/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 108 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 497 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(2(1(2(0(1(x1))))))) -> 2(1(0(2(1(0(1(x1))))))) 3(1(2(4(4(3(5(4(4(2(0(0(0(x1))))))))))))) -> 4(3(2(0(0(1(3(4(0(2(4(4(5(x1))))))))))))) 0(0(4(4(2(0(3(4(4(4(0(3(5(2(4(5(1(x1))))))))))))))))) -> 0(4(1(3(0(4(4(3(2(2(5(0(4(4(5(0(4(x1))))))))))))))))) 0(2(3(0(1(0(2(0(4(3(5(1(5(3(1(4(2(4(1(x1))))))))))))))))))) -> 3(1(1(4(1(0(0(0(3(2(5(1(5(4(2(3(2(4(0(x1))))))))))))))))))) 3(3(2(0(1(1(3(4(3(5(2(2(4(2(3(0(3(0(4(2(3(1(5(x1))))))))))))))))))))))) -> 0(3(2(1(0(3(3(5(3(2(0(1(4(2(4(3(3(2(1(2(3(5(4(x1))))))))))))))))))))))) 5(3(1(3(0(5(4(0(5(2(3(5(0(1(2(5(5(5(0(2(0(4(0(0(x1)))))))))))))))))))))))) -> 5(5(3(0(0(1(5(0(4(0(4(5(1(0(2(5(5(5(0(2(3(3(0(2(x1)))))))))))))))))))))))) 0(3(2(1(3(4(1(2(1(3(3(0(2(4(1(0(3(1(3(4(4(2(4(3(0(3(1(1(0(x1))))))))))))))))))))))))))))) -> 1(1(3(1(3(4(4(2(4(4(3(1(3(2(0(0(0(1(3(2(1(4(3(3(1(0(3(2(0(x1))))))))))))))))))))))))))))) 1(3(5(3(4(0(5(0(2(2(3(5(4(4(1(4(3(4(5(0(5(1(2(3(5(3(4(5(4(2(x1)))))))))))))))))))))))))))))) -> 5(3(2(3(5(4(1(0(3(3(3(4(4(0(1(4(4(5(5(2(5(1(0(5(4(2(4(5(2(3(x1)))))))))))))))))))))))))))))) 2(1(4(0(3(5(1(4(5(3(4(3(1(4(4(5(4(4(2(4(0(2(2(0(3(0(4(2(4(5(x1)))))))))))))))))))))))))))))) -> 4(4(4(5(4(2(5(1(5(4(0(3(0(3(2(3(5(4(0(2(0(3(2(4(2(4(1(4(4(1(x1)))))))))))))))))))))))))))))) 3(2(4(1(1(5(3(3(5(4(5(0(5(0(1(5(4(4(0(4(0(3(1(0(0(0(1(5(3(1(3(x1))))))))))))))))))))))))))))))) -> 3(2(3(5(3(0(3(1(0(0(5(5(4(5(1(0(4(5(0(1(1(5(3(0(4(0(1(1(4(4(3(x1))))))))))))))))))))))))))))))) 4(4(1(2(0(1(1(3(4(0(1(0(0(4(1(5(0(2(2(1(4(0(2(0(4(1(0(3(1(2(2(x1))))))))))))))))))))))))))))))) -> 0(4(0(4(2(1(1(0(2(0(3(2(0(1(1(1(0(0(4(5(2(2(1(0(4(1(3(1(4(2(4(x1))))))))))))))))))))))))))))))) 1(5(0(2(1(3(3(5(2(5(4(0(2(2(0(3(0(5(4(4(5(2(4(3(1(4(2(0(1(2(2(4(x1)))))))))))))))))))))))))))))))) -> 1(0(3(0(1(0(5(5(2(2(4(2(4(2(3(3(3(2(2(4(5(1(4(5(2(1(2(0(4(4(0(5(x1)))))))))))))))))))))))))))))))) 0(5(2(5(0(5(4(2(0(1(4(2(1(5(3(5(3(3(5(5(4(4(3(5(0(2(5(0(5(0(1(0(4(x1))))))))))))))))))))))))))))))))) -> 2(3(2(4(3(4(2(0(4(3(2(3(1(3(0(3(0(1(3(2(5(5(3(3(2(1(0(3(5(3(2(3(1(2(x1)))))))))))))))))))))))))))))))))) 4(3(0(2(2(1(4(0(0(0(5(0(3(1(4(1(5(5(3(1(5(1(5(1(5(3(1(4(3(5(2(2(3(x1))))))))))))))))))))))))))))))))) -> 0(3(0(2(2(1(0(1(3(3(5(0(3(4(5(1(5(1(0(5(1(5(2(1(4(3(1(4(5(5(2(4(3(x1))))))))))))))))))))))))))))))))) 0(3(1(2(5(0(5(5(3(4(3(5(5(5(0(0(5(4(0(0(3(3(5(5(4(4(1(1(0(4(0(4(4(5(x1)))))))))))))))))))))))))))))))))) -> 1(2(3(1(3(4(2(0(3(1(5(2(4(1(3(0(3(2(2(2(2(1(3(3(0(4(3(2(0(4(5(2(3(0(5(x1))))))))))))))))))))))))))))))))))) 1(4(5(5(4(2(0(3(0(0(5(1(4(4(5(0(0(0(1(5(2(2(2(2(2(4(4(5(4(5(5(3(5(5(x1)))))))))))))))))))))))))))))))))) -> 1(4(5(5(2(5(0(0(4(4(5(1(4(4(5(3(0(2(2(0(4(4(5(0(0(2(5(1(2(5(2(3(5(5(x1)))))))))))))))))))))))))))))))))) 3(5(1(1(0(0(5(2(0(3(4(5(3(5(3(3(5(4(3(3(3(1(3(0(1(1(1(2(5(4(5(4(2(5(x1)))))))))))))))))))))))))))))))))) -> 3(5(1(3(0(5(5(2(1(1(4(3(3(3(5(3(0(5(4(3(1(0(5(0(3(1(1(2(5(4(4(3(2(5(x1)))))))))))))))))))))))))))))))))) 3(3(4(1(4(5(0(5(1(0(2(0(5(4(1(4(1(5(0(2(5(5(5(2(0(3(5(0(0(5(4(3(0(5(3(x1))))))))))))))))))))))))))))))))))) -> 0(0(0(0(3(2(1(1(1(2(2(1(3(3(5(2(3(2(1(0(3(2(1(5(2(1(4(3(0(2(4(3(1(3(4(3(x1)))))))))))))))))))))))))))))))))))) 4(2(1(5(3(4(2(2(2(1(4(4(3(0(2(1(5(4(0(2(1(2(2(2(5(2(1(0(2(2(3(3(3(4(4(x1))))))))))))))))))))))))))))))))))) -> 4(2(5(0(0(3(2(2(2(1(4(4(2(3(2(4(5(1(5(2(2(4(3(2(0(3(1(4(2(1(2(2(1(3(4(x1))))))))))))))))))))))))))))))))))) 5(3(4(4(3(4(2(0(5(2(2(3(1(5(2(3(5(3(4(3(4(4(0(4(2(2(4(5(2(4(2(3(4(3(1(x1))))))))))))))))))))))))))))))))))) -> 2(2(4(4(2(4(4(2(1(5(2(3(3(5(5(3(5(3(4(3(2(4(0(1(0(2(4(4(3(4(5(2(4(3(3(x1))))))))))))))))))))))))))))))))))) 3(5(0(1(4(3(4(4(5(4(0(0(2(3(2(2(0(0(5(3(1(5(4(0(4(2(2(2(3(2(4(5(1(2(5(3(x1)))))))))))))))))))))))))))))))))))) -> 3(5(0(2(1(5(0(1(4(4(0(0(2(3(2(0(3(2(5(3(4(5(4(4(4(0(2(2(3(2(5(1(4(2(5(3(x1)))))))))))))))))))))))))))))))))))) 4(4(3(5(3(2(4(3(0(5(0(4(0(3(5(1(4(4(4(5(1(0(4(1(3(5(2(1(0(1(2(2(5(0(0(5(x1)))))))))))))))))))))))))))))))))))) -> 4(4(3(5(3(2(5(3(1(5(0(0(5(0(5(3(0(0(1(5(1(1(4(4(4(4(2(4(0(1(2(2(4(3(0(5(x1)))))))))))))))))))))))))))))))))))) 3(5(1(2(5(4(5(5(4(5(2(2(4(5(1(0(0(5(2(0(2(2(3(3(4(2(4(1(1(1(0(0(0(1(5(1(3(x1))))))))))))))))))))))))))))))))))))) -> 0(3(3(4(1(3(1(4(3(3(1(1(0(1(2(3(3(2(0(4(2(3(3(5(3(2(2(3(2(2(1(3(3(1(5(4(1(3(x1)))))))))))))))))))))))))))))))))))))) 3(4(3(2(0(1(3(3(3(0(1(2(1(0(0(5(0(5(2(2(1(5(4(5(1(2(4(3(3(0(1(2(3(5(3(4(2(0(x1)))))))))))))))))))))))))))))))))))))) -> 3(2(0(3(4(1(2(1(3(4(1(3(0(5(5(1(3(3(2(3(2(0(3(5(5(2(4(3(0(0(1(2(0(5(1(4(2(0(x1)))))))))))))))))))))))))))))))))))))) 4(0(5(4(3(5(3(5(4(0(4(5(0(3(5(5(2(2(5(5(0(3(0(4(4(5(2(5(5(1(2(0(4(5(5(5(5(3(0(3(x1)))))))))))))))))))))))))))))))))))))))) -> 2(0(5(1(2(2(4(3(3(2(1(4(2(3(4(2(1(4(1(4(4(5(3(4(5(5(2(4(2(0(2(2(3(0(0(3(2(3(2(4(3(x1))))))))))))))))))))))))))))))))))))))))) 4(4(2(4(3(3(0(2(5(1(1(0(4(3(5(4(1(4(0(1(0(2(4(4(0(5(3(3(1(3(2(5(0(5(0(4(1(5(1(4(x1)))))))))))))))))))))))))))))))))))))))) -> 4(1(2(2(3(3(0(1(5(3(2(0(4(4(5(1(1(4(0(4(0(0(3(4(1(2(5(3(4(4(4(1(0(5(0(4(3(5(5(1(x1)))))))))))))))))))))))))))))))))))))))) 4(5(2(2(3(3(4(0(2(3(1(3(0(3(4(3(0(5(0(3(4(0(0(1(2(2(2(1(4(4(0(3(2(2(4(2(4(0(1(4(x1)))))))))))))))))))))))))))))))))))))))) -> 4(2(2(1(3(3(2(0(5(3(2(3(0(1(0(3(4(2(2(1(4(3(5(2(4(4(2(0(4(0(0(0(3(2(4(3(4(0(1(4(x1)))))))))))))))))))))))))))))))))))))))) 1(5(4(2(4(4(1(3(5(3(4(0(4(5(1(4(4(5(5(4(5(4(5(5(2(3(1(2(0(5(5(1(4(2(4(0(4(5(4(4(0(x1))))))))))))))))))))))))))))))))))))))))) -> 5(0(0(1(5(0(5(4(2(1(2(2(0(5(3(1(4(0(2(3(2(3(2(3(2(1(1(1(4(3(0(0(3(0(3(0(5(3(3(4(2(2(x1)))))))))))))))))))))))))))))))))))))))))) 5(2(4(5(5(2(5(0(2(3(3(0(0(1(4(1(2(5(5(2(0(5(1(0(3(5(1(2(5(4(0(4(2(5(5(4(4(5(4(2(0(x1))))))))))))))))))))))))))))))))))))))))) -> 2(2(3(3(1(2(5(3(2(3(0(3(0(3(0(3(5(1(2(5(1(3(1(2(5(2(0(2(2(2(1(3(2(4(3(1(4(0(5(5(4(1(x1)))))))))))))))))))))))))))))))))))))))))) 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2(5(5(1(0(4(1(5(3(2(5(5(0(5(4(2(3(4(0(5(5(1(2(0(5(2(4(2(3(4(5(4(0(4(1(4(2(5(5(3(3(5(0(4(3(2(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 3(0(2(3(0(5(5(2(1(3(4(0(4(3(2(3(3(0(2(2(3(2(4(0(2(2(3(3(4(0(1(1(3(2(2(3(3(2(1(2(5(0(3(2(1(1(1(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) 5(0(3(0(0(3(1(0(5(2(3(5(5(2(2(5(3(0(5(4(5(5(0(3(0(5(0(2(5(1(0(0(3(3(0(4(0(1(5(3(5(4(1(5(1(3(1(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 5(0(3(0(3(5(3(0(5(2(0(5(5(3(1(3(5(0(0(4(5(3(2(3(0(1(1(1(5(2(0(1(3(0(5(4(0(5(0(3(5(4(1(0(2(5(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) 0(3(5(4(1(0(1(2(3(1(2(0(2(5(2(4(0(5(0(4(5(5(3(3(5(2(0(1(3(5(5(4(0(0(0(5(0(1(2(0(5(2(0(5(3(4(1(2(0(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 5(2(3(2(2(3(2(1(2(4(0(3(0(5(5(0(0(2(1(4(4(4(3(1(4(2(1(2(5(3(4(3(2(1(1(3(3(2(1(3(3(2(2(3(3(0(0(3(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))) 1(4(5(3(4(3(3(2(2(1(4(4(2(4(5(0(2(3(3(5(3(2(3(0(3(5(4(5(3(3(0(3(0(4(5(4(1(3(4(3(1(0(2(2(3(2(5(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(0(4(3(2(3(2(3(2(1(4(5(1(4(3(0(2(3(3(0(3(3(3(0(4(5(5(4(3(5(5(3(4(4(3(4(4(5(0(1(3(3(2(2(5(2(2(4(3(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_3(x_1)) -> 3(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)) 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(1(2(1(2(0(1(x1))))))) -> 2(1(0(2(1(0(1(x1))))))) 3(1(2(4(4(3(5(4(4(2(0(0(0(x1))))))))))))) -> 4(3(2(0(0(1(3(4(0(2(4(4(5(x1))))))))))))) 0(0(4(4(2(0(3(4(4(4(0(3(5(2(4(5(1(x1))))))))))))))))) -> 0(4(1(3(0(4(4(3(2(2(5(0(4(4(5(0(4(x1))))))))))))))))) 0(2(3(0(1(0(2(0(4(3(5(1(5(3(1(4(2(4(1(x1))))))))))))))))))) -> 3(1(1(4(1(0(0(0(3(2(5(1(5(4(2(3(2(4(0(x1))))))))))))))))))) 3(3(2(0(1(1(3(4(3(5(2(2(4(2(3(0(3(0(4(2(3(1(5(x1))))))))))))))))))))))) -> 0(3(2(1(0(3(3(5(3(2(0(1(4(2(4(3(3(2(1(2(3(5(4(x1))))))))))))))))))))))) 5(3(1(3(0(5(4(0(5(2(3(5(0(1(2(5(5(5(0(2(0(4(0(0(x1)))))))))))))))))))))))) -> 5(5(3(0(0(1(5(0(4(0(4(5(1(0(2(5(5(5(0(2(3(3(0(2(x1)))))))))))))))))))))))) 0(3(2(1(3(4(1(2(1(3(3(0(2(4(1(0(3(1(3(4(4(2(4(3(0(3(1(1(0(x1))))))))))))))))))))))))))))) -> 1(1(3(1(3(4(4(2(4(4(3(1(3(2(0(0(0(1(3(2(1(4(3(3(1(0(3(2(0(x1))))))))))))))))))))))))))))) 1(3(5(3(4(0(5(0(2(2(3(5(4(4(1(4(3(4(5(0(5(1(2(3(5(3(4(5(4(2(x1)))))))))))))))))))))))))))))) -> 5(3(2(3(5(4(1(0(3(3(3(4(4(0(1(4(4(5(5(2(5(1(0(5(4(2(4(5(2(3(x1)))))))))))))))))))))))))))))) 2(1(4(0(3(5(1(4(5(3(4(3(1(4(4(5(4(4(2(4(0(2(2(0(3(0(4(2(4(5(x1)))))))))))))))))))))))))))))) -> 4(4(4(5(4(2(5(1(5(4(0(3(0(3(2(3(5(4(0(2(0(3(2(4(2(4(1(4(4(1(x1)))))))))))))))))))))))))))))) 3(2(4(1(1(5(3(3(5(4(5(0(5(0(1(5(4(4(0(4(0(3(1(0(0(0(1(5(3(1(3(x1))))))))))))))))))))))))))))))) -> 3(2(3(5(3(0(3(1(0(0(5(5(4(5(1(0(4(5(0(1(1(5(3(0(4(0(1(1(4(4(3(x1))))))))))))))))))))))))))))))) 4(4(1(2(0(1(1(3(4(0(1(0(0(4(1(5(0(2(2(1(4(0(2(0(4(1(0(3(1(2(2(x1))))))))))))))))))))))))))))))) -> 0(4(0(4(2(1(1(0(2(0(3(2(0(1(1(1(0(0(4(5(2(2(1(0(4(1(3(1(4(2(4(x1))))))))))))))))))))))))))))))) 1(5(0(2(1(3(3(5(2(5(4(0(2(2(0(3(0(5(4(4(5(2(4(3(1(4(2(0(1(2(2(4(x1)))))))))))))))))))))))))))))))) -> 1(0(3(0(1(0(5(5(2(2(4(2(4(2(3(3(3(2(2(4(5(1(4(5(2(1(2(0(4(4(0(5(x1)))))))))))))))))))))))))))))))) 0(5(2(5(0(5(4(2(0(1(4(2(1(5(3(5(3(3(5(5(4(4(3(5(0(2(5(0(5(0(1(0(4(x1))))))))))))))))))))))))))))))))) -> 2(3(2(4(3(4(2(0(4(3(2(3(1(3(0(3(0(1(3(2(5(5(3(3(2(1(0(3(5(3(2(3(1(2(x1)))))))))))))))))))))))))))))))))) 4(3(0(2(2(1(4(0(0(0(5(0(3(1(4(1(5(5(3(1(5(1(5(1(5(3(1(4(3(5(2(2(3(x1))))))))))))))))))))))))))))))))) -> 0(3(0(2(2(1(0(1(3(3(5(0(3(4(5(1(5(1(0(5(1(5(2(1(4(3(1(4(5(5(2(4(3(x1))))))))))))))))))))))))))))))))) 0(3(1(2(5(0(5(5(3(4(3(5(5(5(0(0(5(4(0(0(3(3(5(5(4(4(1(1(0(4(0(4(4(5(x1)))))))))))))))))))))))))))))))))) -> 1(2(3(1(3(4(2(0(3(1(5(2(4(1(3(0(3(2(2(2(2(1(3(3(0(4(3(2(0(4(5(2(3(0(5(x1))))))))))))))))))))))))))))))))))) 1(4(5(5(4(2(0(3(0(0(5(1(4(4(5(0(0(0(1(5(2(2(2(2(2(4(4(5(4(5(5(3(5(5(x1)))))))))))))))))))))))))))))))))) -> 1(4(5(5(2(5(0(0(4(4(5(1(4(4(5(3(0(2(2(0(4(4(5(0(0(2(5(1(2(5(2(3(5(5(x1)))))))))))))))))))))))))))))))))) 3(5(1(1(0(0(5(2(0(3(4(5(3(5(3(3(5(4(3(3(3(1(3(0(1(1(1(2(5(4(5(4(2(5(x1)))))))))))))))))))))))))))))))))) -> 3(5(1(3(0(5(5(2(1(1(4(3(3(3(5(3(0(5(4(3(1(0(5(0(3(1(1(2(5(4(4(3(2(5(x1)))))))))))))))))))))))))))))))))) 3(3(4(1(4(5(0(5(1(0(2(0(5(4(1(4(1(5(0(2(5(5(5(2(0(3(5(0(0(5(4(3(0(5(3(x1))))))))))))))))))))))))))))))))))) -> 0(0(0(0(3(2(1(1(1(2(2(1(3(3(5(2(3(2(1(0(3(2(1(5(2(1(4(3(0(2(4(3(1(3(4(3(x1)))))))))))))))))))))))))))))))))))) 4(2(1(5(3(4(2(2(2(1(4(4(3(0(2(1(5(4(0(2(1(2(2(2(5(2(1(0(2(2(3(3(3(4(4(x1))))))))))))))))))))))))))))))))))) -> 4(2(5(0(0(3(2(2(2(1(4(4(2(3(2(4(5(1(5(2(2(4(3(2(0(3(1(4(2(1(2(2(1(3(4(x1))))))))))))))))))))))))))))))))))) 5(3(4(4(3(4(2(0(5(2(2(3(1(5(2(3(5(3(4(3(4(4(0(4(2(2(4(5(2(4(2(3(4(3(1(x1))))))))))))))))))))))))))))))))))) -> 2(2(4(4(2(4(4(2(1(5(2(3(3(5(5(3(5(3(4(3(2(4(0(1(0(2(4(4(3(4(5(2(4(3(3(x1))))))))))))))))))))))))))))))))))) 3(5(0(1(4(3(4(4(5(4(0(0(2(3(2(2(0(0(5(3(1(5(4(0(4(2(2(2(3(2(4(5(1(2(5(3(x1)))))))))))))))))))))))))))))))))))) -> 3(5(0(2(1(5(0(1(4(4(0(0(2(3(2(0(3(2(5(3(4(5(4(4(4(0(2(2(3(2(5(1(4(2(5(3(x1)))))))))))))))))))))))))))))))))))) 4(4(3(5(3(2(4(3(0(5(0(4(0(3(5(1(4(4(4(5(1(0(4(1(3(5(2(1(0(1(2(2(5(0(0(5(x1)))))))))))))))))))))))))))))))))))) -> 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0(0(0(0(0(1(5(2(5(0(5(1(1(0(4(4(5(5(1(5(3(2(2(0(0(4(1(5(0(3(2(0(5(5(2(0(3(0(5(0(0(5(x1)))))))))))))))))))))))))))))))))))))))))) -> 1(0(3(2(3(3(3(3(3(3(4(2(1(4(5(1(0(3(2(2(2(3(1(0(1(3(3(5(2(2(4(3(1(3(5(4(1(3(1(0(1(4(0(x1))))))))))))))))))))))))))))))))))))))))))) 1(1(0(0(5(4(5(1(5(2(3(3(5(0(1(4(5(5(5(5(0(4(3(1(1(5(4(2(1(2(2(1(0(2(2(0(5(0(1(0(1(3(x1)))))))))))))))))))))))))))))))))))))))))) -> 1(1(0(3(0(4(5(1(5(5(1(3(0(5(1(5(1(1(5(1(0(0(2(2(1(5(4(2(4(2(2(1(3(5(2(5(5(0(3(0(0(4(x1)))))))))))))))))))))))))))))))))))))))))) 4(5(0(0(0(4(0(3(0(0(0(4(5(4(1(1(3(5(4(5(2(3(4(0(0(3(5(5(1(5(4(1(4(0(4(1(5(2(4(4(5(2(x1)))))))))))))))))))))))))))))))))))))))))) -> 0(0(1(2(2(2(3(2(1(0(3(0(0(2(2(0(1(2(5(0(1(1(1(4(4(1(0(4(3(4(3(4(1(1(0(2(3(3(0(1(2(0(2(x1))))))))))))))))))))))))))))))))))))))))))) 5(2(4(4(2(4(5(4(5(1(0(5(5(4(4(4(1(4(1(3(5(2(2(0(5(0(2(0(2(4(4(5(2(2(5(0(3(1(5(0(4(4(x1)))))))))))))))))))))))))))))))))))))))))) -> 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0(0(3(1(5(0(0(4(4(0(4(5(2(5(0(0(2(2(2(3(1(5(4(0(0(5(2(5(2(5(4(5(5(5(5(5(5(3(2(2(5(5(5(5(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(1(1(3(1(2(3(3(1(2(5(4(4(2(1(2(0(2(3(1(2(1(2(2(1(3(3(4(5(2(3(4(0(1(1(3(2(1(0(5(0(2(1(3(2(x1))))))))))))))))))))))))))))))))))))))))))))) 1(3(2(5(5(3(4(0(1(4(4(2(5(0(3(3(2(5(2(0(2(4(0(1(5(0(0(5(5(0(3(0(3(1(0(1(5(3(1(4(2(0(2(5(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(3(5(5(5(3(4(1(0(4(0(2(5(1(3(3(5(2(4(3(2(4(0(2(5(0(0(5(0(3(2(0(5(1(1(3(5(0(1(4(2(0(2(1(x1)))))))))))))))))))))))))))))))))))))))))))) 4(5(5(5(0(0(4(5(4(5(3(5(4(0(2(5(3(4(5(4(5(3(1(4(5(5(3(1(5(5(4(4(5(5(3(5(5(2(4(4(0(0(4(1(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(2(0(0(2(3(1(2(5(5(5(2(0(0(2(2(3(5(5(4(4(3(4(3(1(2(3(1(4(5(1(5(1(4(0(2(0(3(0(3(0(1(1(5(3(x1))))))))))))))))))))))))))))))))))))))))))))) 0(0(4(4(5(0(0(2(4(0(1(4(5(2(5(4(3(5(3(0(4(5(2(4(4(5(0(4(1(4(0(3(0(0(5(0(4(3(0(5(3(3(5(2(3(x1))))))))))))))))))))))))))))))))))))))))))))) -> 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4(1(5(4(3(4(4(5(2(3(2(3(0(5(0(0(5(2(4(1(5(5(5(4(0(1(5(5(0(5(3(3(4(5(1(3(4(2(3(1(1(0(2(5(5(4(x1)))))))))))))))))))))))))))))))))))))))))))))) -> 4(5(5(4(3(4(4(3(2(2(2(0(4(3(0(1(4(5(4(1(0(5(5(1(5(3(0(5(0(5(3(5(5(2(1(0(3(4(3(4(1(1(2(5(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))) 4(2(1(3(0(2(3(5(0(1(3(1(4(3(0(1(3(5(0(4(2(4(1(3(1(0(4(4(2(0(0(2(1(4(4(3(2(2(5(1(4(0(2(0(2(2(5(x1))))))))))))))))))))))))))))))))))))))))))))))) -> 4(2(5(3(0(2(3(1(1(4(3(3(3(4(1(1(5(5(0(1(2(0(1(1(0(0(3(2(4(0(0(2(1(4(4(3(2(4(5(4(2(4(2(0(2(2(0(x1))))))))))))))))))))))))))))))))))))))))))))))) 5(5(3(0(5(0(1(3(5(2(2(0(0(4(3(2(1(5(0(2(3(0(0(3(4(2(3(4(1(3(5(0(1(2(3(5(1(5(3(4(0(0(4(4(5(5(5(x1))))))))))))))))))))))))))))))))))))))))))))))) -> 3(2(3(2(2(1(1(1(4(4(1(3(2(5(5(2(1(1(3(4(5(1(3(4(3(2(0(1(2(3(0(3(4(2(0(1(4(4(3(2(3(2(5(1(1(5(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))) 2(5(5(1(0(4(1(5(3(2(5(5(0(5(4(2(3(4(0(5(5(1(2(0(5(2(4(2(3(4(5(4(0(4(1(4(2(5(5(3(3(5(0(4(3(2(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 3(0(2(3(0(5(5(2(1(3(4(0(4(3(2(3(3(0(2(2(3(2(4(0(2(2(3(3(4(0(1(1(3(2(2(3(3(2(1(2(5(0(3(2(1(1(1(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) 5(0(3(0(0(3(1(0(5(2(3(5(5(2(2(5(3(0(5(4(5(5(0(3(0(5(0(2(5(1(0(0(3(3(0(4(0(1(5(3(5(4(1(5(1(3(1(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 5(0(3(0(3(5(3(0(5(2(0(5(5(3(1(3(5(0(0(4(5(3(2(3(0(1(1(1(5(2(0(1(3(0(5(4(0(5(0(3(5(4(1(0(2(5(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) 0(3(5(4(1(0(1(2(3(1(2(0(2(5(2(4(0(5(0(4(5(5(3(3(5(2(0(1(3(5(5(4(0(0(0(5(0(1(2(0(5(2(0(5(3(4(1(2(0(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 5(2(3(2(2(3(2(1(2(4(0(3(0(5(5(0(0(2(1(4(4(4(3(1(4(2(1(2(5(3(4(3(2(1(1(3(3(2(1(3(3(2(2(3(3(0(0(3(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))) 1(4(5(3(4(3(3(2(2(1(4(4(2(4(5(0(2(3(3(5(3(2(3(0(3(5(4(5(3(3(0(3(0(4(5(4(1(3(4(3(1(0(2(2(3(2(5(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(0(4(3(2(3(2(3(2(1(4(5(1(4(3(0(2(3(3(0(3(3(3(0(4(5(5(4(3(5(5(3(4(4(3(4(4(5(0(1(3(3(2(2(5(2(2(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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)) 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(1(2(1(2(0(1(x1))))))) -> 2(1(0(2(1(0(1(x1))))))) 3(1(2(4(4(3(5(4(4(2(0(0(0(x1))))))))))))) -> 4(3(2(0(0(1(3(4(0(2(4(4(5(x1))))))))))))) 0(0(4(4(2(0(3(4(4(4(0(3(5(2(4(5(1(x1))))))))))))))))) -> 0(4(1(3(0(4(4(3(2(2(5(0(4(4(5(0(4(x1))))))))))))))))) 0(2(3(0(1(0(2(0(4(3(5(1(5(3(1(4(2(4(1(x1))))))))))))))))))) -> 3(1(1(4(1(0(0(0(3(2(5(1(5(4(2(3(2(4(0(x1))))))))))))))))))) 3(3(2(0(1(1(3(4(3(5(2(2(4(2(3(0(3(0(4(2(3(1(5(x1))))))))))))))))))))))) -> 0(3(2(1(0(3(3(5(3(2(0(1(4(2(4(3(3(2(1(2(3(5(4(x1))))))))))))))))))))))) 5(3(1(3(0(5(4(0(5(2(3(5(0(1(2(5(5(5(0(2(0(4(0(0(x1)))))))))))))))))))))))) -> 5(5(3(0(0(1(5(0(4(0(4(5(1(0(2(5(5(5(0(2(3(3(0(2(x1)))))))))))))))))))))))) 0(3(2(1(3(4(1(2(1(3(3(0(2(4(1(0(3(1(3(4(4(2(4(3(0(3(1(1(0(x1))))))))))))))))))))))))))))) -> 1(1(3(1(3(4(4(2(4(4(3(1(3(2(0(0(0(1(3(2(1(4(3(3(1(0(3(2(0(x1))))))))))))))))))))))))))))) 1(3(5(3(4(0(5(0(2(2(3(5(4(4(1(4(3(4(5(0(5(1(2(3(5(3(4(5(4(2(x1)))))))))))))))))))))))))))))) -> 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1(4(5(3(4(3(3(2(2(1(4(4(2(4(5(0(2(3(3(5(3(2(3(0(3(5(4(5(3(3(0(3(0(4(5(4(1(3(4(3(1(0(2(2(3(2(5(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(0(4(3(2(3(2(3(2(1(4(5(1(4(3(0(2(3(3(0(3(3(3(0(4(5(5(4(3(5(5(3(4(4(3(4(4(5(0(1(3(3(2(2(5(2(2(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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)) 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(1(2(1(2(0(1(x1))))))) -> 2(1(0(2(1(0(1(x1))))))) 3(1(2(4(4(3(5(4(4(2(0(0(0(x1))))))))))))) -> 4(3(2(0(0(1(3(4(0(2(4(4(5(x1))))))))))))) 0(0(4(4(2(0(3(4(4(4(0(3(5(2(4(5(1(x1))))))))))))))))) -> 0(4(1(3(0(4(4(3(2(2(5(0(4(4(5(0(4(x1))))))))))))))))) 0(2(3(0(1(0(2(0(4(3(5(1(5(3(1(4(2(4(1(x1))))))))))))))))))) -> 3(1(1(4(1(0(0(0(3(2(5(1(5(4(2(3(2(4(0(x1))))))))))))))))))) 3(3(2(0(1(1(3(4(3(5(2(2(4(2(3(0(3(0(4(2(3(1(5(x1))))))))))))))))))))))) -> 0(3(2(1(0(3(3(5(3(2(0(1(4(2(4(3(3(2(1(2(3(5(4(x1))))))))))))))))))))))) 5(3(1(3(0(5(4(0(5(2(3(5(0(1(2(5(5(5(0(2(0(4(0(0(x1)))))))))))))))))))))))) -> 5(5(3(0(0(1(5(0(4(0(4(5(1(0(2(5(5(5(0(2(3(3(0(2(x1)))))))))))))))))))))))) 0(3(2(1(3(4(1(2(1(3(3(0(2(4(1(0(3(1(3(4(4(2(4(3(0(3(1(1(0(x1))))))))))))))))))))))))))))) -> 1(1(3(1(3(4(4(2(4(4(3(1(3(2(0(0(0(1(3(2(1(4(3(3(1(0(3(2(0(x1))))))))))))))))))))))))))))) 1(3(5(3(4(0(5(0(2(2(3(5(4(4(1(4(3(4(5(0(5(1(2(3(5(3(4(5(4(2(x1)))))))))))))))))))))))))))))) -> 5(3(2(3(5(4(1(0(3(3(3(4(4(0(1(4(4(5(5(2(5(1(0(5(4(2(4(5(2(3(x1)))))))))))))))))))))))))))))) 2(1(4(0(3(5(1(4(5(3(4(3(1(4(4(5(4(4(2(4(0(2(2(0(3(0(4(2(4(5(x1)))))))))))))))))))))))))))))) -> 4(4(4(5(4(2(5(1(5(4(0(3(0(3(2(3(5(4(0(2(0(3(2(4(2(4(1(4(4(1(x1)))))))))))))))))))))))))))))) 3(2(4(1(1(5(3(3(5(4(5(0(5(0(1(5(4(4(0(4(0(3(1(0(0(0(1(5(3(1(3(x1))))))))))))))))))))))))))))))) -> 3(2(3(5(3(0(3(1(0(0(5(5(4(5(1(0(4(5(0(1(1(5(3(0(4(0(1(1(4(4(3(x1))))))))))))))))))))))))))))))) 4(4(1(2(0(1(1(3(4(0(1(0(0(4(1(5(0(2(2(1(4(0(2(0(4(1(0(3(1(2(2(x1))))))))))))))))))))))))))))))) -> 0(4(0(4(2(1(1(0(2(0(3(2(0(1(1(1(0(0(4(5(2(2(1(0(4(1(3(1(4(2(4(x1))))))))))))))))))))))))))))))) 1(5(0(2(1(3(3(5(2(5(4(0(2(2(0(3(0(5(4(4(5(2(4(3(1(4(2(0(1(2(2(4(x1)))))))))))))))))))))))))))))))) -> 1(0(3(0(1(0(5(5(2(2(4(2(4(2(3(3(3(2(2(4(5(1(4(5(2(1(2(0(4(4(0(5(x1)))))))))))))))))))))))))))))))) 0(5(2(5(0(5(4(2(0(1(4(2(1(5(3(5(3(3(5(5(4(4(3(5(0(2(5(0(5(0(1(0(4(x1))))))))))))))))))))))))))))))))) -> 2(3(2(4(3(4(2(0(4(3(2(3(1(3(0(3(0(1(3(2(5(5(3(3(2(1(0(3(5(3(2(3(1(2(x1)))))))))))))))))))))))))))))))))) 4(3(0(2(2(1(4(0(0(0(5(0(3(1(4(1(5(5(3(1(5(1(5(1(5(3(1(4(3(5(2(2(3(x1))))))))))))))))))))))))))))))))) -> 0(3(0(2(2(1(0(1(3(3(5(0(3(4(5(1(5(1(0(5(1(5(2(1(4(3(1(4(5(5(2(4(3(x1))))))))))))))))))))))))))))))))) 0(3(1(2(5(0(5(5(3(4(3(5(5(5(0(0(5(4(0(0(3(3(5(5(4(4(1(1(0(4(0(4(4(5(x1)))))))))))))))))))))))))))))))))) -> 1(2(3(1(3(4(2(0(3(1(5(2(4(1(3(0(3(2(2(2(2(1(3(3(0(4(3(2(0(4(5(2(3(0(5(x1))))))))))))))))))))))))))))))))))) 1(4(5(5(4(2(0(3(0(0(5(1(4(4(5(0(0(0(1(5(2(2(2(2(2(4(4(5(4(5(5(3(5(5(x1)))))))))))))))))))))))))))))))))) -> 1(4(5(5(2(5(0(0(4(4(5(1(4(4(5(3(0(2(2(0(4(4(5(0(0(2(5(1(2(5(2(3(5(5(x1)))))))))))))))))))))))))))))))))) 3(5(1(1(0(0(5(2(0(3(4(5(3(5(3(3(5(4(3(3(3(1(3(0(1(1(1(2(5(4(5(4(2(5(x1)))))))))))))))))))))))))))))))))) -> 3(5(1(3(0(5(5(2(1(1(4(3(3(3(5(3(0(5(4(3(1(0(5(0(3(1(1(2(5(4(4(3(2(5(x1)))))))))))))))))))))))))))))))))) 3(3(4(1(4(5(0(5(1(0(2(0(5(4(1(4(1(5(0(2(5(5(5(2(0(3(5(0(0(5(4(3(0(5(3(x1))))))))))))))))))))))))))))))))))) -> 0(0(0(0(3(2(1(1(1(2(2(1(3(3(5(2(3(2(1(0(3(2(1(5(2(1(4(3(0(2(4(3(1(3(4(3(x1)))))))))))))))))))))))))))))))))))) 4(2(1(5(3(4(2(2(2(1(4(4(3(0(2(1(5(4(0(2(1(2(2(2(5(2(1(0(2(2(3(3(3(4(4(x1))))))))))))))))))))))))))))))))))) -> 4(2(5(0(0(3(2(2(2(1(4(4(2(3(2(4(5(1(5(2(2(4(3(2(0(3(1(4(2(1(2(2(1(3(4(x1))))))))))))))))))))))))))))))))))) 5(3(4(4(3(4(2(0(5(2(2(3(1(5(2(3(5(3(4(3(4(4(0(4(2(2(4(5(2(4(2(3(4(3(1(x1))))))))))))))))))))))))))))))))))) -> 2(2(4(4(2(4(4(2(1(5(2(3(3(5(5(3(5(3(4(3(2(4(0(1(0(2(4(4(3(4(5(2(4(3(3(x1))))))))))))))))))))))))))))))))))) 3(5(0(1(4(3(4(4(5(4(0(0(2(3(2(2(0(0(5(3(1(5(4(0(4(2(2(2(3(2(4(5(1(2(5(3(x1)))))))))))))))))))))))))))))))))))) -> 3(5(0(2(1(5(0(1(4(4(0(0(2(3(2(0(3(2(5(3(4(5(4(4(4(0(2(2(3(2(5(1(4(2(5(3(x1)))))))))))))))))))))))))))))))))))) 4(4(3(5(3(2(4(3(0(5(0(4(0(3(5(1(4(4(4(5(1(0(4(1(3(5(2(1(0(1(2(2(5(0(0(5(x1)))))))))))))))))))))))))))))))))))) -> 4(4(3(5(3(2(5(3(1(5(0(0(5(0(5(3(0(0(1(5(1(1(4(4(4(4(2(4(0(1(2(2(4(3(0(5(x1)))))))))))))))))))))))))))))))))))) 3(5(1(2(5(4(5(5(4(5(2(2(4(5(1(0(0(5(2(0(2(2(3(3(4(2(4(1(1(1(0(0(0(1(5(1(3(x1))))))))))))))))))))))))))))))))))))) -> 0(3(3(4(1(3(1(4(3(3(1(1(0(1(2(3(3(2(0(4(2(3(3(5(3(2(2(3(2(2(1(3(3(1(5(4(1(3(x1)))))))))))))))))))))))))))))))))))))) 3(4(3(2(0(1(3(3(3(0(1(2(1(0(0(5(0(5(2(2(1(5(4(5(1(2(4(3(3(0(1(2(3(5(3(4(2(0(x1)))))))))))))))))))))))))))))))))))))) -> 3(2(0(3(4(1(2(1(3(4(1(3(0(5(5(1(3(3(2(3(2(0(3(5(5(2(4(3(0(0(1(2(0(5(1(4(2(0(x1)))))))))))))))))))))))))))))))))))))) 4(0(5(4(3(5(3(5(4(0(4(5(0(3(5(5(2(2(5(5(0(3(0(4(4(5(2(5(5(1(2(0(4(5(5(5(5(3(0(3(x1)))))))))))))))))))))))))))))))))))))))) -> 2(0(5(1(2(2(4(3(3(2(1(4(2(3(4(2(1(4(1(4(4(5(3(4(5(5(2(4(2(0(2(2(3(0(0(3(2(3(2(4(3(x1))))))))))))))))))))))))))))))))))))))))) 4(4(2(4(3(3(0(2(5(1(1(0(4(3(5(4(1(4(0(1(0(2(4(4(0(5(3(3(1(3(2(5(0(5(0(4(1(5(1(4(x1)))))))))))))))))))))))))))))))))))))))) -> 4(1(2(2(3(3(0(1(5(3(2(0(4(4(5(1(1(4(0(4(0(0(3(4(1(2(5(3(4(4(4(1(0(5(0(4(3(5(5(1(x1)))))))))))))))))))))))))))))))))))))))) 4(5(2(2(3(3(4(0(2(3(1(3(0(3(4(3(0(5(0(3(4(0(0(1(2(2(2(1(4(4(0(3(2(2(4(2(4(0(1(4(x1)))))))))))))))))))))))))))))))))))))))) -> 4(2(2(1(3(3(2(0(5(3(2(3(0(1(0(3(4(2(2(1(4(3(5(2(4(4(2(0(4(0(0(0(3(2(4(3(4(0(1(4(x1)))))))))))))))))))))))))))))))))))))))) 1(5(4(2(4(4(1(3(5(3(4(0(4(5(1(4(4(5(5(4(5(4(5(5(2(3(1(2(0(5(5(1(4(2(4(0(4(5(4(4(0(x1))))))))))))))))))))))))))))))))))))))))) -> 5(0(0(1(5(0(5(4(2(1(2(2(0(5(3(1(4(0(2(3(2(3(2(3(2(1(1(1(4(3(0(0(3(0(3(0(5(3(3(4(2(2(x1)))))))))))))))))))))))))))))))))))))))))) 5(2(4(5(5(2(5(0(2(3(3(0(0(1(4(1(2(5(5(2(0(5(1(0(3(5(1(2(5(4(0(4(2(5(5(4(4(5(4(2(0(x1))))))))))))))))))))))))))))))))))))))))) -> 2(2(3(3(1(2(5(3(2(3(0(3(0(3(0(3(5(1(2(5(1(3(1(2(5(2(0(2(2(2(1(3(2(4(3(1(4(0(5(5(4(1(x1)))))))))))))))))))))))))))))))))))))))))) 0(0(0(0(0(1(5(2(5(0(5(1(1(0(4(4(5(5(1(5(3(2(2(0(0(4(1(5(0(3(2(0(5(5(2(0(3(0(5(0(0(5(x1)))))))))))))))))))))))))))))))))))))))))) -> 1(0(3(2(3(3(3(3(3(3(4(2(1(4(5(1(0(3(2(2(2(3(1(0(1(3(3(5(2(2(4(3(1(3(5(4(1(3(1(0(1(4(0(x1))))))))))))))))))))))))))))))))))))))))))) 1(1(0(0(5(4(5(1(5(2(3(3(5(0(1(4(5(5(5(5(0(4(3(1(1(5(4(2(1(2(2(1(0(2(2(0(5(0(1(0(1(3(x1)))))))))))))))))))))))))))))))))))))))))) -> 1(1(0(3(0(4(5(1(5(5(1(3(0(5(1(5(1(1(5(1(0(0(2(2(1(5(4(2(4(2(2(1(3(5(2(5(5(0(3(0(0(4(x1)))))))))))))))))))))))))))))))))))))))))) 4(5(0(0(0(4(0(3(0(0(0(4(5(4(1(1(3(5(4(5(2(3(4(0(0(3(5(5(1(5(4(1(4(0(4(1(5(2(4(4(5(2(x1)))))))))))))))))))))))))))))))))))))))))) -> 0(0(1(2(2(2(3(2(1(0(3(0(0(2(2(0(1(2(5(0(1(1(1(4(4(1(0(4(3(4(3(4(1(1(0(2(3(3(0(1(2(0(2(x1))))))))))))))))))))))))))))))))))))))))))) 5(2(4(4(2(4(5(4(5(1(0(5(5(4(4(4(1(4(1(3(5(2(2(0(5(0(2(0(2(4(4(5(2(2(5(0(3(1(5(0(4(4(x1)))))))))))))))))))))))))))))))))))))))))) -> 2(0(4(3(2(3(1(2(5(5(4(1(0(0(1(5(4(1(0(2(2(3(5(1(4(2(1(5(0(3(2(3(3(3(4(1(0(4(3(0(3(2(1(x1))))))))))))))))))))))))))))))))))))))))))) 0(3(2(4(3(4(4(0(3(1(0(5(5(4(5(4(5(1(0(5(2(2(5(2(5(0(3(4(0(4(0(5(3(0(0(2(4(2(5(2(5(0(1(x1))))))))))))))))))))))))))))))))))))))))))) -> 4(3(1(1(3(3(1(3(3(4(3(3(2(5(2(1(0(3(2(0(5(1(2(1(5(0(0(3(0(3(1(4(1(5(3(3(2(0(1(5(2(2(4(1(x1)))))))))))))))))))))))))))))))))))))))))))) 0(5(4(5(1(3(4(1(4(0(0(4(1(1(4(3(5(4(5(4(3(4(5(3(5(4(3(3(0(0(4(5(3(1(0(3(2(5(0(5(0(4(1(x1))))))))))))))))))))))))))))))))))))))))))) -> 0(5(5(5(1(3(4(4(4(3(0(4(4(3(4(0(3(4(5(5(5(1(3(3(4(5(1(4(3(1(4(1(0(0(0(4(2(5(3(5(0(0(1(x1))))))))))))))))))))))))))))))))))))))))))) 5(2(3(3(5(1(2(5(5(5(2(4(5(2(5(5(2(1(0(5(0(0(0(0(0(2(4(0(5(2(2(3(3(3(0(0(2(3(1(3(4(0(4(x1))))))))))))))))))))))))))))))))))))))))))) -> 3(1(3(0(5(4(2(2(0(5(0(4(5(5(2(5(0(5(1(5(3(0(0(2(0(2(4(2(5(5(3(3(3(2(0(5(3(1(0(2(0(2(4(x1))))))))))))))))))))))))))))))))))))))))))) 0(0(3(1(5(0(0(4(4(0(4(5(2(5(0(0(2(2(2(3(1(5(4(0(0(5(2(5(2(5(4(5(5(5(5(5(5(3(2(2(5(5(5(5(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(1(1(3(1(2(3(3(1(2(5(4(4(2(1(2(0(2(3(1(2(1(2(2(1(3(3(4(5(2(3(4(0(1(1(3(2(1(0(5(0(2(1(3(2(x1))))))))))))))))))))))))))))))))))))))))))))) 1(3(2(5(5(3(4(0(1(4(4(2(5(0(3(3(2(5(2(0(2(4(0(1(5(0(0(5(5(0(3(0(3(1(0(1(5(3(1(4(2(0(2(5(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(3(5(5(5(3(4(1(0(4(0(2(5(1(3(3(5(2(4(3(2(4(0(2(5(0(0(5(0(3(2(0(5(1(1(3(5(0(1(4(2(0(2(1(x1)))))))))))))))))))))))))))))))))))))))))))) 4(5(5(5(0(0(4(5(4(5(3(5(4(0(2(5(3(4(5(4(5(3(1(4(5(5(3(1(5(5(4(4(5(5(3(5(5(2(4(4(0(0(4(1(x1)))))))))))))))))))))))))))))))))))))))))))) -> 0(2(0(0(2(3(1(2(5(5(5(2(0(0(2(2(3(5(5(4(4(3(4(3(1(2(3(1(4(5(1(5(1(4(0(2(0(3(0(3(0(1(1(5(3(x1))))))))))))))))))))))))))))))))))))))))))))) 0(0(4(4(5(0(0(2(4(0(1(4(5(2(5(4(3(5(3(0(4(5(2(4(4(5(0(4(1(4(0(3(0(0(5(0(4(3(0(5(3(3(5(2(3(x1))))))))))))))))))))))))))))))))))))))))))))) -> 4(3(5(5(2(2(1(1(0(3(4(3(3(2(3(4(2(2(1(1(4(2(1(5(2(2(0(1(2(3(3(1(5(2(3(1(3(3(3(2(4(5(0(2(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))) 1(4(4(1(5(5(4(0(0(4(5(0(4(3(1(2(1(5(4(4(0(0(5(0(0(5(4(3(3(4(2(2(4(5(2(5(5(2(0(4(2(0(5(3(4(x1))))))))))))))))))))))))))))))))))))))))))))) -> 0(1(4(2(4(1(5(3(1(3(4(1(5(2(2(3(0(3(3(0(3(4(0(1(3(4(2(1(3(0(1(2(1(1(3(2(1(2(2(5(0(2(0(0(4(2(x1)))))))))))))))))))))))))))))))))))))))))))))) 5(0(5(4(1(2(4(3(5(5(2(5(4(3(5(1(3(5(5(1(4(0(0(2(4(0(1(2(5(0(5(0(5(5(3(0(1(3(1(0(0(4(0(5(5(x1))))))))))))))))))))))))))))))))))))))))))))) -> 2(3(0(4(5(0(3(1(0(4(1(2(1(0(2(3(3(1(5(4(5(2(1(1(3(3(0(2(4(1(3(3(3(2(4(5(2(3(0(3(0(1(4(1(0(1(x1)))))))))))))))))))))))))))))))))))))))))))))) 0(4(3(5(5(2(2(0(2(2(1(5(2(1(2(0(0(3(0(0(2(3(0(5(0(0(3(3(4(0(5(3(0(1(0(2(3(5(2(5(0(4(1(5(1(5(x1)))))))))))))))))))))))))))))))))))))))))))))) -> 4(4(3(0(5(5(5(1(5(2(2(0(5(0(3(0(0(3(2(3(2(2(0(5(0(2(1(3(1(2(5(2(0(0(3(0(3(1(2(5(0(4(0(5(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))) 4(1(5(4(3(4(4(5(2(3(2(3(0(5(0(0(5(2(4(1(5(5(5(4(0(1(5(5(0(5(3(3(4(5(1(3(4(2(3(1(1(0(2(5(5(4(x1)))))))))))))))))))))))))))))))))))))))))))))) -> 4(5(5(4(3(4(4(3(2(2(2(0(4(3(0(1(4(5(4(1(0(5(5(1(5(3(0(5(0(5(3(5(5(2(1(0(3(4(3(4(1(1(2(5(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))) 4(2(1(3(0(2(3(5(0(1(3(1(4(3(0(1(3(5(0(4(2(4(1(3(1(0(4(4(2(0(0(2(1(4(4(3(2(2(5(1(4(0(2(0(2(2(5(x1))))))))))))))))))))))))))))))))))))))))))))))) -> 4(2(5(3(0(2(3(1(1(4(3(3(3(4(1(1(5(5(0(1(2(0(1(1(0(0(3(2(4(0(0(2(1(4(4(3(2(4(5(4(2(4(2(0(2(2(0(x1))))))))))))))))))))))))))))))))))))))))))))))) 5(5(3(0(5(0(1(3(5(2(2(0(0(4(3(2(1(5(0(2(3(0(0(3(4(2(3(4(1(3(5(0(1(2(3(5(1(5(3(4(0(0(4(4(5(5(5(x1))))))))))))))))))))))))))))))))))))))))))))))) -> 3(2(3(2(2(1(1(1(4(4(1(3(2(5(5(2(1(1(3(4(5(1(3(4(3(2(0(1(2(3(0(3(4(2(0(1(4(4(3(2(3(2(5(1(1(5(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))) 2(5(5(1(0(4(1(5(3(2(5(5(0(5(4(2(3(4(0(5(5(1(2(0(5(2(4(2(3(4(5(4(0(4(1(4(2(5(5(3(3(5(0(4(3(2(5(5(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 3(0(2(3(0(5(5(2(1(3(4(0(4(3(2(3(3(0(2(2(3(2(4(0(2(2(3(3(4(0(1(1(3(2(2(3(3(2(1(2(5(0(3(2(1(1(1(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) 5(0(3(0(0(3(1(0(5(2(3(5(5(2(2(5(3(0(5(4(5(5(0(3(0(5(0(2(5(1(0(0(3(3(0(4(0(1(5(3(5(4(1(5(1(3(1(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) -> 5(0(3(0(3(5(3(0(5(2(0(5(5(3(1(3(5(0(0(4(5(3(2(3(0(1(1(1(5(2(0(1(3(0(5(4(0(5(0(3(5(4(1(0(2(5(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))) 0(3(5(4(1(0(1(2(3(1(2(0(2(5(2(4(0(5(0(4(5(5(3(3(5(2(0(1(3(5(5(4(0(0(0(5(0(1(2(0(5(2(0(5(3(4(1(2(0(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 5(2(3(2(2(3(2(1(2(4(0(3(0(5(5(0(0(2(1(4(4(4(3(1(4(2(1(2(5(3(4(3(2(1(1(3(3(2(1(3(3(2(2(3(3(0(0(3(5(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))) 1(4(5(3(4(3(3(2(2(1(4(4(2(4(5(0(2(3(3(5(3(2(3(0(3(5(4(5(3(3(0(3(0(4(5(4(1(3(4(3(1(0(2(2(3(2(5(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) -> 1(0(4(3(2(3(2(3(2(1(4(5(1(4(3(0(2(3(3(0(3(3(3(0(4(5(5(4(3(5(5(3(4(4(3(4(4(5(0(1(3(3(2(2(5(2(2(4(3(x1))))))))))))))))))))))))))))))))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(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)) 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 2. The certificate found is represented by the following graph. 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(1792,1793,[0_1|2]), (1793,1794,[2_1|2]), (1794,1795,[2_1|2]), (1795,1796,[3_1|2]), (1796,1797,[0_1|2]), (1797,1798,[0_1|2]), (1798,1799,[3_1|2]), (1799,1800,[2_1|2]), (1800,1801,[3_1|2]), (1801,1802,[2_1|2]), (1802,1803,[4_1|2]), (1802,1652,[0_1|2]), (1803,152,[3_1|2]), (1803,306,[3_1|2]), (1803,667,[3_1|2]), (1803,697,[3_1|2]), (1803,767,[3_1|2]), (1803,802,[3_1|2]), (1803,979,[3_1|2]), (1803,1113,[3_1|2]), (1803,1500,[3_1|2]), (1803,611,[3_1|2]), (1803,731,[3_1|2]), (1803,1190,[3_1|2]), (1803,1653,[3_1|2]), (1803,598,[4_1|2]), (1803,610,[0_1|2]), (1803,632,[0_1|2]), (1803,730,[0_1|2]), (1804,1805,[2_1|2]), (1805,1806,[2_1|2]), (1806,1807,[1_1|2]), (1807,1808,[3_1|2]), (1808,1809,[3_1|2]), (1809,1810,[2_1|2]), (1810,1811,[0_1|2]), (1811,1812,[5_1|2]), (1812,1813,[3_1|2]), (1813,1814,[2_1|2]), (1814,1815,[3_1|2]), (1815,1816,[0_1|2]), (1816,1817,[1_1|2]), (1817,1818,[0_1|2]), (1818,1819,[3_1|2]), (1819,1820,[4_1|2]), (1820,1821,[2_1|2]), (1821,1822,[2_1|2]), (1822,1823,[1_1|2]), (1823,1824,[4_1|2]), (1824,1825,[3_1|2]), (1825,1826,[5_1|2]), (1826,1827,[2_1|2]), (1827,1828,[4_1|2]), (1828,1829,[4_1|2]), (1829,1830,[2_1|2]), (1830,1831,[0_1|2]), (1831,1832,[4_1|2]), (1832,1833,[0_1|2]), (1833,1834,[0_1|2]), (1834,1835,[0_1|2]), (1835,1836,[3_1|2]), (1836,1837,[2_1|2]), (1837,1838,[4_1|2]), (1838,1839,[3_1|2]), (1839,1840,[4_1|2]), (1840,1841,[0_1|2]), (1841,1842,[1_1|2]), (1841,1304,[1_1|2]), (1841,1337,[1_1|2]), (1841,1385,[0_1|2]), (1842,152,[4_1|2]), (1842,175,[4_1|2]), (1842,352,[4_1|2]), (1842,553,[4_1|2]), (1842,598,[4_1|2]), (1842,1471,[4_1|2]), (1842,1578,[4_1|2]), (1842,1613,[4_1|2]), (1842,1684,[4_1|2]), (1842,1718,[4_1|2]), (1842,1804,[4_1|2]), (1842,1929,[4_1|2]), (1842,1305,[4_1|2]), (1842,1387,[4_1|2]), (1842,1548,[0_1|2]), (1842,1652,[0_1|2]), (1842,1764,[2_1|2]), (1842,1843,[0_1|2]), (1842,1885,[0_1|2]), (1843,1844,[0_1|2]), (1844,1845,[1_1|2]), (1845,1846,[2_1|2]), (1846,1847,[2_1|2]), (1847,1848,[2_1|2]), (1848,1849,[3_1|2]), (1849,1850,[2_1|2]), (1850,1851,[1_1|2]), (1851,1852,[0_1|2]), (1852,1853,[3_1|2]), (1853,1854,[0_1|2]), (1854,1855,[0_1|2]), (1855,1856,[2_1|2]), (1856,1857,[2_1|2]), (1857,1858,[0_1|2]), (1858,1859,[1_1|2]), (1859,1860,[2_1|2]), (1860,1861,[5_1|2]), (1861,1862,[0_1|2]), (1862,1863,[1_1|2]), (1863,1864,[1_1|2]), (1864,1865,[1_1|2]), (1865,1866,[4_1|2]), (1866,1867,[4_1|2]), (1867,1868,[1_1|2]), (1868,1869,[0_1|2]), (1869,1870,[4_1|2]), (1870,1871,[3_1|2]), (1871,1872,[4_1|2]), (1872,1873,[3_1|2]), (1873,1874,[4_1|2]), (1874,1875,[1_1|2]), (1875,1876,[1_1|2]), (1876,1877,[0_1|2]), (1877,1878,[2_1|2]), (1878,1879,[3_1|2]), (1879,1880,[3_1|2]), (1880,1881,[0_1|2]), (1881,1882,[1_1|2]), (1882,1883,[2_1|2]), (1883,1884,[0_1|2]), (1883,306,[3_1|2]), (1884,152,[2_1|2]), (1884,153,[2_1|2]), (1884,478,[2_1|2]), (1884,862,[2_1|2]), (1884,896,[2_1|2]), (1884,937,[2_1|2]), (1884,1021,[2_1|2]), (1884,1764,[2_1|2]), (1884,430,[2_1|2]), (1884,1471,[4_1|2]), (1884,1500,[3_1|2]), (1885,1886,[2_1|2]), (1886,1887,[0_1|2]), (1887,1888,[0_1|2]), (1888,1889,[2_1|2]), (1889,1890,[3_1|2]), (1890,1891,[1_1|2]), (1891,1892,[2_1|2]), (1892,1893,[5_1|2]), (1893,1894,[5_1|2]), (1894,1895,[5_1|2]), (1895,1896,[2_1|2]), (1896,1897,[0_1|2]), (1897,1898,[0_1|2]), (1898,1899,[2_1|2]), (1899,1900,[2_1|2]), (1900,1901,[3_1|2]), (1901,1902,[5_1|2]), (1902,1903,[5_1|2]), (1903,1904,[4_1|2]), (1904,1905,[4_1|2]), (1905,1906,[3_1|2]), (1906,1907,[4_1|2]), (1907,1908,[3_1|2]), (1908,1909,[1_1|2]), (1909,1910,[2_1|2]), (1910,1911,[3_1|2]), (1911,1912,[1_1|2]), (1912,1913,[4_1|2]), (1913,1914,[5_1|2]), (1914,1915,[1_1|2]), (1915,1916,[5_1|2]), (1916,1917,[1_1|2]), (1917,1918,[4_1|2]), (1918,1919,[0_1|2]), (1919,1920,[2_1|2]), (1920,1921,[0_1|2]), (1921,1922,[3_1|2]), (1922,1923,[0_1|2]), (1923,1924,[3_1|2]), (1924,1925,[0_1|2]), (1925,1926,[1_1|2]), (1926,1927,[1_1|2]), (1927,1928,[5_1|2]), (1927,839,[5_1|2]), (1927,862,[2_1|2]), (1928,152,[3_1|2]), (1928,220,[3_1|2]), (1928,324,[3_1|2]), (1928,395,[3_1|2]), (1928,1232,[3_1|2]), (1928,1304,[3_1|2]), (1928,1337,[3_1|2]), (1928,1430,[3_1|2]), (1928,1614,[3_1|2]), (1928,161,[3_1|2]), (1928,598,[4_1|2]), (1928,610,[0_1|2]), (1928,632,[0_1|2]), (1928,667,[3_1|2]), (1928,697,[3_1|2]), (1928,730,[0_1|2]), (1928,767,[3_1|2]), (1928,802,[3_1|2]), (1929,1930,[5_1|2]), (1930,1931,[5_1|2]), (1931,1932,[4_1|2]), (1932,1933,[3_1|2]), (1933,1934,[4_1|2]), (1934,1935,[4_1|2]), (1935,1936,[3_1|2]), (1936,1937,[2_1|2]), (1937,1938,[2_1|2]), (1938,1939,[2_1|2]), (1939,1940,[0_1|2]), (1940,1941,[4_1|2]), (1941,1942,[3_1|2]), (1942,1943,[0_1|2]), (1943,1944,[1_1|2]), (1944,1945,[4_1|2]), (1945,1946,[5_1|2]), (1946,1947,[4_1|2]), (1947,1948,[1_1|2]), (1948,1949,[0_1|2]), (1949,1950,[5_1|2]), (1950,1951,[5_1|2]), (1951,1952,[1_1|2]), (1952,1953,[5_1|2]), (1953,1954,[3_1|2]), (1954,1955,[0_1|2]), (1955,1956,[5_1|2]), (1956,1957,[0_1|2]), (1957,1958,[5_1|2]), (1958,1959,[3_1|2]), (1959,1960,[5_1|2]), (1960,1961,[5_1|2]), (1961,1962,[2_1|2]), (1962,1963,[1_1|2]), (1963,1964,[0_1|2]), (1964,1965,[3_1|2]), (1965,1966,[4_1|2]), (1966,1967,[3_1|2]), (1967,1968,[4_1|2]), (1968,1969,[1_1|2]), (1969,1970,[1_1|2]), (1970,1971,[2_1|2]), (1971,1972,[5_1|2]), (1972,1973,[5_1|2]), (1972,1113,[3_1|2]), (1973,152,[5_1|2]), (1973,175,[5_1|2]), (1973,352,[5_1|2]), (1973,553,[5_1|2]), (1973,598,[5_1|2]), (1973,1471,[5_1|2]), (1973,1578,[5_1|2]), (1973,1613,[5_1|2]), (1973,1684,[5_1|2]), (1973,1718,[5_1|2]), (1973,1804,[5_1|2]), (1973,1929,[5_1|2]), (1973,839,[5_1|2]), (1973,862,[2_1|2]), (1973,896,[2_1|2]), (1973,937,[2_1|2]), (1973,979,[3_1|2]), (1973,1021,[2_1|2]), (1973,1066,[5_1|2]), (1973,1113,[3_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)