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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 44 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 118 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: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(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_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(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: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(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: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(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: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(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_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(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. "[78, 79, 91, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 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, 345, 347, 350, 352, 353, 354, 355, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 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, 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, 547, 548, 549, 550, 551, 552, 553, 554, 555] {(78,79,[4_1|0, 1_1|0, 0_1|0, 2_1|0, 3_1|0, 5_1|0, encArg_1|0, encode_4_1|0, encode_0_1|0, encode_1_1|0, encode_3_1|0, encode_2_1|0, encode_5_1|0]), (78,91,[4_1|1, 1_1|1, 0_1|1, 2_1|1, 3_1|1, 5_1|1]), (78,99,[1_1|2]), (78,108,[4_1|2]), (78,117,[5_1|2]), (78,126,[3_1|2]), (78,135,[0_1|2]), (78,144,[4_1|2]), (78,153,[3_1|2]), (78,162,[3_1|2]), (78,171,[5_1|2]), (78,180,[5_1|2]), (78,189,[2_1|2]), (78,198,[5_1|2]), (78,207,[4_1|2]), (78,216,[1_1|2]), (78,225,[5_1|2]), (78,234,[3_1|2]), (78,243,[5_1|2]), (78,252,[5_1|2]), (78,261,[3_1|2]), (78,270,[0_1|2]), (78,279,[5_1|2]), (78,288,[0_1|2]), (78,297,[0_1|2]), (78,306,[0_1|2]), (78,315,[0_1|2]), (78,324,[3_1|2]), (78,333,[0_1|2]), (78,342,[1_1|2]), (78,357,[0_1|2]), (78,366,[0_1|2]), (78,382,[2_1|2]), (78,391,[2_1|2]), (78,400,[2_1|2]), (78,409,[2_1|2]), (78,418,[3_1|2]), (78,427,[0_1|2]), (78,436,[3_1|2]), (78,445,[5_1|2]), (78,454,[5_1|2]), (78,463,[1_1|2]), (78,472,[5_1|2]), (79,79,[cons_4_1|0, cons_1_1|0, cons_0_1|0, cons_2_1|0, cons_3_1|0, cons_5_1|0]), (91,79,[encArg_1|1]), (91,91,[4_1|1, 1_1|1, 0_1|1, 2_1|1, 3_1|1, 5_1|1]), (91,99,[1_1|2]), (91,108,[4_1|2]), (91,117,[5_1|2]), (91,126,[3_1|2]), (91,135,[0_1|2]), (91,144,[4_1|2]), (91,153,[3_1|2]), (91,162,[3_1|2]), (91,171,[5_1|2]), (91,180,[5_1|2]), (91,189,[2_1|2]), (91,198,[5_1|2]), (91,207,[4_1|2]), (91,216,[1_1|2]), (91,225,[5_1|2]), (91,234,[3_1|2]), (91,243,[5_1|2]), (91,252,[5_1|2]), (91,261,[3_1|2]), (91,270,[0_1|2]), (91,279,[5_1|2]), (91,288,[0_1|2]), (91,297,[0_1|2]), (91,306,[0_1|2]), (91,315,[0_1|2]), (91,324,[3_1|2]), (91,333,[0_1|2]), (91,342,[1_1|2]), (91,357,[0_1|2]), (91,366,[0_1|2]), (91,382,[2_1|2]), (91,391,[2_1|2]), (91,400,[2_1|2]), (91,409,[2_1|2]), (91,418,[3_1|2]), (91,427,[0_1|2]), (91,436,[3_1|2]), (91,445,[5_1|2]), (91,454,[5_1|2]), (91,463,[1_1|2]), (91,472,[5_1|2]), (99,100,[3_1|2]), (100,101,[1_1|2]), (101,102,[2_1|2]), (102,103,[4_1|2]), (103,104,[5_1|2]), (104,105,[2_1|2]), 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(161,153,[5_1|2]), (161,162,[5_1|2]), (161,234,[5_1|2]), (161,261,[5_1|2]), (161,324,[5_1|2]), (161,418,[5_1|2]), (161,436,[5_1|2]), (161,118,[5_1|2]), (161,199,[5_1|2]), (161,244,[5_1|2]), (161,454,[5_1|2]), (161,463,[1_1|2]), (161,472,[5_1|2]), (162,163,[3_1|2]), (163,164,[1_1|2]), (164,165,[4_1|2]), (165,166,[5_1|2]), (166,167,[5_1|2]), (167,168,[3_1|2]), (168,169,[4_1|2]), (169,170,[5_1|2]), (170,91,[5_1|2]), (170,108,[5_1|2]), (170,144,[5_1|2]), (170,207,[5_1|2]), (170,217,[5_1|2]), (170,454,[5_1|2]), (170,463,[1_1|2]), (170,472,[5_1|2]), (171,172,[5_1|2]), (172,173,[4_1|2]), (173,174,[5_1|2]), (174,175,[1_1|2]), (175,176,[1_1|2]), (176,177,[2_1|2]), (177,178,[3_1|2]), (178,179,[3_1|2]), (178,436,[3_1|2]), (179,91,[3_1|2]), (179,126,[3_1|2]), (179,153,[3_1|2]), (179,162,[3_1|2]), (179,234,[3_1|2]), (179,261,[3_1|2]), (179,324,[3_1|2]), (179,418,[3_1|2]), (179,436,[3_1|2]), (179,400,[2_1|2]), (179,409,[2_1|2]), (179,427,[0_1|2]), (179,445,[5_1|2]), (180,181,[2_1|2]), 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(206,117,[5_1|2]), (206,171,[5_1|2]), (206,180,[5_1|2]), (206,198,[5_1|2]), (206,225,[5_1|2]), (206,243,[5_1|2]), (206,252,[5_1|2]), (206,279,[5_1|2]), (206,445,[5_1|2]), (206,454,[5_1|2]), (206,472,[5_1|2]), (206,307,[5_1|2]), (206,463,[1_1|2]), (207,208,[1_1|2]), (208,209,[2_1|2]), (209,210,[2_1|2]), (210,211,[2_1|2]), (211,212,[5_1|2]), (212,213,[4_1|2]), (212,525,[4_1|3]), (213,214,[0_1|2]), (214,215,[5_1|2]), (215,91,[3_1|2]), (215,126,[3_1|2]), (215,153,[3_1|2]), (215,162,[3_1|2]), (215,234,[3_1|2]), (215,261,[3_1|2]), (215,324,[3_1|2]), (215,418,[3_1|2]), (215,436,[3_1|2]), (215,118,[3_1|2]), (215,199,[3_1|2]), (215,244,[3_1|2]), (215,400,[2_1|2]), (215,409,[2_1|2]), (215,427,[0_1|2]), (215,445,[5_1|2]), (216,217,[4_1|2]), (217,218,[5_1|2]), (218,219,[5_1|2]), (219,220,[2_1|2]), (220,221,[3_1|2]), (221,222,[1_1|2]), (222,223,[1_1|2]), (223,224,[0_1|2]), (223,366,[0_1|2]), (224,91,[3_1|2]), (224,135,[3_1|2]), (224,270,[3_1|2]), (224,288,[3_1|2]), (224,297,[3_1|2]), (224,306,[3_1|2]), (224,315,[3_1|2]), (224,333,[3_1|2]), (224,357,[3_1|2]), (224,366,[3_1|2]), (224,427,[3_1|2, 0_1|2]), (224,446,[3_1|2]), (224,473,[3_1|2]), (224,400,[2_1|2]), (224,409,[2_1|2]), (224,418,[3_1|2]), (224,436,[3_1|2]), (224,445,[5_1|2]), (225,226,[1_1|2]), (226,227,[0_1|2]), (227,228,[4_1|2]), (228,229,[0_1|2]), (229,230,[3_1|2]), (230,231,[0_1|2]), (231,232,[4_1|2]), (232,233,[5_1|2]), (233,91,[5_1|2]), (233,108,[5_1|2]), (233,144,[5_1|2]), (233,207,[5_1|2]), (233,217,[5_1|2]), (233,454,[5_1|2]), (233,463,[1_1|2]), (233,472,[5_1|2]), (234,235,[2_1|2]), (235,236,[1_1|2]), (236,237,[0_1|2]), (237,238,[1_1|2]), (238,239,[2_1|2]), (239,240,[4_1|2]), (240,241,[2_1|2]), (241,242,[2_1|2]), (242,91,[2_1|2]), (242,189,[2_1|2]), (242,382,[2_1|2]), (242,391,[2_1|2]), (242,400,[2_1|2]), (242,409,[2_1|2]), (242,190,[2_1|2]), (243,244,[3_1|2]), (244,245,[1_1|2]), (245,246,[3_1|2]), (246,247,[2_1|2]), (247,248,[0_1|2]), (248,249,[3_1|2]), (249,250,[4_1|2]), (250,251,[5_1|2]), 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(443,472,[5_1|2]), (444,91,[1_1|2]), (444,99,[1_1|2]), (444,216,[1_1|2]), (444,342,[1_1|2]), (444,463,[1_1|2]), (444,189,[2_1|2]), (444,198,[5_1|2]), (444,207,[4_1|2]), (444,225,[5_1|2]), (444,234,[3_1|2]), (444,243,[5_1|2]), (444,252,[5_1|2]), (444,261,[3_1|2]), (444,270,[0_1|2]), (444,279,[5_1|2]), (444,288,[0_1|2]), (445,446,[0_1|2]), (446,447,[1_1|2]), (447,448,[3_1|2]), (448,449,[3_1|2]), (449,450,[0_1|2]), (450,451,[1_1|2]), (451,452,[1_1|2]), (452,453,[2_1|2]), (452,382,[2_1|2]), (452,391,[2_1|2]), (453,91,[4_1|2]), (453,108,[4_1|2]), (453,144,[4_1|2]), (453,207,[4_1|2]), (453,99,[1_1|2]), (453,117,[5_1|2]), (453,126,[3_1|2]), (453,135,[0_1|2]), (453,153,[3_1|2]), (453,162,[3_1|2]), (453,171,[5_1|2]), (453,180,[5_1|2]), (453,507,[4_1|3]), (453,516,[1_1|3]), (454,455,[1_1|2]), (455,456,[3_1|2]), (456,457,[1_1|2]), (457,458,[3_1|2]), (458,459,[0_1|2]), (459,460,[3_1|2]), (460,461,[0_1|2]), (461,462,[1_1|2]), (461,189,[2_1|2]), (461,198,[5_1|2]), (461,207,[4_1|2]), (461,216,[1_1|2]), (462,91,[1_1|2]), (462,189,[1_1|2, 2_1|2]), (462,382,[1_1|2]), (462,391,[1_1|2]), (462,400,[1_1|2]), (462,409,[1_1|2]), (462,198,[5_1|2]), (462,207,[4_1|2]), (462,216,[1_1|2]), (462,225,[5_1|2]), (462,234,[3_1|2]), (462,243,[5_1|2]), (462,252,[5_1|2]), (462,261,[3_1|2]), (462,270,[0_1|2]), (462,279,[5_1|2]), (462,288,[0_1|2]), (463,464,[3_1|2]), (464,465,[3_1|2]), (465,466,[2_1|2]), (466,467,[1_1|2]), (467,468,[1_1|2]), (468,469,[1_1|2]), (469,470,[2_1|2]), (470,471,[3_1|2]), (470,427,[0_1|2]), (471,91,[1_1|2]), (471,99,[1_1|2]), (471,216,[1_1|2]), (471,342,[1_1|2]), (471,463,[1_1|2]), (471,383,[1_1|2]), (471,189,[2_1|2]), (471,198,[5_1|2]), (471,207,[4_1|2]), (471,225,[5_1|2]), (471,234,[3_1|2]), (471,243,[5_1|2]), (471,252,[5_1|2]), (471,261,[3_1|2]), (471,270,[0_1|2]), (471,279,[5_1|2]), (471,288,[0_1|2]), (472,473,[0_1|2]), (473,474,[0_1|2]), (474,475,[0_1|2]), (475,476,[5_1|2]), (476,477,[0_1|2]), (477,478,[1_1|2]), (478,479,[3_1|2]), (479,480,[4_1|2]), (480,91,[4_1|2]), (480,108,[4_1|2]), (480,144,[4_1|2]), (480,207,[4_1|2]), (480,392,[4_1|2]), (480,99,[1_1|2]), (480,117,[5_1|2]), (480,126,[3_1|2]), (480,135,[0_1|2]), (480,153,[3_1|2]), (480,162,[3_1|2]), (480,171,[5_1|2]), (480,180,[5_1|2]), (480,507,[4_1|3]), (480,516,[1_1|3]), (507,508,[5_1|3]), (508,509,[5_1|3]), (509,510,[2_1|3]), (510,511,[2_1|3]), (511,512,[2_1|3]), (512,513,[4_1|3]), (513,514,[5_1|3]), (514,515,[1_1|3]), (515,307,[1_1|3]), (516,517,[3_1|3]), (517,518,[1_1|3]), (518,519,[2_1|3]), (519,520,[4_1|3]), (520,521,[5_1|3]), (521,522,[2_1|3]), (522,523,[4_1|3]), (523,524,[3_1|3]), (524,316,[4_1|3]), (524,334,[4_1|3]), (525,526,[5_1|3]), (526,527,[5_1|3]), (527,528,[2_1|3]), (528,529,[2_1|3]), (529,530,[2_1|3]), (530,531,[4_1|3]), (531,532,[5_1|3]), (532,533,[1_1|3]), (533,215,[1_1|3]), (547,548,[1_1|3]), (548,549,[0_1|3]), (549,550,[3_1|3]), (550,551,[3_1|3]), (551,552,[2_1|3]), (552,553,[0_1|3]), (553,554,[2_1|3]), (554,555,[1_1|3]), (555,410,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)