(0(5(4(1(5(3(1(x1)))))))))) 1(1(4(2(4(3(4(x1))))))) -> 5(5(4(3(0(0(4(0(0(2(x1)))))))))) 1(2(1(2(2(1(5(x1))))))) -> 5(4(1(0(3(4(3(3(1(3(x1)))))))))) 1(2(3(2(4(2(4(x1))))))) -> 5(5(0(0(3(3(5(0(4(4(x1)))))))))) 1(2(3(5(3(4(5(x1))))))) -> 1(0(5(3(5(1(5(2(0(1(x1)))))))))) 1(3(0(5(2(3(5(x1))))))) -> 1(0(0(4(2(3(5(3(2(5(x1)))))))))) 1(3(4(1(1(2(5(x1))))))) -> 1(5(3(4(2(2(2(0(3(1(x1)))))))))) 1(4(1(1(1(2(0(x1))))))) -> 1(5(0(5(1(5(2(5(2(0(x1)))))))))) 2(0(1(1(1(2(5(x1))))))) -> 5(2(0(3(4(0(0(4(2(5(x1)))))))))) 2(0(1(4(1(1(3(x1))))))) -> 2(0(2(0(0(4(5(0(4(5(x1)))))))))) 2(1(2(5(2(5(1(x1))))))) -> 2(4(5(3(3(5(3(1(0(1(x1)))))))))) 2(1(5(0(2(4(0(x1))))))) -> 2(5(3(2(5(0(3(1(5(0(x1)))))))))) 2(3(1(1(1(1(4(x1))))))) -> 2(0(3(3(1(4(3(5(0(5(x1)))))))))) 2(3(2(0(5(1(0(x1))))))) -> 4(3(5(0(3(3(0(3(5(0(x1)))))))))) 2(3(2(4(0(2(4(x1))))))) -> 4(4(5(4(3(0(3(3(5(4(x1)))))))))) 2(3(3(5(2(5(2(x1))))))) -> 1(0(0(4(2(2(0(5(1(2(x1)))))))))) 3(1(4(2(1(1(2(x1))))))) -> 0(1(2(2(3(2(4(0(0(4(x1)))))))))) 3(2(3(1(1(3(2(x1))))))) -> 0(3(1(2(0(0(2(0(0(3(x1)))))))))) 3(2(3(5(1(3(1(x1))))))) -> 3(1(4(3(0(5(5(5(5(1(x1)))))))))) 3(4(0(2(2(4(0(x1))))))) -> 0(4(5(3(0(3(0(5(3(0(x1)))))))))) 5(2(0(1(4(1(3(x1))))))) -> 5(4(0(0(4(1(4(5(5(4(x1)))))))))) 5(2(0(2(1(1(4(x1))))))) -> 5(1(3(3(3(5(3(0(3(0(x1)))))))))) 5(2(0(5(4(1(5(x1))))))) -> 1(1(5(2(1(5(5(1(1(5(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_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_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(2(x1))) -> 0(0(3(4(0(3(0(3(0(2(x1)))))))))) 0(5(1(4(4(x1))))) -> 0(5(1(5(0(0(1(0(3(2(x1)))))))))) 1(2(5(5(2(x1))))) -> 1(0(5(0(3(5(4(0(3(2(x1)))))))))) 2(5(5(1(3(x1))))) -> 2(5(0(1(1(5(5(1(5(5(x1)))))))))) 3(5(2(5(2(x1))))) -> 3(5(5(3(0(0(3(0(4(3(x1)))))))))) 4(3(1(2(5(x1))))) -> 4(3(0(2(5(0(3(4(3(1(x1)))))))))) 4(5(1(1(2(x1))))) -> 1(0(1(5(5(0(5(2(0(3(x1)))))))))) 5(1(1(0(2(x1))))) -> 5(5(3(1(1(0(0(4(2(2(x1)))))))))) 0(1(2(4(2(4(x1)))))) -> 0(5(3(4(1(0(0(3(2(0(x1)))))))))) 0(1(2(5(1(3(x1)))))) -> 0(4(0(3(0(2(1(5(5(0(x1)))))))))) 0(5(2(0(1(1(x1)))))) -> 0(5(5(5(1(0(3(0(1(1(x1)))))))))) 1(4(1(0(1(2(x1)))))) -> 2(5(2(0(2(0(0(0(3(3(x1)))))))))) 2(1(1(4(1(3(x1)))))) -> 4(1(1(5(5(1(5(2(1(5(x1)))))))))) 2(3(1(2(1(2(x1)))))) -> 1(3(3(3(0(2(0(0(3(4(x1)))))))))) 2(3(5(1(1(2(x1)))))) -> 2(3(4(3(2(2(0(4(0(3(x1)))))))))) 2(3(5(1(1(3(x1)))))) -> 1(0(3(4(0(4(0(1(0(1(x1)))))))))) 2(3(5(1(2(4(x1)))))) -> 1(2(1(5(5(0(3(3(0(0(x1)))))))))) 3(2(5(2(1(2(x1)))))) -> 4(3(1(0(0(4(2(0(4(3(x1)))))))))) 3(3(1(2(5(3(x1)))))) -> 0(4(2(2(2(2(3(3(1(3(x1)))))))))) 4(5(1(2(0(5(x1)))))) -> 1(3(4(0(3(5(5(3(0(3(x1)))))))))) 5(2(4(4(1(2(x1)))))) -> 5(2(3(3(0(0(3(2(2(3(x1)))))))))) 0(4(1(1(4(2(3(x1))))))) -> 0(1(0(0(0(1(2(0(4(3(x1)))))))))) 0(5(2(5(2(3(1(x1))))))) -> 0(0(1(0(2(5(5(1(2(1(x1)))))))))) 1(0(1(2(5(1(2(x1))))))) -> 1(5(2(3(5(1(1(0(0(3(x1)))))))))) 1(1(4(1(2(1(4(x1))))))) -> 1(0(2(3(0(5(0(4(3(0(x1)))))))))) 1(1(4(2(1(4(1(x1))))))) -> 1(3(0(0(5(4(1(5(3(1(x1)))))))))) 1(1(4(2(4(3(4(x1))))))) -> 5(5(4(3(0(0(4(0(0(2(x1)))))))))) 1(2(1(2(2(1(5(x1))))))) -> 5(4(1(0(3(4(3(3(1(3(x1)))))))))) 1(2(3(2(4(2(4(x1))))))) -> 5(5(0(0(3(3(5(0(4(4(x1)))))))))) 1(2(3(5(3(4(5(x1))))))) -> 1(0(5(3(5(1(5(2(0(1(x1)))))))))) 1(3(0(5(2(3(5(x1))))))) -> 1(0(0(4(2(3(5(3(2(5(x1)))))))))) 1(3(4(1(1(2(5(x1))))))) -> 1(5(3(4(2(2(2(0(3(1(x1)))))))))) 1(4(1(1(1(2(0(x1))))))) -> 1(5(0(5(1(5(2(5(2(0(x1)))))))))) 2(0(1(1(1(2(5(x1))))))) -> 5(2(0(3(4(0(0(4(2(5(x1)))))))))) 2(0(1(4(1(1(3(x1))))))) -> 2(0(2(0(0(4(5(0(4(5(x1)))))))))) 2(1(2(5(2(5(1(x1))))))) -> 2(4(5(3(3(5(3(1(0(1(x1)))))))))) 2(1(5(0(2(4(0(x1))))))) -> 2(5(3(2(5(0(3(1(5(0(x1)))))))))) 2(3(1(1(1(1(4(x1))))))) -> 2(0(3(3(1(4(3(5(0(5(x1)))))))))) 2(3(2(0(5(1(0(x1))))))) -> 4(3(5(0(3(3(0(3(5(0(x1)))))))))) 2(3(2(4(0(2(4(x1))))))) -> 4(4(5(4(3(0(3(3(5(4(x1)))))))))) 2(3(3(5(2(5(2(x1))))))) -> 1(0(0(4(2(2(0(5(1(2(x1)))))))))) 3(1(4(2(1(1(2(x1))))))) -> 0(1(2(2(3(2(4(0(0(4(x1)))))))))) 3(2(3(1(1(3(2(x1))))))) -> 0(3(1(2(0(0(2(0(0(3(x1)))))))))) 3(2(3(5(1(3(1(x1))))))) -> 3(1(4(3(0(5(5(5(5(1(x1)))))))))) 3(4(0(2(2(4(0(x1))))))) -> 0(4(5(3(0(3(0(5(3(0(x1)))))))))) 5(2(0(1(4(1(3(x1))))))) -> 5(4(0(0(4(1(4(5(5(4(x1)))))))))) 5(2(0(2(1(1(4(x1))))))) -> 5(1(3(3(3(5(3(0(3(0(x1)))))))))) 5(2(0(5(4(1(5(x1))))))) -> 1(1(5(2(1(5(5(1(1(5(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_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_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(2(x1))) -> 0(0(3(4(0(3(0(3(0(2(x1)))))))))) 0(5(1(4(4(x1))))) -> 0(5(1(5(0(0(1(0(3(2(x1)))))))))) 1(2(5(5(2(x1))))) -> 1(0(5(0(3(5(4(0(3(2(x1)))))))))) 2(5(5(1(3(x1))))) -> 2(5(0(1(1(5(5(1(5(5(x1)))))))))) 3(5(2(5(2(x1))))) -> 3(5(5(3(0(0(3(0(4(3(x1)))))))))) 4(3(1(2(5(x1))))) -> 4(3(0(2(5(0(3(4(3(1(x1)))))))))) 4(5(1(1(2(x1))))) -> 1(0(1(5(5(0(5(2(0(3(x1)))))))))) 5(1(1(0(2(x1))))) -> 5(5(3(1(1(0(0(4(2(2(x1)))))))))) 0(1(2(4(2(4(x1)))))) -> 0(5(3(4(1(0(0(3(2(0(x1)))))))))) 0(1(2(5(1(3(x1)))))) -> 0(4(0(3(0(2(1(5(5(0(x1)))))))))) 0(5(2(0(1(1(x1)))))) -> 0(5(5(5(1(0(3(0(1(1(x1)))))))))) 1(4(1(0(1(2(x1)))))) -> 2(5(2(0(2(0(0(0(3(3(x1)))))))))) 2(1(1(4(1(3(x1)))))) -> 4(1(1(5(5(1(5(2(1(5(x1)))))))))) 2(3(1(2(1(2(x1)))))) -> 1(3(3(3(0(2(0(0(3(4(x1)))))))))) 2(3(5(1(1(2(x1)))))) -> 2(3(4(3(2(2(0(4(0(3(x1)))))))))) 2(3(5(1(1(3(x1)))))) -> 1(0(3(4(0(4(0(1(0(1(x1)))))))))) 2(3(5(1(2(4(x1)))))) -> 1(2(1(5(5(0(3(3(0(0(x1)))))))))) 3(2(5(2(1(2(x1)))))) -> 4(3(1(0(0(4(2(0(4(3(x1)))))))))) 3(3(1(2(5(3(x1)))))) -> 0(4(2(2(2(2(3(3(1(3(x1)))))))))) 4(5(1(2(0(5(x1)))))) -> 1(3(4(0(3(5(5(3(0(3(x1)))))))))) 5(2(4(4(1(2(x1)))))) -> 5(2(3(3(0(0(3(2(2(3(x1)))))))))) 0(4(1(1(4(2(3(x1))))))) -> 0(1(0(0(0(1(2(0(4(3(x1)))))))))) 0(5(2(5(2(3(1(x1))))))) -> 0(0(1(0(2(5(5(1(2(1(x1)))))))))) 1(0(1(2(5(1(2(x1))))))) -> 1(5(2(3(5(1(1(0(0(3(x1)))))))))) 1(1(4(1(2(1(4(x1))))))) -> 1(0(2(3(0(5(0(4(3(0(x1)))))))))) 1(1(4(2(1(4(1(x1))))))) -> 1(3(0(0(5(4(1(5(3(1(x1)))))))))) 1(1(4(2(4(3(4(x1))))))) -> 5(5(4(3(0(0(4(0(0(2(x1)))))))))) 1(2(1(2(2(1(5(x1))))))) -> 5(4(1(0(3(4(3(3(1(3(x1)))))))))) 1(2(3(2(4(2(4(x1))))))) -> 5(5(0(0(3(3(5(0(4(4(x1)))))))))) 1(2(3(5(3(4(5(x1))))))) -> 1(0(5(3(5(1(5(2(0(1(x1)))))))))) 1(3(0(5(2(3(5(x1))))))) -> 1(0(0(4(2(3(5(3(2(5(x1)))))))))) 1(3(4(1(1(2(5(x1))))))) -> 1(5(3(4(2(2(2(0(3(1(x1)))))))))) 1(4(1(1(1(2(0(x1))))))) -> 1(5(0(5(1(5(2(5(2(0(x1)))))))))) 2(0(1(1(1(2(5(x1))))))) -> 5(2(0(3(4(0(0(4(2(5(x1)))))))))) 2(0(1(4(1(1(3(x1))))))) -> 2(0(2(0(0(4(5(0(4(5(x1)))))))))) 2(1(2(5(2(5(1(x1))))))) -> 2(4(5(3(3(5(3(1(0(1(x1)))))))))) 2(1(5(0(2(4(0(x1))))))) -> 2(5(3(2(5(0(3(1(5(0(x1)))))))))) 2(3(1(1(1(1(4(x1))))))) -> 2(0(3(3(1(4(3(5(0(5(x1)))))))))) 2(3(2(0(5(1(0(x1))))))) -> 4(3(5(0(3(3(0(3(5(0(x1)))))))))) 2(3(2(4(0(2(4(x1))))))) -> 4(4(5(4(3(0(3(3(5(4(x1)))))))))) 2(3(3(5(2(5(2(x1))))))) -> 1(0(0(4(2(2(0(5(1(2(x1)))))))))) 3(1(4(2(1(1(2(x1))))))) -> 0(1(2(2(3(2(4(0(0(4(x1)))))))))) 3(2(3(1(1(3(2(x1))))))) -> 0(3(1(2(0(0(2(0(0(3(x1)))))))))) 3(2(3(5(1(3(1(x1))))))) -> 3(1(4(3(0(5(5(5(5(1(x1)))))))))) 3(4(0(2(2(4(0(x1))))))) -> 0(4(5(3(0(3(0(5(3(0(x1)))))))))) 5(2(0(1(4(1(3(x1))))))) -> 5(4(0(0(4(1(4(5(5(4(x1)))))))))) 5(2(0(2(1(1(4(x1))))))) -> 5(1(3(3(3(5(3(0(3(0(x1)))))))))) 5(2(0(5(4(1(5(x1))))))) -> 1(1(5(2(1(5(5(1(1(5(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_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_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(2(x1))) -> 0(0(3(4(0(3(0(3(0(2(x1)))))))))) 0(5(1(4(4(x1))))) -> 0(5(1(5(0(0(1(0(3(2(x1)))))))))) 1(2(5(5(2(x1))))) -> 1(0(5(0(3(5(4(0(3(2(x1)))))))))) 2(5(5(1(3(x1))))) -> 2(5(0(1(1(5(5(1(5(5(x1)))))))))) 3(5(2(5(2(x1))))) -> 3(5(5(3(0(0(3(0(4(3(x1)))))))))) 4(3(1(2(5(x1))))) -> 4(3(0(2(5(0(3(4(3(1(x1)))))))))) 4(5(1(1(2(x1))))) -> 1(0(1(5(5(0(5(2(0(3(x1)))))))))) 5(1(1(0(2(x1))))) -> 5(5(3(1(1(0(0(4(2(2(x1)))))))))) 0(1(2(4(2(4(x1)))))) -> 0(5(3(4(1(0(0(3(2(0(x1)))))))))) 0(1(2(5(1(3(x1)))))) -> 0(4(0(3(0(2(1(5(5(0(x1)))))))))) 0(5(2(0(1(1(x1)))))) -> 0(5(5(5(1(0(3(0(1(1(x1)))))))))) 1(4(1(0(1(2(x1)))))) -> 2(5(2(0(2(0(0(0(3(3(x1)))))))))) 2(1(1(4(1(3(x1)))))) -> 4(1(1(5(5(1(5(2(1(5(x1)))))))))) 2(3(1(2(1(2(x1)))))) -> 1(3(3(3(0(2(0(0(3(4(x1)))))))))) 2(3(5(1(1(2(x1)))))) -> 2(3(4(3(2(2(0(4(0(3(x1)))))))))) 2(3(5(1(1(3(x1)))))) -> 1(0(3(4(0(4(0(1(0(1(x1)))))))))) 2(3(5(1(2(4(x1)))))) -> 1(2(1(5(5(0(3(3(0(0(x1)))))))))) 3(2(5(2(1(2(x1)))))) -> 4(3(1(0(0(4(2(0(4(3(x1)))))))))) 3(3(1(2(5(3(x1)))))) -> 0(4(2(2(2(2(3(3(1(3(x1)))))))))) 4(5(1(2(0(5(x1)))))) -> 1(3(4(0(3(5(5(3(0(3(x1)))))))))) 5(2(4(4(1(2(x1)))))) -> 5(2(3(3(0(0(3(2(2(3(x1)))))))))) 0(4(1(1(4(2(3(x1))))))) -> 0(1(0(0(0(1(2(0(4(3(x1)))))))))) 0(5(2(5(2(3(1(x1))))))) -> 0(0(1(0(2(5(5(1(2(1(x1)))))))))) 1(0(1(2(5(1(2(x1))))))) -> 1(5(2(3(5(1(1(0(0(3(x1)))))))))) 1(1(4(1(2(1(4(x1))))))) -> 1(0(2(3(0(5(0(4(3(0(x1)))))))))) 1(1(4(2(1(4(1(x1))))))) -> 1(3(0(0(5(4(1(5(3(1(x1)))))))))) 1(1(4(2(4(3(4(x1))))))) -> 5(5(4(3(0(0(4(0(0(2(x1)))))))))) 1(2(1(2(2(1(5(x1))))))) -> 5(4(1(0(3(4(3(3(1(3(x1)))))))))) 1(2(3(2(4(2(4(x1))))))) -> 5(5(0(0(3(3(5(0(4(4(x1)))))))))) 1(2(3(5(3(4(5(x1))))))) -> 1(0(5(3(5(1(5(2(0(1(x1)))))))))) 1(3(0(5(2(3(5(x1))))))) -> 1(0(0(4(2(3(5(3(2(5(x1)))))))))) 1(3(4(1(1(2(5(x1))))))) -> 1(5(3(4(2(2(2(0(3(1(x1)))))))))) 1(4(1(1(1(2(0(x1))))))) -> 1(5(0(5(1(5(2(5(2(0(x1)))))))))) 2(0(1(1(1(2(5(x1))))))) -> 5(2(0(3(4(0(0(4(2(5(x1)))))))))) 2(0(1(4(1(1(3(x1))))))) -> 2(0(2(0(0(4(5(0(4(5(x1)))))))))) 2(1(2(5(2(5(1(x1))))))) -> 2(4(5(3(3(5(3(1(0(1(x1)))))))))) 2(1(5(0(2(4(0(x1))))))) -> 2(5(3(2(5(0(3(1(5(0(x1)))))))))) 2(3(1(1(1(1(4(x1))))))) -> 2(0(3(3(1(4(3(5(0(5(x1)))))))))) 2(3(2(0(5(1(0(x1))))))) -> 4(3(5(0(3(3(0(3(5(0(x1)))))))))) 2(3(2(4(0(2(4(x1))))))) -> 4(4(5(4(3(0(3(3(5(4(x1)))))))))) 2(3(3(5(2(5(2(x1))))))) -> 1(0(0(4(2(2(0(5(1(2(x1)))))))))) 3(1(4(2(1(1(2(x1))))))) -> 0(1(2(2(3(2(4(0(0(4(x1)))))))))) 3(2(3(1(1(3(2(x1))))))) -> 0(3(1(2(0(0(2(0(0(3(x1)))))))))) 3(2(3(5(1(3(1(x1))))))) -> 3(1(4(3(0(5(5(5(5(1(x1)))))))))) 3(4(0(2(2(4(0(x1))))))) -> 0(4(5(3(0(3(0(5(3(0(x1)))))))))) 5(2(0(1(4(1(3(x1))))))) -> 5(4(0(0(4(1(4(5(5(4(x1)))))))))) 5(2(0(2(1(1(4(x1))))))) -> 5(1(3(3(3(5(3(0(3(0(x1)))))))))) 5(2(0(5(4(1(5(x1))))))) -> 1(1(5(2(1(5(5(1(1(5(x1)))))))))) encArg(cons_0(x_1)) -> 0(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_4(x_1)) -> 4(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 3. The certificate found is represented by the following graph. 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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, 639, 640, 641, 642, 643] {(137,138,[0_1|0, 1_1|0, 2_1|0, 3_1|0, 4_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]), (137,139,[0_1|1, 1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1]), (137,140,[0_1|2]), (137,149,[0_1|2]), (137,158,[0_1|2]), (137,167,[0_1|2]), (137,176,[0_1|2]), (137,185,[0_1|2]), (137,194,[0_1|2]), (137,203,[1_1|2]), (137,212,[5_1|2]), (137,221,[5_1|2]), (137,230,[1_1|2]), (137,239,[2_1|2]), (137,248,[1_1|2]), 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(526,150,[3_1|2]), (526,168,[3_1|2]), (526,177,[3_1|2]), (526,437,[3_1|2]), (526,446,[4_1|2]), (526,455,[0_1|2]), (526,464,[3_1|2]), (526,473,[0_1|2]), (526,482,[0_1|2]), (526,491,[0_1|2]), (526,572,[0_1|3]), (527,528,[5_1|2]), (528,529,[3_1|2]), (529,530,[1_1|2]), (530,531,[1_1|2]), (531,532,[0_1|2]), (532,533,[0_1|2]), (533,534,[4_1|2]), (534,535,[2_1|2]), (535,139,[2_1|2]), (535,239,[2_1|2]), (535,311,[2_1|2]), (535,329,[2_1|2]), (535,338,[2_1|2]), (535,356,[2_1|2]), (535,365,[2_1|2]), (535,428,[2_1|2]), (535,268,[2_1|2]), (535,320,[4_1|2]), (535,347,[1_1|2]), (535,374,[1_1|2]), (535,383,[1_1|2]), (535,392,[4_1|2]), (535,401,[4_1|2]), (535,410,[1_1|2]), (535,419,[5_1|2]), (536,537,[2_1|2]), (537,538,[3_1|2]), (538,539,[3_1|2]), (539,540,[0_1|2]), (540,541,[0_1|2]), (541,542,[3_1|2]), (542,543,[2_1|2]), (543,544,[2_1|2]), (543,347,[1_1|2]), (543,356,[2_1|2]), (543,365,[2_1|2]), (543,374,[1_1|2]), (543,383,[1_1|2]), (543,392,[4_1|2]), (543,401,[4_1|2]), (543,410,[1_1|2]), (544,139,[3_1|2]), (544,239,[3_1|2]), (544,311,[3_1|2]), (544,329,[3_1|2]), (544,338,[3_1|2]), (544,356,[3_1|2]), (544,365,[3_1|2]), (544,428,[3_1|2]), (544,384,[3_1|2]), (544,437,[3_1|2]), (544,446,[4_1|2]), (544,455,[0_1|2]), (544,464,[3_1|2]), (544,473,[0_1|2]), (544,482,[0_1|2]), (544,491,[0_1|2]), (544,572,[0_1|3]), (545,546,[4_1|2]), (546,547,[0_1|2]), (547,548,[0_1|2]), (548,549,[4_1|2]), (549,550,[1_1|2]), (550,551,[4_1|2]), (551,552,[5_1|2]), (552,553,[5_1|2]), (553,139,[4_1|2]), (553,437,[4_1|2]), (553,464,[4_1|2]), (553,276,[4_1|2]), (553,348,[4_1|2]), (553,519,[4_1|2]), (553,500,[4_1|2]), (553,509,[1_1|2]), (553,518,[1_1|2]), (554,555,[1_1|2]), (555,556,[3_1|2]), (556,557,[3_1|2]), (557,558,[3_1|2]), (558,559,[5_1|2]), (559,560,[3_1|2]), (560,561,[0_1|2]), (561,562,[3_1|2]), (562,139,[0_1|2]), (562,320,[0_1|2]), (562,392,[0_1|2]), (562,401,[0_1|2]), (562,446,[0_1|2]), (562,500,[0_1|2]), (562,140,[0_1|2]), (562,149,[0_1|2]), (562,158,[0_1|2]), (562,167,[0_1|2]), (562,176,[0_1|2]), (562,185,[0_1|2]), (562,194,[0_1|2]), (562,581,[0_1|3]), (563,564,[1_1|2]), (564,565,[5_1|2]), (565,566,[2_1|2]), (566,567,[1_1|2]), (567,568,[5_1|2]), (568,569,[5_1|2]), (569,570,[1_1|2]), (570,571,[1_1|2]), (571,139,[5_1|2]), (571,212,[5_1|2]), (571,221,[5_1|2]), (571,284,[5_1|2]), (571,419,[5_1|2]), (571,527,[5_1|2]), (571,536,[5_1|2]), (571,545,[5_1|2]), (571,554,[5_1|2]), (571,249,[5_1|2]), (571,258,[5_1|2]), (571,303,[5_1|2]), (571,563,[1_1|2]), (572,573,[0_1|3]), (573,574,[3_1|3]), (574,575,[4_1|3]), (575,576,[0_1|3]), (576,577,[3_1|3]), (577,578,[0_1|3]), (578,579,[3_1|3]), (579,580,[0_1|3]), (580,484,[2_1|3]), (581,582,[0_1|3]), (582,583,[3_1|3]), (583,584,[4_1|3]), (584,585,[0_1|3]), (585,586,[3_1|3]), (586,587,[0_1|3]), (587,588,[3_1|3]), (588,589,[0_1|3]), (589,384,[2_1|3]), (589,484,[2_1|3]), (590,591,[0_1|3]), (591,592,[3_1|3]), (592,593,[4_1|3]), (593,594,[0_1|3]), (594,595,[3_1|3]), (595,596,[0_1|3]), (596,597,[3_1|3]), (597,598,[0_1|3]), (598,200,[2_1|3]), (599,600,[0_1|3]), (600,601,[3_1|3]), (601,602,[4_1|3]), (602,603,[0_1|3]), (603,604,[3_1|3]), (604,605,[0_1|3]), (605,606,[3_1|3]), (606,607,[0_1|3]), (607,239,[2_1|3]), (607,311,[2_1|3]), (607,329,[2_1|3]), (607,338,[2_1|3]), (607,356,[2_1|3]), (607,365,[2_1|3]), (607,428,[2_1|3]), (607,420,[2_1|3]), (607,537,[2_1|3]), (607,384,[2_1|3]), (608,609,[5_1|3]), (609,610,[0_1|3]), (610,611,[1_1|3]), (611,612,[1_1|3]), (612,613,[5_1|3]), (613,614,[5_1|3]), (614,615,[1_1|3]), (615,616,[5_1|3]), (616,556,[5_1|3]), (617,618,[5_1|3]), (618,619,[3_1|3]), (619,620,[1_1|3]), (620,621,[1_1|3]), (621,622,[0_1|3]), (622,623,[0_1|3]), (623,624,[4_1|3]), (624,625,[2_1|3]), (625,268,[2_1|3]), (626,627,[3_1|3]), (627,628,[0_1|3]), (628,629,[2_1|3]), (629,630,[5_1|3]), (630,631,[0_1|3]), (631,632,[3_1|3]), (632,633,[4_1|3]), (633,634,[3_1|3]), (634,240,[1_1|3]), (634,312,[1_1|3]), (634,339,[1_1|3]), (635,636,[5_1|3]), (636,637,[1_1|3]), (637,638,[5_1|3]), (638,639,[0_1|3]), (639,640,[0_1|3]), (640,641,[1_1|3]), (641,642,[0_1|3]), (642,643,[3_1|3]), (643,402,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)