WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 84 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 122 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 3(0(0(x1))) -> 3(3(3(1(3(1(3(2(3(3(x1)))))))))) 0(0(1(4(x1)))) -> 5(2(3(5(5(2(5(1(1(4(x1)))))))))) 0(4(0(1(x1)))) -> 5(3(2(2(2(3(2(2(3(1(x1)))))))))) 4(0(0(3(x1)))) -> 5(3(3(4(2(3(5(5(2(3(x1)))))))))) 4(2(4(0(x1)))) -> 1(1(5(0(2(1(3(0(3(4(x1)))))))))) 5(0(0(4(x1)))) -> 5(2(3(1(1(0(5(4(4(4(x1)))))))))) 0(4(0(1(0(x1))))) -> 5(2(3(5(4(1(5(5(3(2(x1)))))))))) 1(0(0(1(0(x1))))) -> 1(0(4(4(5(2(2(3(3(2(x1)))))))))) 2(4(0(0(1(x1))))) -> 2(0(2(3(3(3(3(3(4(1(x1)))))))))) 3(0(4(0(5(x1))))) -> 3(4(1(1(3(4(0(5(2(2(x1)))))))))) 3(2(5(2(1(x1))))) -> 3(2(2(3(1(1(4(3(1(1(x1)))))))))) 5(0(0(4(1(x1))))) -> 1(3(1(1(2(2(3(0(0(3(x1)))))))))) 0(0(5(0(0(0(x1)))))) -> 0(3(4(5(4(4(4(0(4(4(x1)))))))))) 1(0(0(0(4(3(x1)))))) -> 1(5(5(2(2(4(3(2(3(3(x1)))))))))) 1(3(0(5(3(2(x1)))))) -> 1(2(3(0(3(3(1(5(2(2(x1)))))))))) 1(3(2(4(5(0(x1)))))) -> 1(1(1(0(3(3(1(5(2(0(x1)))))))))) 1(4(5(0(2(1(x1)))))) -> 1(2(1(1(2(1(3(5(5(1(x1)))))))))) 2(0(4(2(5(3(x1)))))) -> 2(1(4(2(3(1(5(5(4(1(x1)))))))))) 4(0(1(0(0(1(x1)))))) -> 4(2(4(1(3(4(1(5(5(1(x1)))))))))) 4(1(4(4(0(4(x1)))))) -> 4(1(3(3(1(5(0(2(4(4(x1)))))))))) 5(0(0(0(4(0(x1)))))) -> 5(0(5(3(4(1(1(0(3(4(x1)))))))))) 5(4(2(0(0(5(x1)))))) -> 5(4(5(1(1(0(5(2(1(4(x1)))))))))) 0(0(0(3(2(4(3(x1))))))) -> 5(2(2(5(5(4(0(1(2(3(x1)))))))))) 0(0(2(4(0(5(0(x1))))))) -> 5(2(2(4(1(0(4(4(3(4(x1)))))))))) 0(1(5(2(0(4(3(x1))))))) -> 0(1(1(1(3(2(0(3(4(3(x1)))))))))) 0(4(3(3(0(0(5(x1))))))) -> 2(5(4(1(3(3(3(0(0(5(x1)))))))))) 1(0(0(0(0(0(3(x1))))))) -> 5(2(5(1(1(2(5(1(0(1(x1)))))))))) 1(0(0(0(5(4(1(x1))))))) -> 1(2(3(1(0(2(4(3(4(1(x1)))))))))) 1(3(2(1(2(0(1(x1))))))) -> 1(4(2(1(3(1(1(5(3(1(x1)))))))))) 2(0(2(0(1(4(4(x1))))))) -> 2(0(2(3(5(4(3(0(1(3(x1)))))))))) 2(0(5(3(0(5(0(x1))))))) -> 2(5(5(1(2(1(1(4(4(0(x1)))))))))) 2(1(4(2(5(0(1(x1))))))) -> 2(4(0(2(3(5(5(4(5(3(x1)))))))))) 2(4(3(3(2(5(3(x1))))))) -> 2(4(4(2(3(2(2(4(4(5(x1)))))))))) 2(5(3(0(4(3(2(x1))))))) -> 2(2(5(3(0(3(3(0(3(0(x1)))))))))) 3(3(0(5(3(0(4(x1))))))) -> 3(3(4(1(0(5(5(3(1(4(x1)))))))))) 3(4(1(4(5(4(2(x1))))))) -> 3(5(3(3(1(1(3(1(1(2(x1)))))))))) 3(4(3(2(4(1(2(x1))))))) -> 3(5(1(1(1(5(3(1(3(0(x1)))))))))) 4(2(5(4(0(2(1(x1))))))) -> 1(1(1(1(5(5(1(4(5(1(x1)))))))))) 4(3(2(0(1(0(5(x1))))))) -> 4(3(1(3(3(0(1(0(3(4(x1)))))))))) 4(3(3(2(3(4(3(x1))))))) -> 4(0(3(0(3(3(3(5(4(4(x1)))))))))) 5(0(1(0(5(0(0(x1))))))) -> 5(0(0(1(1(3(4(3(0(0(x1)))))))))) 5(4(0(1(4(0(1(x1))))))) -> 4(1(4(5(3(5(1(2(4(1(x1)))))))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_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_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 3(0(0(x1))) -> 3(3(3(1(3(1(3(2(3(3(x1)))))))))) 0(0(1(4(x1)))) -> 5(2(3(5(5(2(5(1(1(4(x1)))))))))) 0(4(0(1(x1)))) -> 5(3(2(2(2(3(2(2(3(1(x1)))))))))) 4(0(0(3(x1)))) -> 5(3(3(4(2(3(5(5(2(3(x1)))))))))) 4(2(4(0(x1)))) -> 1(1(5(0(2(1(3(0(3(4(x1)))))))))) 5(0(0(4(x1)))) -> 5(2(3(1(1(0(5(4(4(4(x1)))))))))) 0(4(0(1(0(x1))))) -> 5(2(3(5(4(1(5(5(3(2(x1)))))))))) 1(0(0(1(0(x1))))) -> 1(0(4(4(5(2(2(3(3(2(x1)))))))))) 2(4(0(0(1(x1))))) -> 2(0(2(3(3(3(3(3(4(1(x1)))))))))) 3(0(4(0(5(x1))))) -> 3(4(1(1(3(4(0(5(2(2(x1)))))))))) 3(2(5(2(1(x1))))) -> 3(2(2(3(1(1(4(3(1(1(x1)))))))))) 5(0(0(4(1(x1))))) -> 1(3(1(1(2(2(3(0(0(3(x1)))))))))) 0(0(5(0(0(0(x1)))))) -> 0(3(4(5(4(4(4(0(4(4(x1)))))))))) 1(0(0(0(4(3(x1)))))) -> 1(5(5(2(2(4(3(2(3(3(x1)))))))))) 1(3(0(5(3(2(x1)))))) -> 1(2(3(0(3(3(1(5(2(2(x1)))))))))) 1(3(2(4(5(0(x1)))))) -> 1(1(1(0(3(3(1(5(2(0(x1)))))))))) 1(4(5(0(2(1(x1)))))) -> 1(2(1(1(2(1(3(5(5(1(x1)))))))))) 2(0(4(2(5(3(x1)))))) -> 2(1(4(2(3(1(5(5(4(1(x1)))))))))) 4(0(1(0(0(1(x1)))))) -> 4(2(4(1(3(4(1(5(5(1(x1)))))))))) 4(1(4(4(0(4(x1)))))) -> 4(1(3(3(1(5(0(2(4(4(x1)))))))))) 5(0(0(0(4(0(x1)))))) -> 5(0(5(3(4(1(1(0(3(4(x1)))))))))) 5(4(2(0(0(5(x1)))))) -> 5(4(5(1(1(0(5(2(1(4(x1)))))))))) 0(0(0(3(2(4(3(x1))))))) -> 5(2(2(5(5(4(0(1(2(3(x1)))))))))) 0(0(2(4(0(5(0(x1))))))) -> 5(2(2(4(1(0(4(4(3(4(x1)))))))))) 0(1(5(2(0(4(3(x1))))))) -> 0(1(1(1(3(2(0(3(4(3(x1)))))))))) 0(4(3(3(0(0(5(x1))))))) -> 2(5(4(1(3(3(3(0(0(5(x1)))))))))) 1(0(0(0(0(0(3(x1))))))) -> 5(2(5(1(1(2(5(1(0(1(x1)))))))))) 1(0(0(0(5(4(1(x1))))))) -> 1(2(3(1(0(2(4(3(4(1(x1)))))))))) 1(3(2(1(2(0(1(x1))))))) -> 1(4(2(1(3(1(1(5(3(1(x1)))))))))) 2(0(2(0(1(4(4(x1))))))) -> 2(0(2(3(5(4(3(0(1(3(x1)))))))))) 2(0(5(3(0(5(0(x1))))))) -> 2(5(5(1(2(1(1(4(4(0(x1)))))))))) 2(1(4(2(5(0(1(x1))))))) -> 2(4(0(2(3(5(5(4(5(3(x1)))))))))) 2(4(3(3(2(5(3(x1))))))) -> 2(4(4(2(3(2(2(4(4(5(x1)))))))))) 2(5(3(0(4(3(2(x1))))))) -> 2(2(5(3(0(3(3(0(3(0(x1)))))))))) 3(3(0(5(3(0(4(x1))))))) -> 3(3(4(1(0(5(5(3(1(4(x1)))))))))) 3(4(1(4(5(4(2(x1))))))) -> 3(5(3(3(1(1(3(1(1(2(x1)))))))))) 3(4(3(2(4(1(2(x1))))))) -> 3(5(1(1(1(5(3(1(3(0(x1)))))))))) 4(2(5(4(0(2(1(x1))))))) -> 1(1(1(1(5(5(1(4(5(1(x1)))))))))) 4(3(2(0(1(0(5(x1))))))) -> 4(3(1(3(3(0(1(0(3(4(x1)))))))))) 4(3(3(2(3(4(3(x1))))))) -> 4(0(3(0(3(3(3(5(4(4(x1)))))))))) 5(0(1(0(5(0(0(x1))))))) -> 5(0(0(1(1(3(4(3(0(0(x1)))))))))) 5(4(0(1(4(0(1(x1))))))) -> 4(1(4(5(3(5(1(2(4(1(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_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_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 3(0(0(x1))) -> 3(3(3(1(3(1(3(2(3(3(x1)))))))))) 0(0(1(4(x1)))) -> 5(2(3(5(5(2(5(1(1(4(x1)))))))))) 0(4(0(1(x1)))) -> 5(3(2(2(2(3(2(2(3(1(x1)))))))))) 4(0(0(3(x1)))) -> 5(3(3(4(2(3(5(5(2(3(x1)))))))))) 4(2(4(0(x1)))) -> 1(1(5(0(2(1(3(0(3(4(x1)))))))))) 5(0(0(4(x1)))) -> 5(2(3(1(1(0(5(4(4(4(x1)))))))))) 0(4(0(1(0(x1))))) -> 5(2(3(5(4(1(5(5(3(2(x1)))))))))) 1(0(0(1(0(x1))))) -> 1(0(4(4(5(2(2(3(3(2(x1)))))))))) 2(4(0(0(1(x1))))) -> 2(0(2(3(3(3(3(3(4(1(x1)))))))))) 3(0(4(0(5(x1))))) -> 3(4(1(1(3(4(0(5(2(2(x1)))))))))) 3(2(5(2(1(x1))))) -> 3(2(2(3(1(1(4(3(1(1(x1)))))))))) 5(0(0(4(1(x1))))) -> 1(3(1(1(2(2(3(0(0(3(x1)))))))))) 0(0(5(0(0(0(x1)))))) -> 0(3(4(5(4(4(4(0(4(4(x1)))))))))) 1(0(0(0(4(3(x1)))))) -> 1(5(5(2(2(4(3(2(3(3(x1)))))))))) 1(3(0(5(3(2(x1)))))) -> 1(2(3(0(3(3(1(5(2(2(x1)))))))))) 1(3(2(4(5(0(x1)))))) -> 1(1(1(0(3(3(1(5(2(0(x1)))))))))) 1(4(5(0(2(1(x1)))))) -> 1(2(1(1(2(1(3(5(5(1(x1)))))))))) 2(0(4(2(5(3(x1)))))) -> 2(1(4(2(3(1(5(5(4(1(x1)))))))))) 4(0(1(0(0(1(x1)))))) -> 4(2(4(1(3(4(1(5(5(1(x1)))))))))) 4(1(4(4(0(4(x1)))))) -> 4(1(3(3(1(5(0(2(4(4(x1)))))))))) 5(0(0(0(4(0(x1)))))) -> 5(0(5(3(4(1(1(0(3(4(x1)))))))))) 5(4(2(0(0(5(x1)))))) -> 5(4(5(1(1(0(5(2(1(4(x1)))))))))) 0(0(0(3(2(4(3(x1))))))) -> 5(2(2(5(5(4(0(1(2(3(x1)))))))))) 0(0(2(4(0(5(0(x1))))))) -> 5(2(2(4(1(0(4(4(3(4(x1)))))))))) 0(1(5(2(0(4(3(x1))))))) -> 0(1(1(1(3(2(0(3(4(3(x1)))))))))) 0(4(3(3(0(0(5(x1))))))) -> 2(5(4(1(3(3(3(0(0(5(x1)))))))))) 1(0(0(0(0(0(3(x1))))))) -> 5(2(5(1(1(2(5(1(0(1(x1)))))))))) 1(0(0(0(5(4(1(x1))))))) -> 1(2(3(1(0(2(4(3(4(1(x1)))))))))) 1(3(2(1(2(0(1(x1))))))) -> 1(4(2(1(3(1(1(5(3(1(x1)))))))))) 2(0(2(0(1(4(4(x1))))))) -> 2(0(2(3(5(4(3(0(1(3(x1)))))))))) 2(0(5(3(0(5(0(x1))))))) -> 2(5(5(1(2(1(1(4(4(0(x1)))))))))) 2(1(4(2(5(0(1(x1))))))) -> 2(4(0(2(3(5(5(4(5(3(x1)))))))))) 2(4(3(3(2(5(3(x1))))))) -> 2(4(4(2(3(2(2(4(4(5(x1)))))))))) 2(5(3(0(4(3(2(x1))))))) -> 2(2(5(3(0(3(3(0(3(0(x1)))))))))) 3(3(0(5(3(0(4(x1))))))) -> 3(3(4(1(0(5(5(3(1(4(x1)))))))))) 3(4(1(4(5(4(2(x1))))))) -> 3(5(3(3(1(1(3(1(1(2(x1)))))))))) 3(4(3(2(4(1(2(x1))))))) -> 3(5(1(1(1(5(3(1(3(0(x1)))))))))) 4(2(5(4(0(2(1(x1))))))) -> 1(1(1(1(5(5(1(4(5(1(x1)))))))))) 4(3(2(0(1(0(5(x1))))))) -> 4(3(1(3(3(0(1(0(3(4(x1)))))))))) 4(3(3(2(3(4(3(x1))))))) -> 4(0(3(0(3(3(3(5(4(4(x1)))))))))) 5(0(1(0(5(0(0(x1))))))) -> 5(0(0(1(1(3(4(3(0(0(x1)))))))))) 5(4(0(1(4(0(1(x1))))))) -> 4(1(4(5(3(5(1(2(4(1(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_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_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 3(0(0(x1))) -> 3(3(3(1(3(1(3(2(3(3(x1)))))))))) 0(0(1(4(x1)))) -> 5(2(3(5(5(2(5(1(1(4(x1)))))))))) 0(4(0(1(x1)))) -> 5(3(2(2(2(3(2(2(3(1(x1)))))))))) 4(0(0(3(x1)))) -> 5(3(3(4(2(3(5(5(2(3(x1)))))))))) 4(2(4(0(x1)))) -> 1(1(5(0(2(1(3(0(3(4(x1)))))))))) 5(0(0(4(x1)))) -> 5(2(3(1(1(0(5(4(4(4(x1)))))))))) 0(4(0(1(0(x1))))) -> 5(2(3(5(4(1(5(5(3(2(x1)))))))))) 1(0(0(1(0(x1))))) -> 1(0(4(4(5(2(2(3(3(2(x1)))))))))) 2(4(0(0(1(x1))))) -> 2(0(2(3(3(3(3(3(4(1(x1)))))))))) 3(0(4(0(5(x1))))) -> 3(4(1(1(3(4(0(5(2(2(x1)))))))))) 3(2(5(2(1(x1))))) -> 3(2(2(3(1(1(4(3(1(1(x1)))))))))) 5(0(0(4(1(x1))))) -> 1(3(1(1(2(2(3(0(0(3(x1)))))))))) 0(0(5(0(0(0(x1)))))) -> 0(3(4(5(4(4(4(0(4(4(x1)))))))))) 1(0(0(0(4(3(x1)))))) -> 1(5(5(2(2(4(3(2(3(3(x1)))))))))) 1(3(0(5(3(2(x1)))))) -> 1(2(3(0(3(3(1(5(2(2(x1)))))))))) 1(3(2(4(5(0(x1)))))) -> 1(1(1(0(3(3(1(5(2(0(x1)))))))))) 1(4(5(0(2(1(x1)))))) -> 1(2(1(1(2(1(3(5(5(1(x1)))))))))) 2(0(4(2(5(3(x1)))))) -> 2(1(4(2(3(1(5(5(4(1(x1)))))))))) 4(0(1(0(0(1(x1)))))) -> 4(2(4(1(3(4(1(5(5(1(x1)))))))))) 4(1(4(4(0(4(x1)))))) -> 4(1(3(3(1(5(0(2(4(4(x1)))))))))) 5(0(0(0(4(0(x1)))))) -> 5(0(5(3(4(1(1(0(3(4(x1)))))))))) 5(4(2(0(0(5(x1)))))) -> 5(4(5(1(1(0(5(2(1(4(x1)))))))))) 0(0(0(3(2(4(3(x1))))))) -> 5(2(2(5(5(4(0(1(2(3(x1)))))))))) 0(0(2(4(0(5(0(x1))))))) -> 5(2(2(4(1(0(4(4(3(4(x1)))))))))) 0(1(5(2(0(4(3(x1))))))) -> 0(1(1(1(3(2(0(3(4(3(x1)))))))))) 0(4(3(3(0(0(5(x1))))))) -> 2(5(4(1(3(3(3(0(0(5(x1)))))))))) 1(0(0(0(0(0(3(x1))))))) -> 5(2(5(1(1(2(5(1(0(1(x1)))))))))) 1(0(0(0(5(4(1(x1))))))) -> 1(2(3(1(0(2(4(3(4(1(x1)))))))))) 1(3(2(1(2(0(1(x1))))))) -> 1(4(2(1(3(1(1(5(3(1(x1)))))))))) 2(0(2(0(1(4(4(x1))))))) -> 2(0(2(3(5(4(3(0(1(3(x1)))))))))) 2(0(5(3(0(5(0(x1))))))) -> 2(5(5(1(2(1(1(4(4(0(x1)))))))))) 2(1(4(2(5(0(1(x1))))))) -> 2(4(0(2(3(5(5(4(5(3(x1)))))))))) 2(4(3(3(2(5(3(x1))))))) -> 2(4(4(2(3(2(2(4(4(5(x1)))))))))) 2(5(3(0(4(3(2(x1))))))) -> 2(2(5(3(0(3(3(0(3(0(x1)))))))))) 3(3(0(5(3(0(4(x1))))))) -> 3(3(4(1(0(5(5(3(1(4(x1)))))))))) 3(4(1(4(5(4(2(x1))))))) -> 3(5(3(3(1(1(3(1(1(2(x1)))))))))) 3(4(3(2(4(1(2(x1))))))) -> 3(5(1(1(1(5(3(1(3(0(x1)))))))))) 4(2(5(4(0(2(1(x1))))))) -> 1(1(1(1(5(5(1(4(5(1(x1)))))))))) 4(3(2(0(1(0(5(x1))))))) -> 4(3(1(3(3(0(1(0(3(4(x1)))))))))) 4(3(3(2(3(4(3(x1))))))) -> 4(0(3(0(3(3(3(5(4(4(x1)))))))))) 5(0(1(0(5(0(0(x1))))))) -> 5(0(0(1(1(3(4(3(0(0(x1)))))))))) 5(4(0(1(4(0(1(x1))))))) -> 4(1(4(5(3(5(1(2(4(1(x1)))))))))) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_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_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 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] {(148,149,[3_1|0, 0_1|0, 4_1|0, 5_1|0, 1_1|0, 2_1|0, encArg_1|0, encode_3_1|0, encode_0_1|0, encode_1_1|0, encode_2_1|0, encode_4_1|0, encode_5_1|0]), (148,150,[3_1|1, 0_1|1, 4_1|1, 5_1|1, 1_1|1, 2_1|1]), (148,151,[3_1|2]), (148,160,[3_1|2]), (148,169,[3_1|2]), (148,178,[3_1|2]), (148,187,[3_1|2]), (148,196,[3_1|2]), (148,205,[5_1|2]), (148,214,[0_1|2]), (148,223,[5_1|2]), (148,232,[5_1|2]), (148,241,[5_1|2]), (148,250,[5_1|2]), (148,259,[2_1|2]), (148,268,[0_1|2]), (148,277,[5_1|2]), (148,286,[4_1|2]), (148,295,[1_1|2]), (148,304,[1_1|2]), (148,313,[4_1|2]), (148,322,[4_1|2]), (148,331,[4_1|2]), (148,340,[5_1|2]), (148,349,[1_1|2]), (148,358,[5_1|2]), (148,367,[5_1|2]), (148,376,[5_1|2]), (148,385,[4_1|2]), 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(384,358,[4_1|2]), (384,367,[4_1|2]), (384,376,[4_1|2]), (384,412,[4_1|2]), (384,286,[4_1|2]), (384,295,[1_1|2]), (384,304,[1_1|2]), (384,313,[4_1|2]), (384,322,[4_1|2]), (384,331,[4_1|2]), (384,529,[1_1|3]), (385,386,[1_1|2]), (386,387,[4_1|2]), (387,388,[5_1|2]), (388,389,[3_1|2]), (389,390,[5_1|2]), (390,391,[1_1|2]), (391,392,[2_1|2]), (392,393,[4_1|2]), (392,313,[4_1|2]), (393,150,[1_1|2]), (393,295,[1_1|2]), (393,304,[1_1|2]), (393,349,[1_1|2]), (393,394,[1_1|2]), (393,403,[1_1|2]), (393,421,[1_1|2]), (393,430,[1_1|2]), (393,439,[1_1|2]), (393,448,[1_1|2]), (393,457,[1_1|2]), (393,269,[1_1|2]), (393,412,[5_1|2]), (394,395,[0_1|2]), (395,396,[4_1|2]), (396,397,[4_1|2]), (397,398,[5_1|2]), (398,399,[2_1|2]), (399,400,[2_1|2]), (400,401,[3_1|2]), (401,402,[3_1|2]), (401,169,[3_1|2]), (402,150,[2_1|2]), (402,214,[2_1|2]), (402,268,[2_1|2]), (402,395,[2_1|2]), (402,466,[2_1|2]), (402,475,[2_1|2]), (402,484,[2_1|2]), (402,493,[2_1|2]), (402,502,[2_1|2]), (402,511,[2_1|2]), (402,520,[2_1|2]), (403,404,[5_1|2]), (404,405,[5_1|2]), (405,406,[2_1|2]), (406,407,[2_1|2]), (407,408,[4_1|2]), (408,409,[3_1|2]), (409,410,[2_1|2]), (410,411,[3_1|2]), (410,178,[3_1|2]), (411,150,[3_1|2]), (411,151,[3_1|2]), (411,160,[3_1|2]), (411,169,[3_1|2]), (411,178,[3_1|2]), (411,187,[3_1|2]), (411,196,[3_1|2]), (411,323,[3_1|2]), (412,413,[2_1|2]), (413,414,[5_1|2]), (414,415,[1_1|2]), (415,416,[1_1|2]), (416,417,[2_1|2]), (417,418,[5_1|2]), (418,419,[1_1|2]), (419,420,[0_1|2]), (419,268,[0_1|2]), (420,150,[1_1|2]), (420,151,[1_1|2]), (420,160,[1_1|2]), (420,169,[1_1|2]), (420,178,[1_1|2]), (420,187,[1_1|2]), (420,196,[1_1|2]), (420,215,[1_1|2]), (420,394,[1_1|2]), (420,403,[1_1|2]), (420,412,[5_1|2]), (420,421,[1_1|2]), (420,430,[1_1|2]), (420,439,[1_1|2]), (420,448,[1_1|2]), (420,457,[1_1|2]), (421,422,[2_1|2]), (422,423,[3_1|2]), (423,424,[1_1|2]), (424,425,[0_1|2]), (425,426,[2_1|2]), (426,427,[4_1|2]), (427,428,[3_1|2]), (427,187,[3_1|2]), (428,429,[4_1|2]), (428,313,[4_1|2]), (429,150,[1_1|2]), (429,295,[1_1|2]), (429,304,[1_1|2]), (429,349,[1_1|2]), (429,394,[1_1|2]), (429,403,[1_1|2]), (429,421,[1_1|2]), (429,430,[1_1|2]), (429,439,[1_1|2]), (429,448,[1_1|2]), (429,457,[1_1|2]), (429,314,[1_1|2]), (429,386,[1_1|2]), (429,412,[5_1|2]), (430,431,[2_1|2]), (431,432,[3_1|2]), (432,433,[0_1|2]), (433,434,[3_1|2]), (434,435,[3_1|2]), (435,436,[1_1|2]), (436,437,[5_1|2]), (437,438,[2_1|2]), (438,150,[2_1|2]), (438,259,[2_1|2]), (438,466,[2_1|2]), (438,475,[2_1|2]), (438,484,[2_1|2]), (438,493,[2_1|2]), (438,502,[2_1|2]), (438,511,[2_1|2]), (438,520,[2_1|2]), (438,170,[2_1|2]), (438,243,[2_1|2]), (439,440,[1_1|2]), (440,441,[1_1|2]), (441,442,[0_1|2]), (442,443,[3_1|2]), (443,444,[3_1|2]), (444,445,[1_1|2]), (445,446,[5_1|2]), (446,447,[2_1|2]), (446,484,[2_1|2]), (446,493,[2_1|2]), (446,502,[2_1|2]), (447,150,[0_1|2]), (447,214,[0_1|2]), (447,268,[0_1|2]), (447,359,[0_1|2]), (447,368,[0_1|2]), (447,205,[5_1|2]), (447,223,[5_1|2]), (447,232,[5_1|2]), (447,241,[5_1|2]), (447,250,[5_1|2]), (447,259,[2_1|2]), (448,449,[4_1|2]), (449,450,[2_1|2]), (450,451,[1_1|2]), (451,452,[3_1|2]), (452,453,[1_1|2]), (453,454,[1_1|2]), (454,455,[5_1|2]), (455,456,[3_1|2]), (456,150,[1_1|2]), (456,295,[1_1|2]), (456,304,[1_1|2]), (456,349,[1_1|2]), (456,394,[1_1|2]), (456,403,[1_1|2]), (456,421,[1_1|2]), (456,430,[1_1|2]), (456,439,[1_1|2]), (456,448,[1_1|2]), (456,457,[1_1|2]), (456,269,[1_1|2]), (456,412,[5_1|2]), (457,458,[2_1|2]), (458,459,[1_1|2]), (459,460,[1_1|2]), (460,461,[2_1|2]), (461,462,[1_1|2]), (462,463,[3_1|2]), (463,464,[5_1|2]), (464,465,[5_1|2]), (465,150,[1_1|2]), (465,295,[1_1|2]), (465,304,[1_1|2]), (465,349,[1_1|2]), (465,394,[1_1|2]), (465,403,[1_1|2]), (465,421,[1_1|2]), (465,430,[1_1|2]), (465,439,[1_1|2]), (465,448,[1_1|2]), (465,457,[1_1|2]), (465,485,[1_1|2]), (465,412,[5_1|2]), (466,467,[0_1|2]), (467,468,[2_1|2]), (468,469,[3_1|2]), (469,470,[3_1|2]), (470,471,[3_1|2]), (471,472,[3_1|2]), (472,473,[3_1|2]), (472,187,[3_1|2]), (473,474,[4_1|2]), (473,313,[4_1|2]), (474,150,[1_1|2]), (474,295,[1_1|2]), (474,304,[1_1|2]), (474,349,[1_1|2]), (474,394,[1_1|2]), (474,403,[1_1|2]), (474,421,[1_1|2]), (474,430,[1_1|2]), (474,439,[1_1|2]), (474,448,[1_1|2]), (474,457,[1_1|2]), (474,269,[1_1|2]), (474,412,[5_1|2]), (475,476,[4_1|2]), (476,477,[4_1|2]), (477,478,[2_1|2]), (478,479,[3_1|2]), (479,480,[2_1|2]), (480,481,[2_1|2]), (481,482,[4_1|2]), (482,483,[4_1|2]), (483,150,[5_1|2]), (483,151,[5_1|2]), (483,160,[5_1|2]), (483,169,[5_1|2]), (483,178,[5_1|2]), (483,187,[5_1|2]), (483,196,[5_1|2]), (483,242,[5_1|2]), (483,278,[5_1|2]), (483,340,[5_1|2]), (483,349,[1_1|2]), (483,358,[5_1|2]), (483,367,[5_1|2]), (483,376,[5_1|2]), (483,385,[4_1|2]), (484,485,[1_1|2]), (485,486,[4_1|2]), (486,487,[2_1|2]), (487,488,[3_1|2]), (488,489,[1_1|2]), (489,490,[5_1|2]), (490,491,[5_1|2]), (491,492,[4_1|2]), (491,313,[4_1|2]), (492,150,[1_1|2]), (492,151,[1_1|2]), (492,160,[1_1|2]), (492,169,[1_1|2]), (492,178,[1_1|2]), (492,187,[1_1|2]), (492,196,[1_1|2]), (492,242,[1_1|2]), (492,278,[1_1|2]), (492,394,[1_1|2]), (492,403,[1_1|2]), (492,412,[5_1|2]), (492,421,[1_1|2]), (492,430,[1_1|2]), (492,439,[1_1|2]), (492,448,[1_1|2]), (492,457,[1_1|2]), (493,494,[0_1|2]), (494,495,[2_1|2]), (495,496,[3_1|2]), (496,497,[5_1|2]), (497,498,[4_1|2]), (498,499,[3_1|2]), (499,500,[0_1|2]), (500,501,[1_1|2]), (500,430,[1_1|2]), (500,439,[1_1|2]), (500,448,[1_1|2]), (501,150,[3_1|2]), (501,286,[3_1|2]), (501,313,[3_1|2]), (501,322,[3_1|2]), (501,331,[3_1|2]), (501,385,[3_1|2]), (501,151,[3_1|2]), (501,160,[3_1|2]), (501,169,[3_1|2]), (501,178,[3_1|2]), (501,187,[3_1|2]), (501,196,[3_1|2]), (502,503,[5_1|2]), (503,504,[5_1|2]), (504,505,[1_1|2]), (505,506,[2_1|2]), (506,507,[1_1|2]), (507,508,[1_1|2]), (508,509,[4_1|2]), (509,510,[4_1|2]), (509,277,[5_1|2]), (509,286,[4_1|2]), (509,592,[5_1|3]), (510,150,[0_1|2]), (510,214,[0_1|2]), (510,268,[0_1|2]), (510,359,[0_1|2]), (510,368,[0_1|2]), (510,205,[5_1|2]), (510,223,[5_1|2]), (510,232,[5_1|2]), (510,241,[5_1|2]), (510,250,[5_1|2]), (510,259,[2_1|2]), (511,512,[4_1|2]), (512,513,[0_1|2]), (513,514,[2_1|2]), (514,515,[3_1|2]), (515,516,[5_1|2]), (516,517,[5_1|2]), (517,518,[4_1|2]), (518,519,[5_1|2]), (519,150,[3_1|2]), (519,295,[3_1|2]), (519,304,[3_1|2]), (519,349,[3_1|2]), (519,394,[3_1|2]), (519,403,[3_1|2]), (519,421,[3_1|2]), (519,430,[3_1|2]), (519,439,[3_1|2]), (519,448,[3_1|2]), (519,457,[3_1|2]), (519,269,[3_1|2]), (519,151,[3_1|2]), (519,160,[3_1|2]), (519,169,[3_1|2]), (519,178,[3_1|2]), (519,187,[3_1|2]), (519,196,[3_1|2]), (520,521,[2_1|2]), (521,522,[5_1|2]), (522,523,[3_1|2]), (523,524,[0_1|2]), (524,525,[3_1|2]), (525,526,[3_1|2]), (526,527,[0_1|2]), (527,528,[3_1|2]), (527,151,[3_1|2]), (527,160,[3_1|2]), (527,547,[3_1|3]), (528,150,[0_1|2]), (528,259,[0_1|2, 2_1|2]), (528,466,[0_1|2]), (528,475,[0_1|2]), (528,484,[0_1|2]), (528,493,[0_1|2]), (528,502,[0_1|2]), (528,511,[0_1|2]), (528,520,[0_1|2]), (528,170,[0_1|2]), (528,205,[5_1|2]), (528,214,[0_1|2]), (528,223,[5_1|2]), (528,232,[5_1|2]), (528,241,[5_1|2]), (528,250,[5_1|2]), (528,268,[0_1|2]), (529,530,[1_1|3]), (530,531,[5_1|3]), (531,532,[0_1|3]), (532,533,[2_1|3]), (533,534,[1_1|3]), (534,535,[3_1|3]), (535,536,[0_1|3]), (536,537,[3_1|3]), (537,513,[4_1|3]), (538,539,[2_1|3]), (539,540,[3_1|3]), (540,541,[0_1|3]), (541,542,[3_1|3]), (542,543,[3_1|3]), (543,544,[1_1|3]), (544,545,[5_1|3]), (545,546,[2_1|3]), (546,243,[2_1|3]), (547,548,[3_1|3]), (548,549,[3_1|3]), (549,550,[1_1|3]), (550,551,[3_1|3]), (551,552,[1_1|3]), (552,553,[3_1|3]), (553,554,[2_1|3]), (554,555,[3_1|3]), (555,214,[3_1|3]), (555,268,[3_1|3]), (555,467,[3_1|3]), (555,494,[3_1|3]), (556,557,[3_1|3]), (557,558,[3_1|3]), (558,559,[1_1|3]), (559,560,[3_1|3]), (560,561,[1_1|3]), (561,562,[3_1|3]), (562,563,[2_1|3]), (563,564,[3_1|3]), (564,267,[3_1|3]), (565,566,[3_1|3]), (566,567,[3_1|3]), (567,568,[1_1|3]), (568,569,[3_1|3]), (569,570,[1_1|3]), (570,571,[3_1|3]), (571,572,[2_1|3]), (572,573,[3_1|3]), (573,357,[3_1|3]), (573,178,[3_1|2]), (574,575,[3_1|3]), (575,576,[3_1|3]), (576,577,[1_1|3]), (577,578,[3_1|3]), (578,579,[1_1|3]), (579,580,[3_1|3]), (580,581,[2_1|3]), (581,582,[3_1|3]), (581,178,[3_1|2]), (582,150,[3_1|3]), (582,214,[3_1|3]), (582,268,[3_1|3]), (582,369,[3_1|3]), (582,151,[3_1|2]), (582,160,[3_1|2]), (582,169,[3_1|2]), (582,178,[3_1|2]), (582,187,[3_1|2]), (582,196,[3_1|2]), (583,584,[2_1|3]), (584,585,[3_1|3]), (585,586,[5_1|3]), (586,587,[5_1|3]), (587,588,[2_1|3]), (588,589,[5_1|3]), (589,590,[1_1|3]), (590,591,[1_1|3]), (591,449,[4_1|3]), (592,593,[3_1|3]), (593,594,[3_1|3]), (594,595,[4_1|3]), (595,596,[2_1|3]), (596,597,[3_1|3]), (597,598,[5_1|3]), (598,599,[5_1|3]), (599,600,[2_1|3]), (600,215,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)