/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 64 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 153 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). 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4(4(0(3(3(1(3(2(4(4(2(4(4(0(2(5(5(4(1(3(4(3(1(2(5(4(1(5(0(3(4(5(3(4(5(2(4(1(1(1(5(4(3(2(2(3(0(1(4(4(1(3(4(1(1(2(2(2(0(0(5(0(3(0(5(3(1(0(2(4(2(0(5(4(1(0(4(3(0(1(4(3(5(4(5(2(2(5(5(2(1(3(4(3(3(0(5(3(0(3(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_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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). 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4(4(0(3(3(1(3(2(4(4(2(4(4(0(2(5(5(4(1(3(4(3(1(2(5(4(1(5(0(3(4(5(3(4(5(2(4(1(1(1(5(4(3(2(2(3(0(1(4(4(1(3(4(1(1(2(2(2(0(0(5(0(3(0(5(3(1(0(2(4(2(0(5(4(1(0(4(3(0(1(4(3(5(4(5(2(2(5(5(2(1(3(4(3(3(0(5(3(0(3(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: 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: 0(0(1(1(2(3(0(2(1(3(1(1(2(0(1(1(3(4(1(3(2(2(5(2(4(2(2(3(3(2(4(1(3(4(3(1(4(3(2(4(2(2(5(5(1(4(4(2(1(1(3(1(4(3(4(2(0(2(4(5(0(4(2(0(0(4(5(2(0(2(2(3(0(3(3(2(1(0(3(0(4(4(4(5(1(0(3(2(5(2(4(3(2(1(4(4(3(2(1(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(3(3(1(3(4(4(3(0(4(1(0(5(5(3(4(0(2(0(0(0(2(0(2(4(2(3(1(0(4(0(2(3(1(0(3(2(1(3(4(0(2(3(4(5(2(2(4(0(2(3(0(3(3(3(4(1(5(1(1(2(3(3(0(4(5(1(1(5(4(4(0(3(3(5(3(2(5(1(2(4(5(4(2(0(5(5(1(1(3(5(5(2(4(5(4(0(0(1(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0(0(1(3(1(4(0(3(4(1(0(1(4(4(4(4(5(4(2(1(3(4(4(2(4(2(0(2(4(0(5(2(0(0(4(1(1(2(3(5(0(4(0(3(4(4(1(3(5(5(3(4(0(3(4(2(5(2(4(4(2(1(2(0(2(0(1(2(5(1(2(2(5(4(2(4(4(3(2(1(3(1(3(3(3(4(5(3(1(2(4(2(1(3(2(5(5(0(0(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 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5(4(1(0(4(0(3(4(0(0(4(2(3(2(3(5(0(4(3(1(4(5(5(0(0(4(0(0(2(1(3(1(3(4(2(3(5(3(4(3(3(3(4(2(2(4(1(1(4(1(1(3(1(1(4(3(4(1(2(1(0(2(1(5(1(3(1(3(1(1(5(1(1(1(2(0(5(2(1(3(1(2(0(1(5(3(1(3(2(0(4(0(3(4(5(3(5(0(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(5(2(5(0(0(3(5(4(3(5(4(2(5(4(4(0(0(1(1(5(4(4(0(0(3(3(1(5(2(1(4(3(1(2(3(5(0(4(0(0(4(2(3(3(5(4(0(5(5(5(1(5(2(1(5(1(0(2(0(3(0(2(3(1(1(1(1(0(4(3(3(2(0(5(5(1(2(1(2(4(4(2(3(5(3(4(5(1(0(4(5(4(3(3(2(1(3(3(4(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 5(4(4(2(5(0(2(4(3(5(1(2(3(4(1(1(1(0(1(1(5(2(5(4(0(0(4(2(0(0(4(4(1(1(1(4(0(1(3(4(0(3(1(2(2(1(4(2(3(4(5(3(2(2(5(1(5(0(4(3(0(4(4(4(4(2(4(0(3(2(5(2(1(0(4(1(4(4(1(4(5(1(1(5(0(1(3(1(2(2(3(0(4(0(4(4(2(4(4(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 4(4(0(3(3(1(3(2(4(4(2(4(4(0(2(5(5(4(1(3(4(3(1(2(5(4(1(5(0(3(4(5(3(4(5(2(4(1(1(1(5(4(3(2(2(3(0(1(4(4(1(3(4(1(1(2(2(2(0(0(5(0(3(0(5(3(1(0(2(4(2(0(5(4(1(0(4(3(0(1(4(3(5(4(5(2(2(5(5(2(1(3(4(3(3(0(5(3(0(3(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: 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: 0(0(1(1(2(3(0(2(1(3(1(1(2(0(1(1(3(4(1(3(2(2(5(2(4(2(2(3(3(2(4(1(3(4(3(1(4(3(2(4(2(2(5(5(1(4(4(2(1(1(3(1(4(3(4(2(0(2(4(5(0(4(2(0(0(4(5(2(0(2(2(3(0(3(3(2(1(0(3(0(4(4(4(5(1(0(3(2(5(2(4(3(2(1(4(4(3(2(1(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(3(3(1(3(4(4(3(0(4(1(0(5(5(3(4(0(2(0(0(0(2(0(2(4(2(3(1(0(4(0(2(3(1(0(3(2(1(3(4(0(2(3(4(5(2(2(4(0(2(3(0(3(3(3(4(1(5(1(1(2(3(3(0(4(5(1(1(5(4(4(0(3(3(5(3(2(5(1(2(4(5(4(2(0(5(5(1(1(3(5(5(2(4(5(4(0(0(1(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0(0(1(3(1(4(0(3(4(1(0(1(4(4(4(4(5(4(2(1(3(4(4(2(4(2(0(2(4(0(5(2(0(0(4(1(1(2(3(5(0(4(0(3(4(4(1(3(5(5(3(4(0(3(4(2(5(2(4(4(2(1(2(0(2(0(1(2(5(1(2(2(5(4(2(4(4(3(2(1(3(1(3(3(3(4(5(3(1(2(4(2(1(3(2(5(5(0(0(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 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0(1(2(1(5(5(2(3(4(4(1(3(4(4(0(4(3(2(2(1(2(3(3(1(5(4(3(3(1(3(5(4(3(1(4(3(4(2(0(5(1(4(2(0(2(5(1(4(0(1(2(4(1(2(5(3(1(2(2(5(3(3(1(0(3(0(4(5(2(0(3(4(4(1(0(2(0(4(3(3(4(4(0(0(2(2(3(1(3(2(3(0(2(0(3(0(1(1(0(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(2(1(4(2(1(2(0(4(5(1(2(0(4(5(5(2(1(4(2(3(2(5(2(0(3(5(5(4(0(1(5(2(1(3(0(5(0(3(2(1(5(2(5(1(4(2(4(3(0(5(1(4(1(4(0(3(2(4(0(3(3(5(2(0(2(1(5(1(3(4(0(3(4(4(0(0(2(2(5(5(4(1(5(4(1(5(2(1(2(0(0(3(4(0(2(5(2(3(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0(1(3(2(1(4(1(4(2(1(5(4(3(0(4(0(3(3(0(2(0(1(0(1(0(5(4(3(2(0(5(5(3(5(3(3(0(3(3(1(1(4(4(2(2(5(4(2(5(3(1(3(4(4(5(1(2(1(5(4(4(5(3(5(4(0(4(2(4(2(4(2(4(1(0(5(5(3(3(2(2(4(2(3(4(3(5(0(0(0(5(4(1(3(1(5(2(3(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 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1(3(0(2(3(4(2(0(0(1(1(5(4(4(5(0(4(0(1(1(4(4(5(5(2(0(3(4(5(5(5(5(1(2(1(2(5(2(0(0(5(0(3(2(5(2(2(4(4(2(2(1(5(4(2(1(4(3(4(5(0(1(5(1(0(3(3(3(1(4(5(5(3(0(4(1(2(1(3(0(2(3(1(2(2(1(5(3(3(1(0(3(3(4(1(4(5(1(0(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(0(0(4(1(4(4(2(4(5(5(5(0(4(2(3(0(0(2(3(5(0(5(2(3(3(5(1(1(3(3(1(4(5(4(4(2(0(0(0(5(1(3(0(2(1(4(2(4(0(0(4(1(3(4(0(4(3(0(5(5(2(4(5(1(5(1(2(2(2(0(1(3(5(4(4(0(0(5(0(2(4(3(5(4(1(0(1(1(1(0(2(4(2(3(2(1(5(1(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 1(4(0(4(1(1(1(0(0(4(5(0(2(5(4(1(5(3(1(0(5(4(5(3(5(2(0(4(2(3(4(0(4(1(4(2(5(4(1(0(5(0(0(1(0(3(5(3(5(5(2(0(2(3(0(1(2(1(4(0(4(0(0(4(4(0(3(3(5(4(4(5(4(5(3(0(3(2(1(4(0(2(1(4(3(0(5(2(4(3(2(4(0(5(5(2(5(5(1(4(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 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2(1(0(4(5(5(5(0(3(4(1(3(1(5(2(0(0(5(5(2(1(4(5(5(4(3(4(5(1(3(5(1(4(1(3(1(3(3(3(1(2(2(0(2(1(2(2(0(4(3(0(2(5(4(0(2(2(5(2(2(5(0(2(0(2(1(0(0(2(5(3(0(5(2(2(1(5(5(2(4(2(1(0(1(4(3(2(3(0(3(4(3(0(4(1(1(5(4(2(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 3(2(3(2(2(0(4(3(5(3(2(3(5(0(3(2(1(0(5(3(1(4(0(3(5(4(4(1(4(5(1(2(4(2(5(1(0(0(4(1(1(5(1(3(3(5(0(2(4(2(1(3(0(1(5(5(3(0(1(1(1(4(2(1(0(4(0(1(3(0(1(1(0(3(3(5(0(2(0(4(1(1(5(5(4(0(1(2(4(2(5(1(5(2(1(3(0(0(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 2(4(2(3(1(0(2(0(5(0(3(3(2(2(2(5(2(3(1(5(2(3(0(5(5(5(3(4(4(1(1(2(3(5(1(5(0(4(2(2(1(1(5(2(0(2(4(1(5(3(4(2(1(4(1(0(0(5(0(0(1(0(5(5(3(0(5(4(0(4(5(3(3(2(3(2(2(0(1(5(2(1(4(0(5(1(2(4(3(0(3(0(5(2(1(1(4(5(4(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 5(5(0(1(3(5(1(0(3(1(1(5(3(2(2(1(3(2(4(5(5(0(3(1(1(1(5(1(3(2(2(3(1(5(3(1(1(5(5(0(1(0(0(3(3(0(1(3(2(4(5(3(1(5(3(4(3(0(5(2(1(5(5(3(4(5(5(4(4(5(2(0(4(4(1(0(4(1(5(2(3(2(3(0(3(0(3(1(2(1(4(3(5(2(2(1(0(4(4(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3(4(0(1(4(3(0(3(3(2(4(1(4(2(5(1(0(1(4(0(0(4(5(0(3(5(3(0(0(4(3(3(1(5(1(1(2(4(1(3(0(4(2(3(2(1(0(2(5(2(5(1(3(5(5(5(2(1(1(5(3(0(2(4(3(1(2(3(5(4(0(4(3(3(2(1(2(2(1(4(3(1(1(2(4(2(3(4(3(1(1(4(5(0(0(5(2(3(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 3(2(4(2(1(2(3(0(0(1(0(0(1(3(0(0(3(0(4(0(5(1(2(2(0(2(1(4(0(3(5(2(3(3(0(3(5(0(2(5(2(0(0(2(2(5(0(0(3(2(4(1(0(2(3(0(4(1(2(4(3(5(4(2(2(2(2(2(4(0(0(2(3(1(0(2(3(2(2(1(0(2(1(0(5(4(3(4(3(0(0(3(5(3(4(2(1(1(4(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 4(1(3(2(2(1(0(3(4(2(3(4(5(2(3(5(0(4(1(4(1(4(4(0(2(3(5(0(0(4(0(3(1(4(1(0(3(5(2(1(2(4(1(2(3(5(2(5(1(5(1(0(0(2(0(2(5(2(5(0(5(1(5(3(1(3(0(2(3(1(5(1(3(3(2(3(2(5(5(5(4(4(2(1(4(1(4(5(4(0(1(3(0(5(3(0(1(4(2(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 5(4(4(2(0(5(3(3(5(0(2(5(4(2(3(5(2(3(0(0(2(2(3(4(1(2(4(4(0(3(2(0(4(0(0(3(3(1(4(0(3(0(5(4(2(1(0(4(1(0(5(0(0(3(4(1(3(1(4(3(5(0(1(0(5(5(0(1(0(4(5(2(5(2(2(3(5(3(3(5(5(2(4(1(2(2(5(5(5(3(4(5(5(4(3(4(5(5(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 4(4(3(2(0(2(5(2(1(5(1(1(3(1(1(1(2(1(0(1(3(2(1(0(4(0(0(5(0(0(0(0(0(0(3(0(2(3(4(3(4(1(1(5(1(5(1(1(1(4(0(2(5(4(1(4(5(1(2(3(3(1(4(5(4(3(5(5(0(0(5(1(5(1(4(3(0(1(1(2(1(2(4(3(4(3(1(2(3(2(0(3(1(0(4(1(1(1(4(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 3(0(3(2(4(1(0(0(3(4(2(3(0(4(4(3(4(5(3(3(1(1(2(3(4(4(1(4(1(1(4(2(1(4(2(3(0(2(3(1(5(0(0(5(4(4(3(0(1(4(1(2(1(5(2(5(2(5(4(2(4(2(3(5(5(3(0(5(1(5(2(0(3(5(2(4(5(5(3(0(2(5(3(5(5(0(5(3(2(3(0(4(3(3(5(4(2(0(2(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 5(3(1(5(0(4(3(0(5(1(0(4(1(1(2(0(3(2(4(1(3(4(5(2(3(5(4(5(2(2(1(2(3(3(3(3(5(3(4(5(2(1(4(3(3(2(1(4(4(1(5(4(2(0(1(0(1(3(4(3(1(1(0(4(5(0(2(5(3(5(2(1(1(5(3(2(3(2(4(4(0(3(0(2(4(3(2(2(5(3(0(0(1(3(2(0(5(2(4(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(2(1(0(4(2(3(0(1(0(0(5(1(1(5(1(3(0(3(5(4(4(4(5(3(0(0(1(3(5(3(2(5(0(2(4(5(3(0(3(3(1(4(1(5(4(0(2(5(2(1(4(3(0(2(3(1(2(2(3(3(5(3(4(5(2(1(5(5(0(4(2(5(5(5(1(4(3(3(0(5(1(5(2(1(5(4(3(0(4(2(3(5(3(2(2(4(3(2(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 5(4(1(0(4(0(3(4(0(0(4(2(3(2(3(5(0(4(3(1(4(5(5(0(0(4(0(0(2(1(3(1(3(4(2(3(5(3(4(3(3(3(4(2(2(4(1(1(4(1(1(3(1(1(4(3(4(1(2(1(0(2(1(5(1(3(1(3(1(1(5(1(1(1(2(0(5(2(1(3(1(2(0(1(5(3(1(3(2(0(4(0(3(4(5(3(5(0(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(5(2(5(0(0(3(5(4(3(5(4(2(5(4(4(0(0(1(1(5(4(4(0(0(3(3(1(5(2(1(4(3(1(2(3(5(0(4(0(0(4(2(3(3(5(4(0(5(5(5(1(5(2(1(5(1(0(2(0(3(0(2(3(1(1(1(1(0(4(3(3(2(0(5(5(1(2(1(2(4(4(2(3(5(3(4(5(1(0(4(5(4(3(3(2(1(3(3(4(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 5(4(4(2(5(0(2(4(3(5(1(2(3(4(1(1(1(0(1(1(5(2(5(4(0(0(4(2(0(0(4(4(1(1(1(4(0(1(3(4(0(3(1(2(2(1(4(2(3(4(5(3(2(2(5(1(5(0(4(3(0(4(4(4(4(2(4(0(3(2(5(2(1(0(4(1(4(4(1(4(5(1(1(5(0(1(3(1(2(2(3(0(4(0(4(4(2(4(4(5(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 4(4(0(3(3(1(3(2(4(4(2(4(4(0(2(5(5(4(1(3(4(3(1(2(5(4(1(5(0(3(4(5(3(4(5(2(4(1(1(1(5(4(3(2(2(3(0(1(4(4(1(3(4(1(1(2(2(2(0(0(5(0(3(0(5(3(1(0(2(4(2(0(5(4(1(0(4(3(0(1(4(3(5(4(5(2(2(5(5(2(1(3(4(3(3(0(5(3(0(3(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: 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 2. The certificate found is represented by the following graph. 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1955, 1956, 1957, 1958, 1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030, 2031, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2039, 2040, 2041, 2042, 2043, 2044, 2045, 2046, 2047, 2048, 2049, 2050, 2051, 2052, 2053, 2054, 2055, 2056, 2057, 2058, 2059, 2060, 2061, 2062, 2063, 2064, 2065, 2066, 2067, 2068, 2069, 2070, 2071, 2072, 2073, 2074, 2075, 2076, 2077, 2078, 2079, 2080, 2081, 2082, 2083, 2084, 2085, 2086, 2087, 2088, 2089, 2090, 2091, 2092, 2093, 2094, 2095, 2096, 2097, 2098, 2099, 2100, 2101, 2102, 2103, 2104, 2105, 2106, 2107, 2108, 2109, 2110, 2111, 2112, 2113, 2114, 2115, 2116, 2117, 2118, 2119, 2120, 2121, 2122, 2123, 2124, 2125, 2126, 2127, 2128, 2129, 2130, 2131, 2132, 2133] {(151,152,[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]), (151,153,[0_1|1, 1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1]), (151,154,[1_1|2]), (151,253,[5_1|2]), (151,352,[5_1|2]), (151,451,[1_1|2]), (151,550,[4_1|2]), (151,649,[4_1|2]), (151,748,[3_1|2]), (151,847,[3_1|2]), (151,946,[2_1|2]), (151,1045,[0_1|2]), (151,1144,[5_1|2]), (151,1243,[4_1|2]), (151,1342,[3_1|2]), (151,1441,[5_1|2]), (151,1540,[3_1|2]), (151,1639,[5_1|2]), (151,1738,[3_1|2]), (151,1837,[2_1|2]), (151,1936,[2_1|2]), (151,2035,[4_1|2]), (152,152,[cons_0_1|0, cons_1_1|0, cons_2_1|0, cons_3_1|0, cons_4_1|0, cons_5_1|0]), (153,152,[encArg_1|1]), (153,153,[0_1|1, 1_1|1, 2_1|1, 3_1|1, 4_1|1, 5_1|1]), (153,154,[1_1|2]), (153,253,[5_1|2]), (153,352,[5_1|2]), (153,451,[1_1|2]), (153,550,[4_1|2]), (153,649,[4_1|2]), (153,748,[3_1|2]), (153,847,[3_1|2]), 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(1842,1843,[3_1|2]), (1843,1844,[0_1|2]), (1844,1845,[1_1|2]), (1845,1846,[0_1|2]), (1846,1847,[0_1|2]), (1847,1848,[5_1|2]), (1848,1849,[1_1|2]), (1849,1850,[1_1|2]), (1850,1851,[5_1|2]), (1851,1852,[1_1|2]), (1852,1853,[3_1|2]), (1853,1854,[0_1|2]), (1854,1855,[3_1|2]), (1855,1856,[5_1|2]), (1856,1857,[4_1|2]), (1857,1858,[4_1|2]), (1858,1859,[4_1|2]), (1859,1860,[5_1|2]), (1860,1861,[3_1|2]), (1861,1862,[0_1|2]), (1862,1863,[0_1|2]), (1863,1864,[1_1|2]), (1864,1865,[3_1|2]), (1865,1866,[5_1|2]), (1866,1867,[3_1|2]), (1867,1868,[2_1|2]), (1868,1869,[5_1|2]), (1869,1870,[0_1|2]), (1870,1871,[2_1|2]), (1871,1872,[4_1|2]), (1872,1873,[5_1|2]), (1873,1874,[3_1|2]), (1874,1875,[0_1|2]), (1875,1876,[3_1|2]), (1876,1877,[3_1|2]), (1877,1878,[1_1|2]), (1878,1879,[4_1|2]), (1879,1880,[1_1|2]), (1880,1881,[5_1|2]), (1881,1882,[4_1|2]), (1882,1883,[0_1|2]), (1883,1884,[2_1|2]), (1884,1885,[5_1|2]), (1885,1886,[2_1|2]), (1886,1887,[1_1|2]), (1887,1888,[4_1|2]), (1888,1889,[3_1|2]), (1889,1890,[0_1|2]), (1890,1891,[2_1|2]), (1891,1892,[3_1|2]), (1892,1893,[1_1|2]), (1893,1894,[2_1|2]), (1894,1895,[2_1|2]), (1895,1896,[3_1|2]), (1896,1897,[3_1|2]), (1897,1898,[5_1|2]), (1898,1899,[3_1|2]), (1899,1900,[4_1|2]), (1900,1901,[5_1|2]), (1901,1902,[2_1|2]), (1902,1903,[1_1|2]), (1903,1904,[5_1|2]), (1904,1905,[5_1|2]), (1905,1906,[0_1|2]), (1906,1907,[4_1|2]), (1907,1908,[2_1|2]), (1908,1909,[5_1|2]), (1909,1910,[5_1|2]), (1910,1911,[5_1|2]), (1911,1912,[1_1|2]), (1912,1913,[4_1|2]), (1913,1914,[3_1|2]), (1914,1915,[3_1|2]), (1915,1916,[0_1|2]), (1916,1917,[5_1|2]), (1917,1918,[1_1|2]), (1918,1919,[5_1|2]), (1919,1920,[2_1|2]), (1920,1921,[1_1|2]), (1921,1922,[5_1|2]), (1922,1923,[4_1|2]), (1923,1924,[3_1|2]), (1924,1925,[0_1|2]), (1925,1926,[4_1|2]), (1926,1927,[2_1|2]), (1927,1928,[3_1|2]), (1928,1929,[5_1|2]), (1929,1930,[3_1|2]), (1930,1931,[2_1|2]), (1931,1932,[2_1|2]), (1932,1933,[4_1|2]), (1933,1934,[3_1|2]), (1934,1935,[2_1|2]), (1935,153,[3_1|2]), (1935,748,[3_1|2]), (1935,847,[3_1|2]), (1935,1342,[3_1|2]), (1935,1540,[3_1|2]), (1935,1738,[3_1|2]), (1936,1937,[5_1|2]), (1937,1938,[2_1|2]), (1938,1939,[5_1|2]), (1939,1940,[0_1|2]), (1940,1941,[0_1|2]), (1941,1942,[3_1|2]), (1942,1943,[5_1|2]), (1943,1944,[4_1|2]), (1944,1945,[3_1|2]), (1945,1946,[5_1|2]), (1946,1947,[4_1|2]), (1947,1948,[2_1|2]), (1948,1949,[5_1|2]), (1949,1950,[4_1|2]), (1950,1951,[4_1|2]), (1951,1952,[0_1|2]), (1952,1953,[0_1|2]), (1953,1954,[1_1|2]), (1954,1955,[1_1|2]), (1955,1956,[5_1|2]), (1956,1957,[4_1|2]), (1957,1958,[4_1|2]), (1958,1959,[0_1|2]), (1959,1960,[0_1|2]), (1960,1961,[3_1|2]), (1961,1962,[3_1|2]), (1962,1963,[1_1|2]), (1963,1964,[5_1|2]), (1964,1965,[2_1|2]), (1965,1966,[1_1|2]), (1966,1967,[4_1|2]), (1967,1968,[3_1|2]), (1968,1969,[1_1|2]), (1969,1970,[2_1|2]), (1970,1971,[3_1|2]), (1971,1972,[5_1|2]), (1972,1973,[0_1|2]), (1973,1974,[4_1|2]), (1974,1975,[0_1|2]), (1975,1976,[0_1|2]), (1976,1977,[4_1|2]), (1977,1978,[2_1|2]), (1978,1979,[3_1|2]), (1979,1980,[3_1|2]), (1980,1981,[5_1|2]), (1981,1982,[4_1|2]), (1982,1983,[0_1|2]), (1983,1984,[5_1|2]), (1984,1985,[5_1|2]), (1985,1986,[5_1|2]), (1986,1987,[1_1|2]), (1987,1988,[5_1|2]), (1988,1989,[2_1|2]), (1989,1990,[1_1|2]), (1990,1991,[5_1|2]), (1991,1992,[1_1|2]), (1992,1993,[0_1|2]), (1993,1994,[2_1|2]), (1994,1995,[0_1|2]), (1995,1996,[3_1|2]), (1996,1997,[0_1|2]), (1997,1998,[2_1|2]), (1998,1999,[3_1|2]), (1999,2000,[1_1|2]), (2000,2001,[1_1|2]), (2001,2002,[1_1|2]), (2002,2003,[1_1|2]), (2003,2004,[0_1|2]), (2004,2005,[4_1|2]), (2005,2006,[3_1|2]), (2006,2007,[3_1|2]), (2007,2008,[2_1|2]), (2008,2009,[0_1|2]), (2009,2010,[5_1|2]), (2010,2011,[5_1|2]), (2011,2012,[1_1|2]), (2012,2013,[2_1|2]), (2013,2014,[1_1|2]), (2014,2015,[2_1|2]), (2015,2016,[4_1|2]), (2016,2017,[4_1|2]), (2017,2018,[2_1|2]), (2018,2019,[3_1|2]), (2019,2020,[5_1|2]), (2020,2021,[3_1|2]), (2021,2022,[4_1|2]), (2022,2023,[5_1|2]), (2023,2024,[1_1|2]), (2024,2025,[0_1|2]), (2025,2026,[4_1|2]), (2026,2027,[5_1|2]), (2027,2028,[4_1|2]), (2028,2029,[3_1|2]), (2029,2030,[3_1|2]), (2030,2031,[2_1|2]), (2031,2032,[1_1|2]), (2032,2033,[3_1|2]), (2033,2034,[3_1|2]), (2033,1540,[3_1|2]), (2034,153,[4_1|2]), (2034,946,[4_1|2]), (2034,1837,[4_1|2]), (2034,1936,[4_1|2]), (2034,1639,[5_1|2]), (2034,1738,[3_1|2]), (2035,2036,[4_1|2]), (2036,2037,[0_1|2]), (2037,2038,[3_1|2]), (2038,2039,[3_1|2]), (2039,2040,[1_1|2]), (2040,2041,[3_1|2]), (2041,2042,[2_1|2]), (2042,2043,[4_1|2]), (2043,2044,[4_1|2]), (2044,2045,[2_1|2]), (2045,2046,[4_1|2]), (2046,2047,[4_1|2]), (2047,2048,[0_1|2]), (2048,2049,[2_1|2]), (2049,2050,[5_1|2]), (2050,2051,[5_1|2]), (2051,2052,[4_1|2]), (2052,2053,[1_1|2]), (2053,2054,[3_1|2]), (2054,2055,[4_1|2]), (2055,2056,[3_1|2]), (2056,2057,[1_1|2]), (2057,2058,[2_1|2]), (2058,2059,[5_1|2]), (2059,2060,[4_1|2]), (2060,2061,[1_1|2]), (2061,2062,[5_1|2]), (2062,2063,[0_1|2]), (2063,2064,[3_1|2]), (2064,2065,[4_1|2]), (2065,2066,[5_1|2]), (2066,2067,[3_1|2]), (2067,2068,[4_1|2]), (2068,2069,[5_1|2]), (2069,2070,[2_1|2]), (2070,2071,[4_1|2]), (2071,2072,[1_1|2]), (2072,2073,[1_1|2]), (2073,2074,[1_1|2]), (2074,2075,[5_1|2]), (2075,2076,[4_1|2]), (2076,2077,[3_1|2]), (2077,2078,[2_1|2]), (2078,2079,[2_1|2]), (2079,2080,[3_1|2]), (2080,2081,[0_1|2]), (2081,2082,[1_1|2]), (2082,2083,[4_1|2]), (2083,2084,[4_1|2]), (2084,2085,[1_1|2]), (2085,2086,[3_1|2]), (2086,2087,[4_1|2]), (2087,2088,[1_1|2]), (2088,2089,[1_1|2]), (2089,2090,[2_1|2]), (2090,2091,[2_1|2]), (2091,2092,[2_1|2]), (2092,2093,[0_1|2]), (2093,2094,[0_1|2]), (2094,2095,[5_1|2]), (2095,2096,[0_1|2]), (2096,2097,[3_1|2]), (2097,2098,[0_1|2]), (2098,2099,[5_1|2]), (2099,2100,[3_1|2]), (2100,2101,[1_1|2]), (2101,2102,[0_1|2]), (2102,2103,[2_1|2]), (2103,2104,[4_1|2]), (2104,2105,[2_1|2]), (2105,2106,[0_1|2]), (2106,2107,[5_1|2]), (2107,2108,[4_1|2]), (2108,2109,[1_1|2]), (2109,2110,[0_1|2]), (2110,2111,[4_1|2]), (2111,2112,[3_1|2]), (2112,2113,[0_1|2]), (2113,2114,[1_1|2]), (2114,2115,[4_1|2]), (2115,2116,[3_1|2]), (2116,2117,[5_1|2]), (2117,2118,[4_1|2]), (2118,2119,[5_1|2]), (2119,2120,[2_1|2]), (2120,2121,[2_1|2]), (2121,2122,[5_1|2]), (2122,2123,[5_1|2]), (2123,2124,[2_1|2]), (2124,2125,[1_1|2]), (2125,2126,[3_1|2]), (2126,2127,[4_1|2]), (2127,2128,[3_1|2]), (2128,2129,[3_1|2]), (2129,2130,[0_1|2]), (2130,2131,[5_1|2]), (2131,2132,[3_1|2]), (2132,2133,[0_1|2]), (2133,153,[3_1|2]), (2133,253,[3_1|2]), (2133,352,[3_1|2]), (2133,1144,[3_1|2]), (2133,1441,[3_1|2]), (2133,1639,[3_1|2]), (2133,551,[3_1|2]), (2133,650,[3_1|2]), (2133,1540,[3_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)