/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 52 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 145 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: 0(0(1(1(x1)))) -> 0(2(1(x1))) 0(0(3(3(0(0(x1)))))) -> 0(0(4(0(4(0(x1)))))) 1(5(5(2(1(0(x1)))))) -> 1(5(4(3(1(0(x1)))))) 4(0(3(4(2(2(x1)))))) -> 4(5(2(2(3(x1))))) 2(2(4(1(3(4(1(x1))))))) -> 5(2(4(5(3(x1))))) 0(3(1(5(2(3(1(5(x1)))))))) -> 0(3(3(5(3(5(1(5(x1)))))))) 3(1(5(1(5(0(4(2(x1)))))))) -> 3(1(4(5(0(5(2(x1))))))) 3(0(0(2(2(0(4(4(0(x1))))))))) -> 2(5(5(0(2(0(2(0(x1)))))))) 3(0(1(3(1(5(1(4(2(x1))))))))) -> 1(1(2(4(2(4(1(0(2(x1))))))))) 3(0(4(4(0(2(4(4(4(x1))))))))) -> 2(5(5(0(3(3(4(x1))))))) 0(2(5(2(4(3(4(0(4(3(x1)))))))))) -> 0(3(5(3(1(3(4(0(3(x1))))))))) 1(4(5(3(1(1(1(0(3(2(x1)))))))))) -> 1(3(2(5(3(5(2(1(x1)))))))) 2(4(0(4(5(3(3(3(2(0(x1)))))))))) -> 0(3(1(5(4(5(5(0(x1)))))))) 1(3(2(1(5(1(1(0(5(2(2(x1))))))))))) -> 1(0(2(5(4(2(2(0(5(3(x1)))))))))) 3(2(4(0(1(0(0(2(0(1(3(x1))))))))))) -> 2(4(0(3(5(5(1(3(3(x1))))))))) 4(4(1(0(1(3(3(1(4(1(2(x1))))))))))) -> 3(1(2(4(4(4(2(1(2(2(x1)))))))))) 4(0(4(2(2(0(4(2(1(4(1(4(x1)))))))))))) -> 4(5(0(5(4(5(5(4(x1)))))))) 4(3(5(4(3(4(0(0(4(2(5(0(x1)))))))))))) -> 4(0(2(0(3(3(3(5(3(1(2(4(x1)))))))))))) 2(5(5(1(5(1(2(0(2(1(3(3(4(x1))))))))))))) -> 3(1(1(1(2(5(2(1(4(0(0(4(4(x1))))))))))))) 3(4(1(1(3(1(1(1(5(4(3(4(1(x1))))))))))))) -> 3(4(0(0(3(5(3(5(0(4(4(4(x1)))))))))))) 1(1(2(0(2(4(1(1(3(3(3(5(1(3(x1)))))))))))))) -> 5(1(3(4(5(1(1(0(3(4(0(2(0(x1))))))))))))) 1(4(0(0(5(2(2(5(2(2(3(0(2(5(x1)))))))))))))) -> 1(5(4(2(3(0(1(3(0(3(2(2(0(5(x1)))))))))))))) 2(2(0(2(0(0(5(3(2(3(2(0(3(2(x1)))))))))))))) -> 4(5(1(5(3(3(3(4(4(2(0(4(x1)))))))))))) 4(2(3(5(4(2(5(5(1(1(4(4(0(4(1(0(x1)))))))))))))))) -> 4(1(2(5(4(0(5(3(2(5(0(4(2(4(0(x1))))))))))))))) 5(0(3(4(0(0(0(4(3(4(2(4(3(3(2(0(4(x1))))))))))))))))) -> 5(2(3(3(1(2(0(4(3(0(1(5(5(2(3(4(x1)))))))))))))))) 5(5(4(4(0(5(5(2(4(0(5(1(2(3(2(4(0(x1))))))))))))))))) -> 0(3(2(0(4(3(1(2(4(5(3(3(0(1(1(5(1(3(x1)))))))))))))))))) 0(1(4(3(5(3(0(4(1(1(2(3(3(1(4(0(5(2(3(x1))))))))))))))))))) -> 0(3(0(0(5(4(5(0(4(0(2(2(1(4(0(0(0(4(x1)))))))))))))))))) 4(2(1(0(0(4(4(4(3(0(5(1(3(1(2(3(5(1(0(2(x1)))))))))))))))))))) -> 4(4(1(4(5(0(4(2(4(5(2(3(4(2(2(5(5(1(x1)))))))))))))))))) 4(4(4(1(4(5(2(2(0(1(4(5(2(2(1(4(5(0(0(4(x1)))))))))))))))))))) -> 2(1(2(1(4(0(4(5(3(2(5(0(3(2(0(2(2(5(1(4(x1)))))))))))))))))))) 5(3(2(0(4(1(4(1(4(2(5(2(3(4(4(4(3(2(5(2(4(x1))))))))))))))))))))) -> 5(0(4(3(0(0(2(1(5(4(4(2(5(3(4(3(0(5(5(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_4(x_1)) -> 4(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_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). The TRS R consists of the following rules: 0(0(1(1(x1)))) -> 0(2(1(x1))) 0(0(3(3(0(0(x1)))))) -> 0(0(4(0(4(0(x1)))))) 1(5(5(2(1(0(x1)))))) -> 1(5(4(3(1(0(x1)))))) 4(0(3(4(2(2(x1)))))) -> 4(5(2(2(3(x1))))) 2(2(4(1(3(4(1(x1))))))) -> 5(2(4(5(3(x1))))) 0(3(1(5(2(3(1(5(x1)))))))) -> 0(3(3(5(3(5(1(5(x1)))))))) 3(1(5(1(5(0(4(2(x1)))))))) -> 3(1(4(5(0(5(2(x1))))))) 3(0(0(2(2(0(4(4(0(x1))))))))) -> 2(5(5(0(2(0(2(0(x1)))))))) 3(0(1(3(1(5(1(4(2(x1))))))))) -> 1(1(2(4(2(4(1(0(2(x1))))))))) 3(0(4(4(0(2(4(4(4(x1))))))))) -> 2(5(5(0(3(3(4(x1))))))) 0(2(5(2(4(3(4(0(4(3(x1)))))))))) -> 0(3(5(3(1(3(4(0(3(x1))))))))) 1(4(5(3(1(1(1(0(3(2(x1)))))))))) -> 1(3(2(5(3(5(2(1(x1)))))))) 2(4(0(4(5(3(3(3(2(0(x1)))))))))) -> 0(3(1(5(4(5(5(0(x1)))))))) 1(3(2(1(5(1(1(0(5(2(2(x1))))))))))) -> 1(0(2(5(4(2(2(0(5(3(x1)))))))))) 3(2(4(0(1(0(0(2(0(1(3(x1))))))))))) -> 2(4(0(3(5(5(1(3(3(x1))))))))) 4(4(1(0(1(3(3(1(4(1(2(x1))))))))))) -> 3(1(2(4(4(4(2(1(2(2(x1)))))))))) 4(0(4(2(2(0(4(2(1(4(1(4(x1)))))))))))) -> 4(5(0(5(4(5(5(4(x1)))))))) 4(3(5(4(3(4(0(0(4(2(5(0(x1)))))))))))) -> 4(0(2(0(3(3(3(5(3(1(2(4(x1)))))))))))) 2(5(5(1(5(1(2(0(2(1(3(3(4(x1))))))))))))) -> 3(1(1(1(2(5(2(1(4(0(0(4(4(x1))))))))))))) 3(4(1(1(3(1(1(1(5(4(3(4(1(x1))))))))))))) -> 3(4(0(0(3(5(3(5(0(4(4(4(x1)))))))))))) 1(1(2(0(2(4(1(1(3(3(3(5(1(3(x1)))))))))))))) -> 5(1(3(4(5(1(1(0(3(4(0(2(0(x1))))))))))))) 1(4(0(0(5(2(2(5(2(2(3(0(2(5(x1)))))))))))))) -> 1(5(4(2(3(0(1(3(0(3(2(2(0(5(x1)))))))))))))) 2(2(0(2(0(0(5(3(2(3(2(0(3(2(x1)))))))))))))) -> 4(5(1(5(3(3(3(4(4(2(0(4(x1)))))))))))) 4(2(3(5(4(2(5(5(1(1(4(4(0(4(1(0(x1)))))))))))))))) -> 4(1(2(5(4(0(5(3(2(5(0(4(2(4(0(x1))))))))))))))) 5(0(3(4(0(0(0(4(3(4(2(4(3(3(2(0(4(x1))))))))))))))))) -> 5(2(3(3(1(2(0(4(3(0(1(5(5(2(3(4(x1)))))))))))))))) 5(5(4(4(0(5(5(2(4(0(5(1(2(3(2(4(0(x1))))))))))))))))) -> 0(3(2(0(4(3(1(2(4(5(3(3(0(1(1(5(1(3(x1)))))))))))))))))) 0(1(4(3(5(3(0(4(1(1(2(3(3(1(4(0(5(2(3(x1))))))))))))))))))) -> 0(3(0(0(5(4(5(0(4(0(2(2(1(4(0(0(0(4(x1)))))))))))))))))) 4(2(1(0(0(4(4(4(3(0(5(1(3(1(2(3(5(1(0(2(x1)))))))))))))))))))) -> 4(4(1(4(5(0(4(2(4(5(2(3(4(2(2(5(5(1(x1)))))))))))))))))) 4(4(4(1(4(5(2(2(0(1(4(5(2(2(1(4(5(0(0(4(x1)))))))))))))))))))) -> 2(1(2(1(4(0(4(5(3(2(5(0(3(2(0(2(2(5(1(4(x1)))))))))))))))))))) 5(3(2(0(4(1(4(1(4(2(5(2(3(4(4(4(3(2(5(2(4(x1))))))))))))))))))))) -> 5(0(4(3(0(0(2(1(5(4(4(2(5(3(4(3(0(5(5(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_4(x_1)) -> 4(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_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(x1)))) -> 0(2(1(x1))) 0(0(3(3(0(0(x1)))))) -> 0(0(4(0(4(0(x1)))))) 1(5(5(2(1(0(x1)))))) -> 1(5(4(3(1(0(x1)))))) 4(0(3(4(2(2(x1)))))) -> 4(5(2(2(3(x1))))) 2(2(4(1(3(4(1(x1))))))) -> 5(2(4(5(3(x1))))) 0(3(1(5(2(3(1(5(x1)))))))) -> 0(3(3(5(3(5(1(5(x1)))))))) 3(1(5(1(5(0(4(2(x1)))))))) -> 3(1(4(5(0(5(2(x1))))))) 3(0(0(2(2(0(4(4(0(x1))))))))) -> 2(5(5(0(2(0(2(0(x1)))))))) 3(0(1(3(1(5(1(4(2(x1))))))))) -> 1(1(2(4(2(4(1(0(2(x1))))))))) 3(0(4(4(0(2(4(4(4(x1))))))))) -> 2(5(5(0(3(3(4(x1))))))) 0(2(5(2(4(3(4(0(4(3(x1)))))))))) -> 0(3(5(3(1(3(4(0(3(x1))))))))) 1(4(5(3(1(1(1(0(3(2(x1)))))))))) -> 1(3(2(5(3(5(2(1(x1)))))))) 2(4(0(4(5(3(3(3(2(0(x1)))))))))) -> 0(3(1(5(4(5(5(0(x1)))))))) 1(3(2(1(5(1(1(0(5(2(2(x1))))))))))) -> 1(0(2(5(4(2(2(0(5(3(x1)))))))))) 3(2(4(0(1(0(0(2(0(1(3(x1))))))))))) -> 2(4(0(3(5(5(1(3(3(x1))))))))) 4(4(1(0(1(3(3(1(4(1(2(x1))))))))))) -> 3(1(2(4(4(4(2(1(2(2(x1)))))))))) 4(0(4(2(2(0(4(2(1(4(1(4(x1)))))))))))) -> 4(5(0(5(4(5(5(4(x1)))))))) 4(3(5(4(3(4(0(0(4(2(5(0(x1)))))))))))) -> 4(0(2(0(3(3(3(5(3(1(2(4(x1)))))))))))) 2(5(5(1(5(1(2(0(2(1(3(3(4(x1))))))))))))) -> 3(1(1(1(2(5(2(1(4(0(0(4(4(x1))))))))))))) 3(4(1(1(3(1(1(1(5(4(3(4(1(x1))))))))))))) -> 3(4(0(0(3(5(3(5(0(4(4(4(x1)))))))))))) 1(1(2(0(2(4(1(1(3(3(3(5(1(3(x1)))))))))))))) -> 5(1(3(4(5(1(1(0(3(4(0(2(0(x1))))))))))))) 1(4(0(0(5(2(2(5(2(2(3(0(2(5(x1)))))))))))))) -> 1(5(4(2(3(0(1(3(0(3(2(2(0(5(x1)))))))))))))) 2(2(0(2(0(0(5(3(2(3(2(0(3(2(x1)))))))))))))) -> 4(5(1(5(3(3(3(4(4(2(0(4(x1)))))))))))) 4(2(3(5(4(2(5(5(1(1(4(4(0(4(1(0(x1)))))))))))))))) -> 4(1(2(5(4(0(5(3(2(5(0(4(2(4(0(x1))))))))))))))) 5(0(3(4(0(0(0(4(3(4(2(4(3(3(2(0(4(x1))))))))))))))))) -> 5(2(3(3(1(2(0(4(3(0(1(5(5(2(3(4(x1)))))))))))))))) 5(5(4(4(0(5(5(2(4(0(5(1(2(3(2(4(0(x1))))))))))))))))) -> 0(3(2(0(4(3(1(2(4(5(3(3(0(1(1(5(1(3(x1)))))))))))))))))) 0(1(4(3(5(3(0(4(1(1(2(3(3(1(4(0(5(2(3(x1))))))))))))))))))) -> 0(3(0(0(5(4(5(0(4(0(2(2(1(4(0(0(0(4(x1)))))))))))))))))) 4(2(1(0(0(4(4(4(3(0(5(1(3(1(2(3(5(1(0(2(x1)))))))))))))))))))) -> 4(4(1(4(5(0(4(2(4(5(2(3(4(2(2(5(5(1(x1)))))))))))))))))) 4(4(4(1(4(5(2(2(0(1(4(5(2(2(1(4(5(0(0(4(x1)))))))))))))))))))) -> 2(1(2(1(4(0(4(5(3(2(5(0(3(2(0(2(2(5(1(4(x1)))))))))))))))))))) 5(3(2(0(4(1(4(1(4(2(5(2(3(4(4(4(3(2(5(2(4(x1))))))))))))))))))))) -> 5(0(4(3(0(0(2(1(5(4(4(2(5(3(4(3(0(5(5(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_4(x_1)) -> 4(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_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(x1)))) -> 0(2(1(x1))) 0(0(3(3(0(0(x1)))))) -> 0(0(4(0(4(0(x1)))))) 1(5(5(2(1(0(x1)))))) -> 1(5(4(3(1(0(x1)))))) 4(0(3(4(2(2(x1)))))) -> 4(5(2(2(3(x1))))) 2(2(4(1(3(4(1(x1))))))) -> 5(2(4(5(3(x1))))) 0(3(1(5(2(3(1(5(x1)))))))) -> 0(3(3(5(3(5(1(5(x1)))))))) 3(1(5(1(5(0(4(2(x1)))))))) -> 3(1(4(5(0(5(2(x1))))))) 3(0(0(2(2(0(4(4(0(x1))))))))) -> 2(5(5(0(2(0(2(0(x1)))))))) 3(0(1(3(1(5(1(4(2(x1))))))))) -> 1(1(2(4(2(4(1(0(2(x1))))))))) 3(0(4(4(0(2(4(4(4(x1))))))))) -> 2(5(5(0(3(3(4(x1))))))) 0(2(5(2(4(3(4(0(4(3(x1)))))))))) -> 0(3(5(3(1(3(4(0(3(x1))))))))) 1(4(5(3(1(1(1(0(3(2(x1)))))))))) -> 1(3(2(5(3(5(2(1(x1)))))))) 2(4(0(4(5(3(3(3(2(0(x1)))))))))) -> 0(3(1(5(4(5(5(0(x1)))))))) 1(3(2(1(5(1(1(0(5(2(2(x1))))))))))) -> 1(0(2(5(4(2(2(0(5(3(x1)))))))))) 3(2(4(0(1(0(0(2(0(1(3(x1))))))))))) -> 2(4(0(3(5(5(1(3(3(x1))))))))) 4(4(1(0(1(3(3(1(4(1(2(x1))))))))))) -> 3(1(2(4(4(4(2(1(2(2(x1)))))))))) 4(0(4(2(2(0(4(2(1(4(1(4(x1)))))))))))) -> 4(5(0(5(4(5(5(4(x1)))))))) 4(3(5(4(3(4(0(0(4(2(5(0(x1)))))))))))) -> 4(0(2(0(3(3(3(5(3(1(2(4(x1)))))))))))) 2(5(5(1(5(1(2(0(2(1(3(3(4(x1))))))))))))) -> 3(1(1(1(2(5(2(1(4(0(0(4(4(x1))))))))))))) 3(4(1(1(3(1(1(1(5(4(3(4(1(x1))))))))))))) -> 3(4(0(0(3(5(3(5(0(4(4(4(x1)))))))))))) 1(1(2(0(2(4(1(1(3(3(3(5(1(3(x1)))))))))))))) -> 5(1(3(4(5(1(1(0(3(4(0(2(0(x1))))))))))))) 1(4(0(0(5(2(2(5(2(2(3(0(2(5(x1)))))))))))))) -> 1(5(4(2(3(0(1(3(0(3(2(2(0(5(x1)))))))))))))) 2(2(0(2(0(0(5(3(2(3(2(0(3(2(x1)))))))))))))) -> 4(5(1(5(3(3(3(4(4(2(0(4(x1)))))))))))) 4(2(3(5(4(2(5(5(1(1(4(4(0(4(1(0(x1)))))))))))))))) -> 4(1(2(5(4(0(5(3(2(5(0(4(2(4(0(x1))))))))))))))) 5(0(3(4(0(0(0(4(3(4(2(4(3(3(2(0(4(x1))))))))))))))))) -> 5(2(3(3(1(2(0(4(3(0(1(5(5(2(3(4(x1)))))))))))))))) 5(5(4(4(0(5(5(2(4(0(5(1(2(3(2(4(0(x1))))))))))))))))) -> 0(3(2(0(4(3(1(2(4(5(3(3(0(1(1(5(1(3(x1)))))))))))))))))) 0(1(4(3(5(3(0(4(1(1(2(3(3(1(4(0(5(2(3(x1))))))))))))))))))) -> 0(3(0(0(5(4(5(0(4(0(2(2(1(4(0(0(0(4(x1)))))))))))))))))) 4(2(1(0(0(4(4(4(3(0(5(1(3(1(2(3(5(1(0(2(x1)))))))))))))))))))) -> 4(4(1(4(5(0(4(2(4(5(2(3(4(2(2(5(5(1(x1)))))))))))))))))) 4(4(4(1(4(5(2(2(0(1(4(5(2(2(1(4(5(0(0(4(x1)))))))))))))))))))) -> 2(1(2(1(4(0(4(5(3(2(5(0(3(2(0(2(2(5(1(4(x1)))))))))))))))))))) 5(3(2(0(4(1(4(1(4(2(5(2(3(4(4(4(3(2(5(2(4(x1))))))))))))))))))))) -> 5(0(4(3(0(0(2(1(5(4(4(2(5(3(4(3(0(5(5(x1))))))))))))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(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_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. "[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] {(150,151,[0_1|0, 1_1|0, 4_1|0, 2_1|0, 3_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]), (150,152,[0_1|1, 1_1|1, 4_1|1, 2_1|1, 3_1|1, 5_1|1]), (150,153,[0_1|2]), (150,155,[0_1|2]), (150,160,[0_1|2]), (150,167,[0_1|2]), (150,175,[0_1|2]), (150,192,[1_1|2]), (150,197,[1_1|2]), (150,204,[1_1|2]), (150,217,[1_1|2]), (150,226,[5_1|2]), (150,238,[4_1|2]), (150,242,[4_1|2]), (150,249,[3_1|2]), (150,258,[2_1|2]), (150,277,[4_1|2]), (150,288,[4_1|2]), (150,302,[4_1|2]), (150,319,[5_1|2]), (150,323,[4_1|2]), (150,334,[0_1|2]), (150,341,[3_1|2]), (150,353,[3_1|2]), (150,359,[2_1|2]), (150,366,[1_1|2]), (150,374,[2_1|2]), (150,380,[2_1|2]), (150,388,[3_1|2]), (150,399,[5_1|2]), (150,414,[0_1|2]), (150,431,[5_1|2]), (151,151,[cons_0_1|0, cons_1_1|0, cons_4_1|0, cons_2_1|0, cons_3_1|0, cons_5_1|0]), (152,151,[encArg_1|1]), (152,152,[0_1|1, 1_1|1, 4_1|1, 2_1|1, 3_1|1, 5_1|1]), (152,153,[0_1|2]), (152,155,[0_1|2]), (152,160,[0_1|2]), (152,167,[0_1|2]), (152,175,[0_1|2]), (152,192,[1_1|2]), (152,197,[1_1|2]), (152,204,[1_1|2]), (152,217,[1_1|2]), (152,226,[5_1|2]), (152,238,[4_1|2]), (152,242,[4_1|2]), (152,249,[3_1|2]), (152,258,[2_1|2]), (152,277,[4_1|2]), (152,288,[4_1|2]), (152,302,[4_1|2]), (152,319,[5_1|2]), (152,323,[4_1|2]), (152,334,[0_1|2]), (152,341,[3_1|2]), (152,353,[3_1|2]), (152,359,[2_1|2]), (152,366,[1_1|2]), (152,374,[2_1|2]), (152,380,[2_1|2]), (152,388,[3_1|2]), (152,399,[5_1|2]), (152,414,[0_1|2]), (152,431,[5_1|2]), (153,154,[2_1|2]), (154,152,[1_1|2]), (154,192,[1_1|2]), (154,197,[1_1|2]), (154,204,[1_1|2]), (154,217,[1_1|2]), (154,366,[1_1|2]), (154,367,[1_1|2]), (154,226,[5_1|2]), (155,156,[0_1|2]), (156,157,[4_1|2]), (157,158,[0_1|2]), (158,159,[4_1|2]), (158,238,[4_1|2]), (158,242,[4_1|2]), (159,152,[0_1|2]), (159,153,[0_1|2]), (159,155,[0_1|2]), (159,160,[0_1|2]), (159,167,[0_1|2]), (159,175,[0_1|2]), (159,334,[0_1|2]), (159,414,[0_1|2]), (159,156,[0_1|2]), (160,161,[3_1|2]), (161,162,[3_1|2]), (162,163,[5_1|2]), (163,164,[3_1|2]), (164,165,[5_1|2]), (165,166,[1_1|2]), (165,192,[1_1|2]), (166,152,[5_1|2]), (166,226,[5_1|2]), (166,319,[5_1|2]), (166,399,[5_1|2]), (166,431,[5_1|2]), (166,193,[5_1|2]), (166,205,[5_1|2]), (166,414,[0_1|2]), (167,168,[3_1|2]), (168,169,[5_1|2]), (169,170,[3_1|2]), (170,171,[1_1|2]), (171,172,[3_1|2]), (172,173,[4_1|2]), (172,238,[4_1|2]), (173,174,[0_1|2]), (173,160,[0_1|2]), (174,152,[3_1|2]), (174,249,[3_1|2]), (174,341,[3_1|2]), (174,353,[3_1|2]), (174,388,[3_1|2]), (174,359,[2_1|2]), (174,366,[1_1|2]), (174,374,[2_1|2]), (174,380,[2_1|2]), (175,176,[3_1|2]), (176,177,[0_1|2]), (177,178,[0_1|2]), (178,179,[5_1|2]), (179,180,[4_1|2]), (180,181,[5_1|2]), (181,182,[0_1|2]), (182,183,[4_1|2]), (183,184,[0_1|2]), (184,185,[2_1|2]), (185,186,[2_1|2]), (186,187,[1_1|2]), (187,188,[4_1|2]), (188,189,[0_1|2]), (189,190,[0_1|2]), (190,191,[0_1|2]), (191,152,[4_1|2]), (191,249,[4_1|2, 3_1|2]), (191,341,[4_1|2]), (191,353,[4_1|2]), (191,388,[4_1|2]), (191,401,[4_1|2]), (191,238,[4_1|2]), (191,242,[4_1|2]), (191,258,[2_1|2]), (191,277,[4_1|2]), (191,288,[4_1|2]), (191,302,[4_1|2]), (192,193,[5_1|2]), (193,194,[4_1|2]), (194,195,[3_1|2]), (195,196,[1_1|2]), (196,152,[0_1|2]), (196,153,[0_1|2]), (196,155,[0_1|2]), (196,160,[0_1|2]), (196,167,[0_1|2]), (196,175,[0_1|2]), (196,334,[0_1|2]), (196,414,[0_1|2]), (196,218,[0_1|2]), (197,198,[3_1|2]), (198,199,[2_1|2]), (199,200,[5_1|2]), (200,201,[3_1|2]), (201,202,[5_1|2]), (202,203,[2_1|2]), (203,152,[1_1|2]), (203,258,[1_1|2]), (203,359,[1_1|2]), (203,374,[1_1|2]), (203,380,[1_1|2]), (203,416,[1_1|2]), (203,192,[1_1|2]), (203,197,[1_1|2]), (203,204,[1_1|2]), (203,217,[1_1|2]), (203,226,[5_1|2]), (204,205,[5_1|2]), (205,206,[4_1|2]), (206,207,[2_1|2]), (207,208,[3_1|2]), (208,209,[0_1|2]), (209,210,[1_1|2]), (210,211,[3_1|2]), (211,212,[0_1|2]), (212,213,[3_1|2]), (213,214,[2_1|2]), (214,215,[2_1|2]), (215,216,[0_1|2]), (216,152,[5_1|2]), (216,226,[5_1|2]), (216,319,[5_1|2]), (216,399,[5_1|2]), (216,431,[5_1|2]), (216,360,[5_1|2]), (216,375,[5_1|2]), (216,414,[0_1|2]), (217,218,[0_1|2]), (218,219,[2_1|2]), (219,220,[5_1|2]), (220,221,[4_1|2]), (221,222,[2_1|2]), (222,223,[2_1|2]), (223,224,[0_1|2]), (224,225,[5_1|2]), (224,431,[5_1|2]), (225,152,[3_1|2]), (225,258,[3_1|2]), (225,359,[3_1|2, 2_1|2]), (225,374,[3_1|2, 2_1|2]), (225,380,[3_1|2, 2_1|2]), (225,353,[3_1|2]), (225,366,[1_1|2]), (225,388,[3_1|2]), (226,227,[1_1|2]), (227,228,[3_1|2]), (228,229,[4_1|2]), (229,230,[5_1|2]), (230,231,[1_1|2]), (231,232,[1_1|2]), (232,233,[0_1|2]), (233,234,[3_1|2]), (234,235,[4_1|2]), (235,236,[0_1|2]), (236,237,[2_1|2]), (237,152,[0_1|2]), (237,249,[0_1|2]), (237,341,[0_1|2]), (237,353,[0_1|2]), (237,388,[0_1|2]), (237,198,[0_1|2]), (237,228,[0_1|2]), (237,153,[0_1|2]), (237,155,[0_1|2]), (237,160,[0_1|2]), (237,167,[0_1|2]), (237,175,[0_1|2]), (238,239,[5_1|2]), (239,240,[2_1|2]), (240,241,[2_1|2]), (241,152,[3_1|2]), (241,258,[3_1|2]), (241,359,[3_1|2, 2_1|2]), (241,374,[3_1|2, 2_1|2]), (241,380,[3_1|2, 2_1|2]), (241,353,[3_1|2]), (241,366,[1_1|2]), (241,388,[3_1|2]), (242,243,[5_1|2]), (243,244,[0_1|2]), (244,245,[5_1|2]), (245,246,[4_1|2]), (246,247,[5_1|2]), (246,414,[0_1|2]), (247,248,[5_1|2]), (248,152,[4_1|2]), (248,238,[4_1|2]), (248,242,[4_1|2]), (248,277,[4_1|2]), (248,288,[4_1|2]), (248,302,[4_1|2]), (248,323,[4_1|2]), (248,249,[3_1|2]), (248,258,[2_1|2]), (249,250,[1_1|2]), (250,251,[2_1|2]), (251,252,[4_1|2]), (252,253,[4_1|2]), (253,254,[4_1|2]), (254,255,[2_1|2]), (255,256,[1_1|2]), (256,257,[2_1|2]), (256,319,[5_1|2]), (256,323,[4_1|2]), (257,152,[2_1|2]), (257,258,[2_1|2]), (257,359,[2_1|2]), (257,374,[2_1|2]), (257,380,[2_1|2]), (257,290,[2_1|2]), (257,319,[5_1|2]), (257,323,[4_1|2]), (257,334,[0_1|2]), (257,341,[3_1|2]), (258,259,[1_1|2]), (259,260,[2_1|2]), (260,261,[1_1|2]), (261,262,[4_1|2]), (262,263,[0_1|2]), (263,264,[4_1|2]), (264,265,[5_1|2]), (265,266,[3_1|2]), (266,267,[2_1|2]), (267,268,[5_1|2]), (268,269,[0_1|2]), (269,270,[3_1|2]), (270,271,[2_1|2]), (271,272,[0_1|2]), (272,273,[2_1|2]), (273,274,[2_1|2]), (274,275,[5_1|2]), (275,276,[1_1|2]), (275,197,[1_1|2]), (275,204,[1_1|2]), (276,152,[4_1|2]), (276,238,[4_1|2]), (276,242,[4_1|2]), (276,277,[4_1|2]), (276,288,[4_1|2]), (276,302,[4_1|2]), (276,323,[4_1|2]), (276,157,[4_1|2]), (276,249,[3_1|2]), (276,258,[2_1|2]), (277,278,[0_1|2]), (278,279,[2_1|2]), (279,280,[0_1|2]), (280,281,[3_1|2]), (281,282,[3_1|2]), (282,283,[3_1|2]), (283,284,[5_1|2]), (284,285,[3_1|2]), (285,286,[1_1|2]), (286,287,[2_1|2]), (286,334,[0_1|2]), (287,152,[4_1|2]), (287,153,[4_1|2]), (287,155,[4_1|2]), (287,160,[4_1|2]), (287,167,[4_1|2]), (287,175,[4_1|2]), (287,334,[4_1|2]), (287,414,[4_1|2]), (287,432,[4_1|2]), (287,238,[4_1|2]), (287,242,[4_1|2]), (287,249,[3_1|2]), (287,258,[2_1|2]), (287,277,[4_1|2]), (287,288,[4_1|2]), (287,302,[4_1|2]), (288,289,[1_1|2]), (289,290,[2_1|2]), (290,291,[5_1|2]), (291,292,[4_1|2]), (292,293,[0_1|2]), (293,294,[5_1|2]), (294,295,[3_1|2]), (295,296,[2_1|2]), (296,297,[5_1|2]), (297,298,[0_1|2]), (298,299,[4_1|2]), (299,300,[2_1|2]), (299,334,[0_1|2]), (300,301,[4_1|2]), (300,238,[4_1|2]), (300,242,[4_1|2]), (301,152,[0_1|2]), (301,153,[0_1|2]), (301,155,[0_1|2]), (301,160,[0_1|2]), (301,167,[0_1|2]), (301,175,[0_1|2]), (301,334,[0_1|2]), (301,414,[0_1|2]), (301,218,[0_1|2]), (302,303,[4_1|2]), (303,304,[1_1|2]), (304,305,[4_1|2]), (305,306,[5_1|2]), (306,307,[0_1|2]), (307,308,[4_1|2]), (308,309,[2_1|2]), (309,310,[4_1|2]), (310,311,[5_1|2]), (311,312,[2_1|2]), (312,313,[3_1|2]), (313,314,[4_1|2]), (314,315,[2_1|2]), (315,316,[2_1|2]), (315,341,[3_1|2]), (316,317,[5_1|2]), (317,318,[5_1|2]), (318,152,[1_1|2]), (318,258,[1_1|2]), (318,359,[1_1|2]), (318,374,[1_1|2]), (318,380,[1_1|2]), (318,154,[1_1|2]), (318,219,[1_1|2]), (318,192,[1_1|2]), (318,197,[1_1|2]), (318,204,[1_1|2]), (318,217,[1_1|2]), (318,226,[5_1|2]), (319,320,[2_1|2]), (320,321,[4_1|2]), (321,322,[5_1|2]), (321,431,[5_1|2]), (322,152,[3_1|2]), (322,192,[3_1|2]), (322,197,[3_1|2]), (322,204,[3_1|2]), (322,217,[3_1|2]), (322,366,[3_1|2, 1_1|2]), (322,289,[3_1|2]), (322,353,[3_1|2]), (322,359,[2_1|2]), (322,374,[2_1|2]), (322,380,[2_1|2]), (322,388,[3_1|2]), (323,324,[5_1|2]), (324,325,[1_1|2]), (325,326,[5_1|2]), (326,327,[3_1|2]), (327,328,[3_1|2]), (328,329,[3_1|2]), (329,330,[4_1|2]), (330,331,[4_1|2]), (331,332,[2_1|2]), (332,333,[0_1|2]), (333,152,[4_1|2]), (333,258,[4_1|2, 2_1|2]), (333,359,[4_1|2]), (333,374,[4_1|2]), (333,380,[4_1|2]), (333,416,[4_1|2]), (333,238,[4_1|2]), (333,242,[4_1|2]), (333,249,[3_1|2]), (333,277,[4_1|2]), (333,288,[4_1|2]), (333,302,[4_1|2]), (334,335,[3_1|2]), (335,336,[1_1|2]), (336,337,[5_1|2]), (337,338,[4_1|2]), (338,339,[5_1|2]), (339,340,[5_1|2]), (339,399,[5_1|2]), (340,152,[0_1|2]), (340,153,[0_1|2]), (340,155,[0_1|2]), (340,160,[0_1|2]), (340,167,[0_1|2]), (340,175,[0_1|2]), (340,334,[0_1|2]), (340,414,[0_1|2]), (341,342,[1_1|2]), (342,343,[1_1|2]), (343,344,[1_1|2]), (344,345,[2_1|2]), (345,346,[5_1|2]), (346,347,[2_1|2]), (347,348,[1_1|2]), (348,349,[4_1|2]), (349,350,[0_1|2]), (350,351,[0_1|2]), (351,352,[4_1|2]), (351,249,[3_1|2]), (351,258,[2_1|2]), (352,152,[4_1|2]), (352,238,[4_1|2]), (352,242,[4_1|2]), (352,277,[4_1|2]), (352,288,[4_1|2]), (352,302,[4_1|2]), (352,323,[4_1|2]), (352,389,[4_1|2]), (352,249,[3_1|2]), (352,258,[2_1|2]), (353,354,[1_1|2]), (354,355,[4_1|2]), (355,356,[5_1|2]), (356,357,[0_1|2]), (357,358,[5_1|2]), (358,152,[2_1|2]), (358,258,[2_1|2]), (358,359,[2_1|2]), (358,374,[2_1|2]), (358,380,[2_1|2]), (358,319,[5_1|2]), (358,323,[4_1|2]), (358,334,[0_1|2]), (358,341,[3_1|2]), (359,360,[5_1|2]), (360,361,[5_1|2]), (361,362,[0_1|2]), (362,363,[2_1|2]), (363,364,[0_1|2]), (364,365,[2_1|2]), (365,152,[0_1|2]), (365,153,[0_1|2]), (365,155,[0_1|2]), (365,160,[0_1|2]), (365,167,[0_1|2]), (365,175,[0_1|2]), (365,334,[0_1|2]), (365,414,[0_1|2]), (365,278,[0_1|2]), (366,367,[1_1|2]), (367,368,[2_1|2]), (368,369,[4_1|2]), (369,370,[2_1|2]), (370,371,[4_1|2]), (371,372,[1_1|2]), (372,373,[0_1|2]), (372,167,[0_1|2]), (373,152,[2_1|2]), (373,258,[2_1|2]), (373,359,[2_1|2]), (373,374,[2_1|2]), (373,380,[2_1|2]), (373,319,[5_1|2]), (373,323,[4_1|2]), (373,334,[0_1|2]), (373,341,[3_1|2]), (374,375,[5_1|2]), (375,376,[5_1|2]), (376,377,[0_1|2]), (377,378,[3_1|2]), (378,379,[3_1|2]), (378,388,[3_1|2]), (379,152,[4_1|2]), (379,238,[4_1|2]), (379,242,[4_1|2]), (379,277,[4_1|2]), (379,288,[4_1|2]), (379,302,[4_1|2]), (379,323,[4_1|2]), (379,303,[4_1|2]), (379,249,[3_1|2]), (379,258,[2_1|2]), (380,381,[4_1|2]), (381,382,[0_1|2]), (382,383,[3_1|2]), (383,384,[5_1|2]), (384,385,[5_1|2]), (385,386,[1_1|2]), (386,387,[3_1|2]), (387,152,[3_1|2]), (387,249,[3_1|2]), (387,341,[3_1|2]), (387,353,[3_1|2]), (387,388,[3_1|2]), (387,198,[3_1|2]), (387,359,[2_1|2]), (387,366,[1_1|2]), (387,374,[2_1|2]), (387,380,[2_1|2]), (388,389,[4_1|2]), (389,390,[0_1|2]), (390,391,[0_1|2]), (391,392,[3_1|2]), (392,393,[5_1|2]), (393,394,[3_1|2]), (394,395,[5_1|2]), (395,396,[0_1|2]), (396,397,[4_1|2]), (396,258,[2_1|2]), (397,398,[4_1|2]), (397,249,[3_1|2]), (397,258,[2_1|2]), (398,152,[4_1|2]), (398,192,[4_1|2]), (398,197,[4_1|2]), (398,204,[4_1|2]), (398,217,[4_1|2]), (398,366,[4_1|2]), (398,289,[4_1|2]), (398,238,[4_1|2]), (398,242,[4_1|2]), (398,249,[3_1|2]), (398,258,[2_1|2]), (398,277,[4_1|2]), (398,288,[4_1|2]), (398,302,[4_1|2]), (399,400,[2_1|2]), (400,401,[3_1|2]), (401,402,[3_1|2]), (402,403,[1_1|2]), (403,404,[2_1|2]), (404,405,[0_1|2]), (405,406,[4_1|2]), (406,407,[3_1|2]), (407,408,[0_1|2]), (408,409,[1_1|2]), (409,410,[5_1|2]), (410,411,[5_1|2]), (411,412,[2_1|2]), (412,413,[3_1|2]), (412,388,[3_1|2]), (413,152,[4_1|2]), (413,238,[4_1|2]), (413,242,[4_1|2]), (413,277,[4_1|2]), (413,288,[4_1|2]), (413,302,[4_1|2]), (413,323,[4_1|2]), (413,249,[3_1|2]), (413,258,[2_1|2]), (414,415,[3_1|2]), (415,416,[2_1|2]), (416,417,[0_1|2]), (417,418,[4_1|2]), (418,419,[3_1|2]), (419,420,[1_1|2]), (420,421,[2_1|2]), (421,422,[4_1|2]), (422,423,[5_1|2]), (423,424,[3_1|2]), (424,425,[3_1|2]), (425,426,[0_1|2]), (426,427,[1_1|2]), (427,428,[1_1|2]), (428,429,[5_1|2]), (429,430,[1_1|2]), (429,217,[1_1|2]), (430,152,[3_1|2]), (430,153,[3_1|2]), (430,155,[3_1|2]), (430,160,[3_1|2]), (430,167,[3_1|2]), (430,175,[3_1|2]), (430,334,[3_1|2]), (430,414,[3_1|2]), (430,278,[3_1|2]), (430,382,[3_1|2]), (430,353,[3_1|2]), (430,359,[2_1|2]), (430,366,[1_1|2]), (430,374,[2_1|2]), (430,380,[2_1|2]), (430,388,[3_1|2]), (431,432,[0_1|2]), (432,433,[4_1|2]), (433,434,[3_1|2]), (434,435,[0_1|2]), (435,436,[0_1|2]), (436,437,[2_1|2]), (437,438,[1_1|2]), (438,439,[5_1|2]), (439,440,[4_1|2]), (440,441,[4_1|2]), (441,442,[2_1|2]), (442,443,[5_1|2]), (443,444,[3_1|2]), (444,445,[4_1|2]), (445,446,[3_1|2]), (446,447,[0_1|2]), (447,448,[5_1|2]), (447,414,[0_1|2]), (448,152,[5_1|2]), (448,238,[5_1|2]), (448,242,[5_1|2]), (448,277,[5_1|2]), (448,288,[5_1|2]), (448,302,[5_1|2]), (448,323,[5_1|2]), (448,381,[5_1|2]), (448,321,[5_1|2]), (448,399,[5_1|2]), (448,414,[0_1|2]), (448,431,[5_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)