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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 117 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 164 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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(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_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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(0(x1)) -> 1(1(2(3(1(4(3(1(5(4(x1)))))))))) 0(0(5(x1))) -> 0(2(0(1(4(1(1(1(5(1(x1)))))))))) 0(3(0(5(x1)))) -> 1(4(1(2(3(2(0(5(1(1(x1)))))))))) 2(4(0(5(x1)))) -> 2(1(0(1(1(4(1(5(5(5(x1)))))))))) 5(3(0(5(x1)))) -> 0(3(3(4(1(0(1(4(1(4(x1)))))))))) 5(3(2(5(x1)))) -> 1(2(1(1(1(3(3(4(1(4(x1)))))))))) 0(0(4(0(3(x1))))) -> 1(1(1(4(0(2(4(5(0(3(x1)))))))))) 0(3(4(3(0(x1))))) -> 0(0(4(1(1(2(2(3(1(2(x1)))))))))) 0(5(2(5(2(x1))))) -> 0(3(5(3(2(1(1(0(5(1(x1)))))))))) 1(0(0(0(0(x1))))) -> 1(3(1(3(4(1(0(5(2(1(x1)))))))))) 2(0(5(2(0(x1))))) -> 2(5(2(2(2(1(3(1(1(1(x1)))))))))) 2(3(0(5(4(x1))))) -> 2(3(1(1(2(4(3(0(1(0(x1)))))))))) 2(5(0(0(1(x1))))) -> 2(1(4(5(5(0(1(3(1(2(x1)))))))))) 3(3(0(0(3(x1))))) -> 3(3(5(5(5(5(1(1(4(0(x1)))))))))) 4(0(0(0(2(x1))))) -> 2(0(1(1(1(0(0(1(4(1(x1)))))))))) 4(2(3(2(5(x1))))) -> 2(1(0(4(2(3(4(1(1(4(x1)))))))))) 5(2(0(0(0(x1))))) -> 3(1(1(5(4(1(4(5(2(2(x1)))))))))) 5(3(0(0(5(x1))))) -> 1(1(4(1(3(2(0(4(5(2(x1)))))))))) 5(5(2(5(4(x1))))) -> 5(2(0(2(3(1(2(4(3(3(x1)))))))))) 5(5(3(3(0(x1))))) -> 5(5(5(5(5(2(2(4(3(1(x1)))))))))) 0(0(5(3(2(0(x1)))))) -> 0(1(2(5(2(2(1(4(1(4(x1)))))))))) 0(5(3(0(1(2(x1)))))) -> 1(0(1(0(3(1(0(2(1(2(x1)))))))))) 0(5(4(3(5(4(x1)))))) -> 0(0(2(3(1(0(1(0(2(2(x1)))))))))) 3(0(0(0(0(0(x1)))))) -> 1(1(0(3(2(0(3(4(5(2(x1)))))))))) 4(0(3(5(2(5(x1)))))) -> 1(2(5(0(0(4(5(3(4(1(x1)))))))))) 4(2(3(5(0(0(x1)))))) -> 3(5(1(1(1(4(0(2(2(0(x1)))))))))) 5(0(5(0(5(2(x1)))))) -> 1(5(3(1(5(3(3(2(5(1(x1)))))))))) 5(2(2(0(4(2(x1)))))) -> 5(5(5(5(1(1(1(2(0(2(x1)))))))))) 5(3(0(0(3(4(x1)))))) -> 1(2(1(0(2(4(1(5(5(3(x1)))))))))) 5(5(3(5(4(5(x1)))))) -> 1(3(2(0(1(2(2(1(1(4(x1)))))))))) 5(5(5(0(5(4(x1)))))) -> 5(5(1(2(1(1(3(4(5(1(x1)))))))))) 0(0(0(4(0(3(2(x1))))))) -> 1(1(1(4(0(2(3(5(4(1(x1)))))))))) 0(3(5(4(0(2(3(x1))))))) -> 0(2(4(0(3(4(1(4(4(3(x1)))))))))) 0(5(4(0(3(0(3(x1))))))) -> 3(1(4(5(1(2(3(0(4(3(x1)))))))))) 0(5(4(3(4(3(0(x1))))))) -> 1(3(4(5(0(5(3(4(3(0(x1)))))))))) 1(2(5(3(0(2(5(x1))))))) -> 1(2(5(4(1(1(0(1(5(5(x1)))))))))) 1(3(0(4(0(3(0(x1))))))) -> 1(5(4(4(5(3(3(4(1(4(x1)))))))))) 1(3(4(0(5(0(3(x1))))))) -> 1(3(1(5(4(1(1(5(5(4(x1)))))))))) 1(5(2(4(3(0(5(x1))))))) -> 1(3(1(2(3(3(2(4(1(5(x1)))))))))) 2(2(4(1(3(3(0(x1))))))) -> 2(3(2(3(2(2(1(0(4(2(x1)))))))))) 2(4(5(4(3(0(0(x1))))))) -> 4(5(3(1(0(2(0(4(3(4(x1)))))))))) 2(5(0(3(2(5(0(x1))))))) -> 3(4(5(0(5(4(0(3(1(1(x1)))))))))) 4(2(4(4(0(0(3(x1))))))) -> 2(0(4(5(3(4(4(3(3(3(x1)))))))))) 4(5(0(3(0(0(2(x1))))))) -> 4(1(4(1(2(1(0(4(4(1(x1)))))))))) 5(0(0(0(5(5(0(x1))))))) -> 5(0(0(1(3(4(1(0(5(0(x1)))))))))) 5(0(3(0(5(3(0(x1))))))) -> 1(3(0(0(1(1(3(1(2(0(x1)))))))))) 5(2(3(0(2(2(5(x1))))))) -> 5(0(4(0(5(1(0(2(4(1(x1)))))))))) 5(4(0(0(5(4(0(x1))))))) -> 5(5(3(4(4(4(2(4(3(2(x1)))))))))) 5(4(3(2(2(0(0(x1))))))) -> 5(4(3(5(2(4(1(0(4(1(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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 4. The certificate found is represented by the following graph. "[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, 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, 644, 645, 646, 647, 648, 649, 650] {(99,100,[0_1|0, 2_1|0, 5_1|0, 1_1|0, 3_1|0, 4_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]), (99,101,[0_1|1, 2_1|1, 5_1|1, 1_1|1, 3_1|1, 4_1|1]), (99,102,[1_1|2]), (99,111,[0_1|2]), (99,120,[0_1|2]), (99,129,[1_1|2]), (99,138,[1_1|2]), (99,147,[1_1|2]), (99,156,[0_1|2]), (99,165,[0_1|2]), (99,174,[0_1|2]), (99,183,[1_1|2]), (99,192,[0_1|2]), (99,201,[1_1|2]), (99,210,[3_1|2]), (99,219,[2_1|2]), (99,228,[4_1|2]), (99,237,[2_1|2]), (99,246,[2_1|2]), (99,255,[2_1|2]), (99,264,[3_1|2]), (99,273,[2_1|2]), (99,282,[0_1|2]), (99,291,[1_1|2]), (99,300,[1_1|2]), (99,309,[1_1|2]), (99,318,[3_1|2]), (99,327,[5_1|2]), (99,336,[5_1|2]), (99,345,[5_1|2]), (99,354,[5_1|2]), (99,363,[1_1|2]), (99,372,[5_1|2]), (99,381,[1_1|2]), (99,390,[5_1|2]), (99,399,[1_1|2]), (99,408,[5_1|2]), (99,417,[5_1|2]), (99,426,[1_1|2]), (99,435,[1_1|2]), (99,444,[1_1|2]), (99,453,[1_1|2]), (99,462,[1_1|2]), (99,471,[3_1|2]), (99,480,[1_1|2]), (99,489,[2_1|2]), (99,498,[1_1|2]), (99,507,[2_1|2]), (99,516,[3_1|2]), (99,525,[2_1|2]), (99,534,[4_1|2]), (99,543,[1_1|3]), (100,100,[cons_0_1|0, cons_2_1|0, cons_5_1|0, cons_1_1|0, cons_3_1|0, cons_4_1|0]), (101,100,[encArg_1|1]), (101,101,[0_1|1, 2_1|1, 5_1|1, 1_1|1, 3_1|1, 4_1|1]), (101,102,[1_1|2]), (101,111,[0_1|2]), (101,120,[0_1|2]), (101,129,[1_1|2]), (101,138,[1_1|2]), (101,147,[1_1|2]), (101,156,[0_1|2]), (101,165,[0_1|2]), (101,174,[0_1|2]), (101,183,[1_1|2]), (101,192,[0_1|2]), (101,201,[1_1|2]), (101,210,[3_1|2]), (101,219,[2_1|2]), (101,228,[4_1|2]), (101,237,[2_1|2]), (101,246,[2_1|2]), (101,255,[2_1|2]), (101,264,[3_1|2]), (101,273,[2_1|2]), (101,282,[0_1|2]), (101,291,[1_1|2]), (101,300,[1_1|2]), (101,309,[1_1|2]), (101,318,[3_1|2]), (101,327,[5_1|2]), (101,336,[5_1|2]), (101,345,[5_1|2]), (101,354,[5_1|2]), (101,363,[1_1|2]), (101,372,[5_1|2]), (101,381,[1_1|2]), (101,390,[5_1|2]), (101,399,[1_1|2]), (101,408,[5_1|2]), (101,417,[5_1|2]), (101,426,[1_1|2]), (101,435,[1_1|2]), (101,444,[1_1|2]), (101,453,[1_1|2]), (101,462,[1_1|2]), (101,471,[3_1|2]), (101,480,[1_1|2]), (101,489,[2_1|2]), (101,498,[1_1|2]), (101,507,[2_1|2]), (101,516,[3_1|2]), (101,525,[2_1|2]), (101,534,[4_1|2]), (101,543,[1_1|3]), (102,103,[1_1|2]), (103,104,[2_1|2]), (104,105,[3_1|2]), (105,106,[1_1|2]), (106,107,[4_1|2]), (107,108,[3_1|2]), (108,109,[1_1|2]), (109,110,[5_1|2]), (109,408,[5_1|2]), (109,417,[5_1|2]), (110,101,[4_1|2]), (110,111,[4_1|2]), (110,120,[4_1|2]), (110,156,[4_1|2]), (110,165,[4_1|2]), (110,174,[4_1|2]), (110,192,[4_1|2]), (110,282,[4_1|2]), (110,489,[2_1|2]), (110,498,[1_1|2]), (110,507,[2_1|2]), (110,516,[3_1|2]), (110,525,[2_1|2]), (110,534,[4_1|2]), (111,112,[2_1|2]), (112,113,[0_1|2]), (113,114,[1_1|2]), (114,115,[4_1|2]), (115,116,[1_1|2]), (116,117,[1_1|2]), (117,118,[1_1|2]), (118,119,[5_1|2]), (119,101,[1_1|2]), (119,327,[1_1|2]), (119,336,[1_1|2]), (119,345,[1_1|2]), (119,354,[1_1|2]), (119,372,[1_1|2]), (119,390,[1_1|2]), (119,408,[1_1|2]), (119,417,[1_1|2]), (119,426,[1_1|2]), (119,435,[1_1|2]), (119,444,[1_1|2]), (119,453,[1_1|2]), (119,462,[1_1|2]), (120,121,[1_1|2]), (121,122,[2_1|2]), (122,123,[5_1|2]), (123,124,[2_1|2]), (124,125,[2_1|2]), (125,126,[1_1|2]), (126,127,[4_1|2]), 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(164,237,[2_1|2]), (164,246,[2_1|2]), (164,255,[2_1|2]), (164,264,[3_1|2]), (164,273,[2_1|2]), (164,561,[2_1|3]), (165,166,[2_1|2]), (166,167,[4_1|2]), (167,168,[0_1|2]), (168,169,[3_1|2]), (169,170,[4_1|2]), (170,171,[1_1|2]), (171,172,[4_1|2]), (172,173,[4_1|2]), (173,101,[3_1|2]), (173,210,[3_1|2]), (173,264,[3_1|2]), (173,318,[3_1|2]), (173,471,[3_1|2]), (173,516,[3_1|2]), (173,247,[3_1|2]), (173,274,[3_1|2]), (173,480,[1_1|2]), (174,175,[3_1|2]), (175,176,[5_1|2]), (176,177,[3_1|2]), (177,178,[2_1|2]), (178,179,[1_1|2]), (179,180,[1_1|2]), (180,181,[0_1|2]), (181,182,[5_1|2]), (182,101,[1_1|2]), (182,219,[1_1|2]), (182,237,[1_1|2]), (182,246,[1_1|2]), (182,255,[1_1|2]), (182,273,[1_1|2]), (182,489,[1_1|2]), (182,507,[1_1|2]), (182,525,[1_1|2]), (182,346,[1_1|2]), (182,239,[1_1|2]), (182,426,[1_1|2]), (182,435,[1_1|2]), (182,444,[1_1|2]), (182,453,[1_1|2]), (182,462,[1_1|2]), (183,184,[0_1|2]), (184,185,[1_1|2]), (185,186,[0_1|2]), (186,187,[3_1|2]), (187,188,[1_1|2]), 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(507,508,[1_1|2]), (508,509,[0_1|2]), (509,510,[4_1|2]), (510,511,[2_1|2]), (511,512,[3_1|2]), (512,513,[4_1|2]), (513,514,[1_1|2]), (514,515,[1_1|2]), (515,101,[4_1|2]), (515,327,[4_1|2]), (515,336,[4_1|2]), (515,345,[4_1|2]), (515,354,[4_1|2]), (515,372,[4_1|2]), (515,390,[4_1|2]), (515,408,[4_1|2]), (515,417,[4_1|2]), (515,238,[4_1|2]), (515,489,[2_1|2]), (515,498,[1_1|2]), (515,507,[2_1|2]), (515,516,[3_1|2]), (515,525,[2_1|2]), (515,534,[4_1|2]), (516,517,[5_1|2]), (517,518,[1_1|2]), (518,519,[1_1|2]), (519,520,[1_1|2]), (520,521,[4_1|2]), (521,522,[0_1|2]), (522,523,[2_1|2]), (523,524,[2_1|2]), (523,237,[2_1|2]), (523,606,[2_1|3]), (524,101,[0_1|2]), (524,111,[0_1|2]), (524,120,[0_1|2]), (524,156,[0_1|2]), (524,165,[0_1|2]), (524,174,[0_1|2]), (524,192,[0_1|2]), (524,282,[0_1|2]), (524,157,[0_1|2]), (524,193,[0_1|2]), (524,392,[0_1|2]), (524,102,[1_1|2]), (524,129,[1_1|2]), (524,138,[1_1|2]), (524,147,[1_1|2]), (524,183,[1_1|2]), (524,201,[1_1|2]), (524,210,[3_1|2]), (524,570,[1_1|3]), (525,526,[0_1|2]), (526,527,[4_1|2]), (527,528,[5_1|2]), (528,529,[3_1|2]), (529,530,[4_1|2]), (530,531,[4_1|2]), (531,532,[3_1|2]), (532,533,[3_1|2]), (532,471,[3_1|2]), (533,101,[3_1|2]), (533,210,[3_1|2]), (533,264,[3_1|2]), (533,318,[3_1|2]), (533,471,[3_1|2]), (533,516,[3_1|2]), (533,175,[3_1|2]), (533,283,[3_1|2]), (533,480,[1_1|2]), (534,535,[1_1|2]), (535,536,[4_1|2]), (536,537,[1_1|2]), (537,538,[2_1|2]), (538,539,[1_1|2]), (539,540,[0_1|2]), (540,541,[4_1|2]), (541,542,[4_1|2]), (542,101,[1_1|2]), (542,219,[1_1|2]), (542,237,[1_1|2]), (542,246,[1_1|2]), (542,255,[1_1|2]), (542,273,[1_1|2]), (542,489,[1_1|2]), (542,507,[1_1|2]), (542,525,[1_1|2]), (542,112,[1_1|2]), (542,166,[1_1|2]), (542,194,[1_1|2]), (542,426,[1_1|2]), (542,435,[1_1|2]), (542,444,[1_1|2]), (542,453,[1_1|2]), (542,462,[1_1|2]), (543,544,[1_1|3]), (544,545,[2_1|3]), (545,546,[3_1|3]), (546,547,[1_1|3]), (547,548,[4_1|3]), (548,549,[3_1|3]), (549,550,[1_1|3]), (550,551,[5_1|3]), (551,157,[4_1|3]), (551,193,[4_1|3]), (552,553,[1_1|3]), (553,554,[4_1|3]), (554,555,[1_1|3]), (555,556,[2_1|3]), (556,557,[1_1|3]), (557,558,[0_1|3]), (558,559,[4_1|3]), (559,560,[4_1|3]), (560,194,[1_1|3]), (561,562,[1_1|3]), (562,563,[4_1|3]), (563,564,[5_1|3]), (564,565,[5_1|3]), (565,566,[0_1|3]), (566,567,[1_1|3]), (567,568,[3_1|3]), (568,569,[1_1|3]), (569,393,[2_1|3]), (570,571,[1_1|3]), (571,572,[2_1|3]), (572,573,[3_1|3]), (573,574,[1_1|3]), (574,575,[4_1|3]), (575,576,[3_1|3]), (576,577,[1_1|3]), (577,578,[5_1|3]), (578,111,[4_1|3]), (578,120,[4_1|3]), (578,156,[4_1|3]), (578,165,[4_1|3]), (578,174,[4_1|3]), (578,192,[4_1|3]), (578,282,[4_1|3]), (578,157,[4_1|3]), (578,193,[4_1|3]), (578,392,[4_1|3]), (579,580,[2_1|3]), (580,581,[1_1|3]), (581,582,[1_1|3]), (582,583,[1_1|3]), (583,584,[3_1|3]), (584,585,[3_1|3]), (585,586,[4_1|3]), (586,587,[1_1|3]), (587,238,[4_1|3]), (588,589,[1_1|3]), (589,590,[2_1|3]), (590,591,[3_1|3]), (591,592,[1_1|3]), (592,593,[4_1|3]), (593,594,[3_1|3]), (594,595,[1_1|3]), (595,596,[5_1|3]), (596,392,[4_1|3]), (597,598,[1_1|3]), (598,599,[2_1|3]), (599,600,[3_1|3]), (600,601,[1_1|3]), (601,602,[4_1|3]), (602,603,[3_1|3]), (603,604,[1_1|3]), (604,605,[5_1|3]), (605,402,[4_1|3]), (606,607,[5_1|3]), (607,608,[2_1|3]), (608,609,[2_1|3]), (609,610,[2_1|3]), (610,611,[1_1|3]), (611,612,[3_1|3]), (612,613,[1_1|3]), (613,614,[1_1|3]), (614,347,[1_1|3]), (615,616,[0_1|3]), (616,617,[1_1|3]), (617,618,[1_1|3]), (618,619,[1_1|3]), (619,620,[0_1|3]), (619,642,[1_1|4]), (620,621,[0_1|3]), (621,622,[1_1|3]), (622,623,[4_1|3]), (623,194,[1_1|3]), (624,625,[1_1|3]), (625,626,[2_1|3]), (626,627,[3_1|3]), (627,628,[1_1|3]), (628,629,[4_1|3]), (629,630,[3_1|3]), (630,631,[1_1|3]), (631,632,[5_1|3]), (632,495,[4_1|3]), (633,634,[1_1|3]), (634,635,[2_1|3]), (635,636,[3_1|3]), (636,637,[1_1|3]), (637,638,[4_1|3]), (638,639,[3_1|3]), (639,640,[1_1|3]), (640,641,[5_1|3]), (641,502,[4_1|3]), (642,643,[1_1|4]), (643,644,[2_1|4]), (644,645,[3_1|4]), (645,646,[1_1|4]), (646,647,[4_1|4]), (647,648,[3_1|4]), (648,649,[1_1|4]), (649,650,[5_1|4]), (650,621,[4_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1)