/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 45 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 121 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 4(0(4(x1))) -> 1(3(1(2(4(5(2(4(3(4(x1)))))))))) 4(0(5(x1))) -> 4(5(5(2(2(2(4(5(1(1(x1)))))))))) 1(1(4(4(x1)))) -> 2(2(1(3(4(4(2(2(1(3(x1)))))))))) 0(2(5(3(0(x1))))) -> 0(1(2(0(1(0(3(2(0(0(x1)))))))))) 1(4(1(1(4(x1))))) -> 5(1(0(4(0(3(0(4(5(5(x1)))))))))) 1(4(2(2(2(x1))))) -> 3(2(1(0(1(2(4(2(2(2(x1)))))))))) 2(4(2(4(3(x1))))) -> 2(1(2(3(5(5(4(1(3(5(x1)))))))))) 3(0(1(4(3(x1))))) -> 2(0(2(0(5(5(4(5(1(0(x1)))))))))) 3(1(4(0(3(x1))))) -> 0(1(2(1(2(1(4(3(3(3(x1)))))))))) 4(1(4(3(2(x1))))) -> 3(3(2(0(5(3(0(0(3(1(x1)))))))))) 4(1(4(4(0(x1))))) -> 0(0(4(0(0(0(1(2(5(0(x1)))))))))) 4(2(4(1(4(x1))))) -> 3(3(1(4(5(5(3(4(5(5(x1)))))))))) 4(2(5(4(3(x1))))) -> 5(5(4(5(1(1(2(3(3(3(x1)))))))))) 0(2(1(1(5(0(x1)))))) -> 0(5(1(2(1(0(2(3(5(1(x1)))))))))) 0(2(4(1(4(0(x1)))))) -> 0(4(5(2(2(4(3(3(2(5(x1)))))))))) 0(2(4(3(4(1(x1)))))) -> 3(3(1(0(4(3(4(1(2(1(x1)))))))))) 0(4(2(5(4(3(x1)))))) -> 1(2(4(0(2(4(1(1(2(3(x1)))))))))) 0(4(3(2(5(3(x1)))))) -> 0(3(0(5(3(3(3(0(0(0(x1)))))))))) 1(1(1(5(5(3(x1)))))) -> 4(1(2(2(2(5(4(0(5(3(x1)))))))))) 1(4(2(4(0(2(x1)))))) -> 5(1(3(1(0(3(3(5(1(1(x1)))))))))) 1(4(3(2(5(4(x1)))))) -> 3(2(3(3(5(1(0(3(1(2(x1)))))))))) 1(4(3(3(2(3(x1)))))) -> 5(1(0(3(3(2(0(2(1(3(x1)))))))))) 1(5(4(1(1(4(x1)))))) -> 0(0(2(5(4(5(3(4(1(2(x1)))))))))) 3(0(2(1(5(5(x1)))))) -> 2(3(0(5(3(4(1(0(5(5(x1)))))))))) 3(0(4(2(4(2(x1)))))) -> 3(0(1(3(0(5(5(2(0(1(x1)))))))))) 5(2(3(0(4(2(x1)))))) -> 5(1(3(1(3(0(3(0(1(1(x1)))))))))) 0(2(4(2(5(4(2(x1))))))) -> 0(4(1(5(0(3(0(1(0(2(x1)))))))))) 0(3(3(1(4(0(4(x1))))))) -> 0(3(3(5(4(3(1(2(5(3(x1)))))))))) 1(1(4(5(3(0(5(x1))))))) -> 5(3(5(5(2(2(3(3(0(5(x1)))))))))) 1(1(5(4(2(5(0(x1))))))) -> 1(4(5(5(2(3(1(1(0(3(x1)))))))))) 1(4(2(2(0(4(3(x1))))))) -> 5(3(1(3(2(0(3(4(5(4(x1)))))))))) 1(4(3(2(4(0(5(x1))))))) -> 0(0(5(5(2(5(0(1(0(2(x1)))))))))) 2(4(0(5(5(5(1(x1))))))) -> 2(4(5(5(2(2(4(0(0(2(x1)))))))))) 3(3(2(5(4(3(1(x1))))))) -> 3(2(2(1(3(3(0(0(5(1(x1)))))))))) 3(5(0(0(2(3(4(x1))))))) -> 5(0(1(3(3(0(1(1(2(4(x1)))))))))) 4(0(1(1(4(1(5(x1))))))) -> 5(3(1(2(2(3(0(4(3(1(x1)))))))))) 4(1(1(1(5(5(3(x1))))))) -> 3(5(0(1(3(0(5(3(0(5(x1)))))))))) 4(1(4(1(1(5(4(x1))))))) -> 4(5(4(4(0(1(4(5(2(5(x1)))))))))) 4(2(1(4(0(5(0(x1))))))) -> 5(2(1(0(2(5(2(0(5(3(x1)))))))))) 5(1(4(0(4(2(1(x1))))))) -> 1(3(3(2(1(1(1(2(3(1(x1)))))))))) 5(1(5(5(4(2(4(x1))))))) -> 5(0(0(0(5(0(1(3(4(4(x1)))))))))) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[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] {(150,151,[4_1|0, 1_1|0, 0_1|0, 2_1|0, 3_1|0, 5_1|0, encArg_1|0, encode_4_1|0, encode_0_1|0, encode_1_1|0, encode_3_1|0, encode_2_1|0, encode_5_1|0]), (150,152,[4_1|1, 1_1|1, 0_1|1, 2_1|1, 3_1|1, 5_1|1]), (150,153,[1_1|2]), (150,162,[4_1|2]), (150,171,[5_1|2]), (150,180,[3_1|2]), (150,189,[0_1|2]), (150,198,[4_1|2]), (150,207,[3_1|2]), (150,216,[3_1|2]), (150,225,[5_1|2]), (150,234,[5_1|2]), (150,243,[2_1|2]), (150,252,[5_1|2]), (150,261,[4_1|2]), (150,270,[1_1|2]), (150,279,[5_1|2]), (150,288,[3_1|2]), (150,297,[5_1|2]), (150,306,[5_1|2]), (150,315,[3_1|2]), (150,324,[0_1|2]), (150,333,[5_1|2]), (150,342,[0_1|2]), (150,351,[0_1|2]), (150,360,[0_1|2]), (150,369,[0_1|2]), (150,378,[3_1|2]), (150,387,[0_1|2]), (150,396,[1_1|2]), (150,405,[0_1|2]), (150,414,[0_1|2]), (150,423,[2_1|2]), (150,432,[2_1|2]), (150,441,[2_1|2]), (150,450,[2_1|2]), (150,459,[3_1|2]), (150,468,[0_1|2]), (150,477,[3_1|2]), (150,486,[5_1|2]), (150,495,[5_1|2]), (150,504,[1_1|2]), (150,513,[5_1|2]), (151,151,[cons_4_1|0, cons_1_1|0, cons_0_1|0, cons_2_1|0, cons_3_1|0, cons_5_1|0]), (152,151,[encArg_1|1]), (152,152,[4_1|1, 1_1|1, 0_1|1, 2_1|1, 3_1|1, 5_1|1]), (152,153,[1_1|2]), (152,162,[4_1|2]), (152,171,[5_1|2]), (152,180,[3_1|2]), (152,189,[0_1|2]), (152,198,[4_1|2]), (152,207,[3_1|2]), (152,216,[3_1|2]), (152,225,[5_1|2]), (152,234,[5_1|2]), (152,243,[2_1|2]), (152,252,[5_1|2]), (152,261,[4_1|2]), (152,270,[1_1|2]), (152,279,[5_1|2]), (152,288,[3_1|2]), (152,297,[5_1|2]), (152,306,[5_1|2]), (152,315,[3_1|2]), (152,324,[0_1|2]), (152,333,[5_1|2]), (152,342,[0_1|2]), (152,351,[0_1|2]), (152,360,[0_1|2]), (152,369,[0_1|2]), (152,378,[3_1|2]), (152,387,[0_1|2]), (152,396,[1_1|2]), (152,405,[0_1|2]), (152,414,[0_1|2]), (152,423,[2_1|2]), (152,432,[2_1|2]), (152,441,[2_1|2]), (152,450,[2_1|2]), (152,459,[3_1|2]), (152,468,[0_1|2]), (152,477,[3_1|2]), (152,486,[5_1|2]), (152,495,[5_1|2]), (152,504,[1_1|2]), (152,513,[5_1|2]), (153,154,[3_1|2]), 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(399,400,[2_1|2]), (400,401,[4_1|2]), (401,402,[1_1|2]), (402,403,[1_1|2]), (403,404,[2_1|2]), (404,152,[3_1|2]), (404,180,[3_1|2]), (404,207,[3_1|2]), (404,216,[3_1|2]), (404,288,[3_1|2]), (404,315,[3_1|2]), (404,378,[3_1|2]), (404,459,[3_1|2]), (404,477,[3_1|2]), (404,441,[2_1|2]), (404,450,[2_1|2]), (404,468,[0_1|2]), (404,486,[5_1|2]), (405,406,[3_1|2]), (406,407,[0_1|2]), (407,408,[5_1|2]), (408,409,[3_1|2]), (409,410,[3_1|2]), (410,411,[3_1|2]), (411,412,[0_1|2]), (412,413,[0_1|2]), (413,152,[0_1|2]), (413,180,[0_1|2]), (413,207,[0_1|2]), (413,216,[0_1|2]), (413,288,[0_1|2]), (413,315,[0_1|2]), (413,378,[0_1|2, 3_1|2]), (413,459,[0_1|2]), (413,477,[0_1|2]), (413,172,[0_1|2]), (413,253,[0_1|2]), (413,298,[0_1|2]), (413,351,[0_1|2]), (413,360,[0_1|2]), (413,369,[0_1|2]), (413,387,[0_1|2]), (413,396,[1_1|2]), (413,405,[0_1|2]), (413,414,[0_1|2]), (414,415,[3_1|2]), (415,416,[3_1|2]), (416,417,[5_1|2]), (417,418,[4_1|2]), (418,419,[3_1|2]), (419,420,[1_1|2]), (420,421,[2_1|2]), (421,422,[5_1|2]), (422,152,[3_1|2]), (422,162,[3_1|2]), (422,198,[3_1|2]), (422,261,[3_1|2]), (422,370,[3_1|2]), (422,388,[3_1|2]), (422,441,[2_1|2]), (422,450,[2_1|2]), (422,459,[3_1|2]), (422,468,[0_1|2]), (422,477,[3_1|2]), (422,486,[5_1|2]), (423,424,[1_1|2]), (424,425,[2_1|2]), (425,426,[3_1|2]), (426,427,[5_1|2]), (427,428,[5_1|2]), (428,429,[4_1|2]), (429,430,[1_1|2]), (430,431,[3_1|2]), (430,486,[5_1|2]), (431,152,[5_1|2]), (431,180,[5_1|2]), (431,207,[5_1|2]), (431,216,[5_1|2]), (431,288,[5_1|2]), (431,315,[5_1|2]), (431,378,[5_1|2]), (431,459,[5_1|2]), (431,477,[5_1|2]), (431,495,[5_1|2]), (431,504,[1_1|2]), (431,513,[5_1|2]), (432,433,[4_1|2]), (433,434,[5_1|2]), (434,435,[5_1|2]), (435,436,[2_1|2]), (436,437,[2_1|2]), (437,438,[4_1|2]), (438,439,[0_1|2]), (439,440,[0_1|2]), (439,351,[0_1|2]), (439,360,[0_1|2]), (439,369,[0_1|2]), (439,378,[3_1|2]), (439,387,[0_1|2]), (440,152,[2_1|2]), (440,153,[2_1|2]), (440,270,[2_1|2]), (440,396,[2_1|2]), (440,504,[2_1|2]), (440,280,[2_1|2]), (440,307,[2_1|2]), (440,334,[2_1|2]), (440,496,[2_1|2]), (440,423,[2_1|2]), (440,432,[2_1|2]), (441,442,[0_1|2]), (442,443,[2_1|2]), (443,444,[0_1|2]), (444,445,[5_1|2]), (445,446,[5_1|2]), (446,447,[4_1|2]), (447,448,[5_1|2]), (448,449,[1_1|2]), (449,152,[0_1|2]), (449,180,[0_1|2]), (449,207,[0_1|2]), (449,216,[0_1|2]), (449,288,[0_1|2]), (449,315,[0_1|2]), (449,378,[0_1|2, 3_1|2]), (449,459,[0_1|2]), (449,477,[0_1|2]), (449,351,[0_1|2]), (449,360,[0_1|2]), (449,369,[0_1|2]), (449,387,[0_1|2]), (449,396,[1_1|2]), (449,405,[0_1|2]), (449,414,[0_1|2]), (450,451,[3_1|2]), (451,452,[0_1|2]), (452,453,[5_1|2]), (453,454,[3_1|2]), (454,455,[4_1|2]), (455,456,[1_1|2]), (456,457,[0_1|2]), (457,458,[5_1|2]), (458,152,[5_1|2]), (458,171,[5_1|2]), (458,225,[5_1|2]), (458,234,[5_1|2]), (458,252,[5_1|2]), (458,279,[5_1|2]), (458,297,[5_1|2]), (458,306,[5_1|2]), (458,333,[5_1|2]), (458,486,[5_1|2]), (458,495,[5_1|2]), (458,513,[5_1|2]), (458,226,[5_1|2]), (458,504,[1_1|2]), (459,460,[0_1|2]), (460,461,[1_1|2]), (461,462,[3_1|2]), (462,463,[0_1|2]), (463,464,[5_1|2]), (464,465,[5_1|2]), (465,466,[2_1|2]), (466,467,[0_1|2]), (467,152,[1_1|2]), (467,243,[1_1|2, 2_1|2]), (467,423,[1_1|2]), (467,432,[1_1|2]), (467,441,[1_1|2]), (467,450,[1_1|2]), (467,252,[5_1|2]), (467,261,[4_1|2]), (467,270,[1_1|2]), (467,279,[5_1|2]), (467,288,[3_1|2]), (467,297,[5_1|2]), (467,306,[5_1|2]), (467,315,[3_1|2]), (467,324,[0_1|2]), (467,333,[5_1|2]), (467,342,[0_1|2]), (468,469,[1_1|2]), (469,470,[2_1|2]), (470,471,[1_1|2]), (471,472,[2_1|2]), (472,473,[1_1|2]), (472,549,[5_1|3]), (473,474,[4_1|2]), (474,475,[3_1|2]), (475,476,[3_1|2]), (475,477,[3_1|2]), (476,152,[3_1|2]), (476,180,[3_1|2]), (476,207,[3_1|2]), (476,216,[3_1|2]), (476,288,[3_1|2]), (476,315,[3_1|2]), (476,378,[3_1|2]), (476,459,[3_1|2]), (476,477,[3_1|2]), (476,406,[3_1|2]), (476,415,[3_1|2]), (476,441,[2_1|2]), (476,450,[2_1|2]), (476,468,[0_1|2]), (476,486,[5_1|2]), (477,478,[2_1|2]), (478,479,[2_1|2]), (479,480,[1_1|2]), (480,481,[3_1|2]), (481,482,[3_1|2]), (482,483,[0_1|2]), (483,484,[0_1|2]), (484,485,[5_1|2]), (484,504,[1_1|2]), (484,513,[5_1|2]), (485,152,[1_1|2]), (485,153,[1_1|2]), (485,270,[1_1|2]), (485,396,[1_1|2]), (485,504,[1_1|2]), (485,243,[2_1|2]), (485,252,[5_1|2]), (485,261,[4_1|2]), (485,279,[5_1|2]), (485,288,[3_1|2]), (485,297,[5_1|2]), (485,306,[5_1|2]), (485,315,[3_1|2]), (485,324,[0_1|2]), (485,333,[5_1|2]), (485,342,[0_1|2]), (486,487,[0_1|2]), (487,488,[1_1|2]), (488,489,[3_1|2]), (489,490,[3_1|2]), (490,491,[0_1|2]), (491,492,[1_1|2]), (492,493,[1_1|2]), (493,494,[2_1|2]), (493,423,[2_1|2]), (493,432,[2_1|2]), (494,152,[4_1|2]), (494,162,[4_1|2]), (494,198,[4_1|2]), (494,261,[4_1|2]), (494,153,[1_1|2]), (494,171,[5_1|2]), (494,180,[3_1|2]), (494,189,[0_1|2]), (494,207,[3_1|2]), (494,216,[3_1|2]), (494,225,[5_1|2]), (494,234,[5_1|2]), (494,522,[4_1|3]), (494,531,[1_1|3]), (495,496,[1_1|2]), (496,497,[3_1|2]), (497,498,[1_1|2]), (498,499,[3_1|2]), (499,500,[0_1|2]), (500,501,[3_1|2]), (501,502,[0_1|2]), (502,503,[1_1|2]), (502,243,[2_1|2]), (502,252,[5_1|2]), (502,261,[4_1|2]), (502,270,[1_1|2]), (503,152,[1_1|2]), (503,243,[1_1|2, 2_1|2]), (503,423,[1_1|2]), (503,432,[1_1|2]), (503,441,[1_1|2]), (503,450,[1_1|2]), (503,252,[5_1|2]), (503,261,[4_1|2]), (503,270,[1_1|2]), (503,279,[5_1|2]), (503,288,[3_1|2]), (503,297,[5_1|2]), (503,306,[5_1|2]), (503,315,[3_1|2]), (503,324,[0_1|2]), (503,333,[5_1|2]), (503,342,[0_1|2]), (504,505,[3_1|2]), (505,506,[3_1|2]), (506,507,[2_1|2]), (507,508,[1_1|2]), (508,509,[1_1|2]), (509,510,[1_1|2]), (510,511,[2_1|2]), (511,512,[3_1|2]), (511,468,[0_1|2]), (512,152,[1_1|2]), (512,153,[1_1|2]), (512,270,[1_1|2]), (512,396,[1_1|2]), (512,504,[1_1|2]), (512,424,[1_1|2]), (512,243,[2_1|2]), (512,252,[5_1|2]), (512,261,[4_1|2]), (512,279,[5_1|2]), (512,288,[3_1|2]), (512,297,[5_1|2]), (512,306,[5_1|2]), (512,315,[3_1|2]), (512,324,[0_1|2]), (512,333,[5_1|2]), (512,342,[0_1|2]), (513,514,[0_1|2]), (514,515,[0_1|2]), (515,516,[0_1|2]), (516,517,[5_1|2]), (517,518,[0_1|2]), (518,519,[1_1|2]), (519,520,[3_1|2]), (520,521,[4_1|2]), (521,152,[4_1|2]), (521,162,[4_1|2]), (521,198,[4_1|2]), (521,261,[4_1|2]), (521,433,[4_1|2]), (521,153,[1_1|2]), (521,171,[5_1|2]), (521,180,[3_1|2]), (521,189,[0_1|2]), (521,207,[3_1|2]), (521,216,[3_1|2]), (521,225,[5_1|2]), (521,234,[5_1|2]), (521,522,[4_1|3]), (521,531,[1_1|3]), (522,523,[5_1|3]), (523,524,[5_1|3]), (524,525,[2_1|3]), (525,526,[2_1|3]), (526,527,[2_1|3]), (527,528,[4_1|3]), (528,529,[5_1|3]), (529,530,[1_1|3]), (530,361,[1_1|3]), (531,532,[3_1|3]), (532,533,[1_1|3]), (533,534,[2_1|3]), (534,535,[4_1|3]), (535,536,[5_1|3]), (536,537,[2_1|3]), (537,538,[4_1|3]), (538,539,[3_1|3]), (539,370,[4_1|3]), (539,388,[4_1|3]), (540,541,[5_1|3]), (541,542,[5_1|3]), (542,543,[2_1|3]), (543,544,[2_1|3]), (544,545,[2_1|3]), (545,546,[4_1|3]), (546,547,[5_1|3]), (547,548,[1_1|3]), (548,269,[1_1|3]), (549,550,[1_1|3]), (550,551,[0_1|3]), (551,552,[3_1|3]), (552,553,[3_1|3]), (553,554,[2_1|3]), (554,555,[0_1|3]), (555,556,[2_1|3]), (556,557,[1_1|3]), (557,451,[3_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)