/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 56 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 177 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: 3(3(0(x1))) -> 3(0(2(1(0(0(1(2(1(0(x1)))))))))) 5(1(5(x1))) -> 4(4(1(0(0(3(4(1(0(2(x1)))))))))) 0(5(0(0(x1)))) -> 2(3(0(3(5(2(2(3(4(1(x1)))))))))) 3(1(0(0(x1)))) -> 3(5(5(3(5(0(0(0(0(0(x1)))))))))) 0(2(2(5(5(x1))))) -> 2(5(4(3(0(4(1(3(5(4(x1)))))))))) 2(3(3(2(4(x1))))) -> 2(1(2(5(0(1(0(5(5(4(x1)))))))))) 4(0(3(0(0(x1))))) -> 4(3(5(2(4(1(2(1(2(0(x1)))))))))) 4(5(2(5(2(x1))))) -> 5(2(1(1(1(0(2(5(0(2(x1)))))))))) 0(0(2(2(5(0(x1)))))) -> 0(2(5(3(4(0(1(2(5(0(x1)))))))))) 1(3(2(2(5(2(x1)))))) -> 1(3(5(2(4(5(2(3(4(4(x1)))))))))) 1(5(2(5(3(1(x1)))))) -> 2(5(4(1(1(0(2(5(3(2(x1)))))))))) 2(2(4(5(2(5(x1)))))) -> 2(2(4(3(5(4(5(5(2(5(x1)))))))))) 2(4(0(2(0(0(x1)))))) -> 2(0(0(2(5(4(1(1(2(1(x1)))))))))) 2(4(5(2(5(4(x1)))))) -> 2(4(1(0(0(4(4(3(5(4(x1)))))))))) 3(1(5(1(3(2(x1)))))) -> 1(1(5(3(5(1(2(4(1(5(x1)))))))))) 3(2(0(3(0(0(x1)))))) -> 3(4(1(3(4(0(2(0(4(1(x1)))))))))) 4(5(2(3(1(2(x1)))))) -> 4(4(4(0(1(2(4(2(0(2(x1)))))))))) 4(5(2(5(3(2(x1)))))) -> 4(1(3(4(3(2(4(4(0(2(x1)))))))))) 5(0(5(2(5(2(x1)))))) -> 5(4(2(4(4(3(4(5(0(0(x1)))))))))) 5(2(5(2(2(5(x1)))))) -> 4(4(5(5(1(0(5(2(1(2(x1)))))))))) 0(0(5(2(4(2(5(x1))))))) -> 0(2(0(4(2(1(2(1(3(3(x1)))))))))) 0(5(2(0(3(2(2(x1))))))) -> 2(1(2(4(3(5(2(2(2(2(x1)))))))))) 0(5(2(0(4(3(0(x1))))))) -> 1(1(1(2(4(3(3(4(5(4(x1)))))))))) 1(0(3(2(1(3(2(x1))))))) -> 1(1(1(1(5(5(3(5(1(5(x1)))))))))) 2(0(1(4(5(2(0(x1))))))) -> 2(2(1(0(0(0(3(3(4(4(x1)))))))))) 2(1(5(5(0(1(1(x1))))))) -> 0(1(2(3(4(4(3(3(4(0(x1)))))))))) 2(4(1(3(0(5(2(x1))))))) -> 2(4(4(1(5(3(4(4(4(0(x1)))))))))) 2(4(4(3(1(0(0(x1))))))) -> 2(4(2(1(3(5(4(4(0(1(x1)))))))))) 3(1(4(2(3(0(0(x1))))))) -> 1(0(3(4(3(3(4(2(4(1(x1)))))))))) 3(2(1(4(2(1(4(x1))))))) -> 3(5(4(2(3(0(1(1(5(5(x1)))))))))) 3(2(2(2(2(4(2(x1))))))) -> 3(1(3(0(1(1(4(2(1(0(x1)))))))))) 3(2(5(0(0(3(0(x1))))))) -> 4(3(4(1(2(2(0(2(5(4(x1)))))))))) 3(3(5(2(2(5(2(x1))))))) -> 3(0(1(0(5(1(5(5(4(2(x1)))))))))) 4(2(1(5(1(5(0(x1))))))) -> 4(4(5(4(0(2(3(4(5(0(x1)))))))))) 5(0(0(3(1(5(0(x1))))))) -> 4(0(0(4(0(5(4(4(1(0(x1)))))))))) 5(0(0(4(4(5(0(x1))))))) -> 4(2(1(1(3(4(4(0(5(0(x1)))))))))) 5(0(3(0(5(0(1(x1))))))) -> 5(1(3(3(4(5(3(0(4(1(x1)))))))))) 5(0(3(2(0(5(0(x1))))))) -> 5(1(2(4(0(0(1(5(1(0(x1)))))))))) 5(0(3(3(2(2(0(x1))))))) -> 4(2(2(2(2(5(3(5(1(0(x1)))))))))) 5(1(4(2(0(5(2(x1))))))) -> 4(0(5(1(0(4(4(0(4(2(x1)))))))))) 5(3(3(2(1(5(5(x1))))))) -> 1(3(5(0(2(1(2(0(2(3(x1)))))))))) 5(5(0(3(2(0(3(x1))))))) -> 0(4(4(0(2(5(4(1(0(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(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: 3(3(0(x1))) -> 3(0(2(1(0(0(1(2(1(0(x1)))))))))) 5(1(5(x1))) -> 4(4(1(0(0(3(4(1(0(2(x1)))))))))) 0(5(0(0(x1)))) -> 2(3(0(3(5(2(2(3(4(1(x1)))))))))) 3(1(0(0(x1)))) -> 3(5(5(3(5(0(0(0(0(0(x1)))))))))) 0(2(2(5(5(x1))))) -> 2(5(4(3(0(4(1(3(5(4(x1)))))))))) 2(3(3(2(4(x1))))) -> 2(1(2(5(0(1(0(5(5(4(x1)))))))))) 4(0(3(0(0(x1))))) -> 4(3(5(2(4(1(2(1(2(0(x1)))))))))) 4(5(2(5(2(x1))))) -> 5(2(1(1(1(0(2(5(0(2(x1)))))))))) 0(0(2(2(5(0(x1)))))) -> 0(2(5(3(4(0(1(2(5(0(x1)))))))))) 1(3(2(2(5(2(x1)))))) -> 1(3(5(2(4(5(2(3(4(4(x1)))))))))) 1(5(2(5(3(1(x1)))))) -> 2(5(4(1(1(0(2(5(3(2(x1)))))))))) 2(2(4(5(2(5(x1)))))) -> 2(2(4(3(5(4(5(5(2(5(x1)))))))))) 2(4(0(2(0(0(x1)))))) -> 2(0(0(2(5(4(1(1(2(1(x1)))))))))) 2(4(5(2(5(4(x1)))))) -> 2(4(1(0(0(4(4(3(5(4(x1)))))))))) 3(1(5(1(3(2(x1)))))) -> 1(1(5(3(5(1(2(4(1(5(x1)))))))))) 3(2(0(3(0(0(x1)))))) -> 3(4(1(3(4(0(2(0(4(1(x1)))))))))) 4(5(2(3(1(2(x1)))))) -> 4(4(4(0(1(2(4(2(0(2(x1)))))))))) 4(5(2(5(3(2(x1)))))) -> 4(1(3(4(3(2(4(4(0(2(x1)))))))))) 5(0(5(2(5(2(x1)))))) -> 5(4(2(4(4(3(4(5(0(0(x1)))))))))) 5(2(5(2(2(5(x1)))))) -> 4(4(5(5(1(0(5(2(1(2(x1)))))))))) 0(0(5(2(4(2(5(x1))))))) -> 0(2(0(4(2(1(2(1(3(3(x1)))))))))) 0(5(2(0(3(2(2(x1))))))) -> 2(1(2(4(3(5(2(2(2(2(x1)))))))))) 0(5(2(0(4(3(0(x1))))))) -> 1(1(1(2(4(3(3(4(5(4(x1)))))))))) 1(0(3(2(1(3(2(x1))))))) -> 1(1(1(1(5(5(3(5(1(5(x1)))))))))) 2(0(1(4(5(2(0(x1))))))) -> 2(2(1(0(0(0(3(3(4(4(x1)))))))))) 2(1(5(5(0(1(1(x1))))))) -> 0(1(2(3(4(4(3(3(4(0(x1)))))))))) 2(4(1(3(0(5(2(x1))))))) -> 2(4(4(1(5(3(4(4(4(0(x1)))))))))) 2(4(4(3(1(0(0(x1))))))) -> 2(4(2(1(3(5(4(4(0(1(x1)))))))))) 3(1(4(2(3(0(0(x1))))))) -> 1(0(3(4(3(3(4(2(4(1(x1)))))))))) 3(2(1(4(2(1(4(x1))))))) -> 3(5(4(2(3(0(1(1(5(5(x1)))))))))) 3(2(2(2(2(4(2(x1))))))) -> 3(1(3(0(1(1(4(2(1(0(x1)))))))))) 3(2(5(0(0(3(0(x1))))))) -> 4(3(4(1(2(2(0(2(5(4(x1)))))))))) 3(3(5(2(2(5(2(x1))))))) -> 3(0(1(0(5(1(5(5(4(2(x1)))))))))) 4(2(1(5(1(5(0(x1))))))) -> 4(4(5(4(0(2(3(4(5(0(x1)))))))))) 5(0(0(3(1(5(0(x1))))))) -> 4(0(0(4(0(5(4(4(1(0(x1)))))))))) 5(0(0(4(4(5(0(x1))))))) -> 4(2(1(1(3(4(4(0(5(0(x1)))))))))) 5(0(3(0(5(0(1(x1))))))) -> 5(1(3(3(4(5(3(0(4(1(x1)))))))))) 5(0(3(2(0(5(0(x1))))))) -> 5(1(2(4(0(0(1(5(1(0(x1)))))))))) 5(0(3(3(2(2(0(x1))))))) -> 4(2(2(2(2(5(3(5(1(0(x1)))))))))) 5(1(4(2(0(5(2(x1))))))) -> 4(0(5(1(0(4(4(0(4(2(x1)))))))))) 5(3(3(2(1(5(5(x1))))))) -> 1(3(5(0(2(1(2(0(2(3(x1)))))))))) 5(5(0(3(2(0(3(x1))))))) -> 0(4(4(0(2(5(4(1(0(3(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(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: 3(3(0(x1))) -> 3(0(2(1(0(0(1(2(1(0(x1)))))))))) 5(1(5(x1))) -> 4(4(1(0(0(3(4(1(0(2(x1)))))))))) 0(5(0(0(x1)))) -> 2(3(0(3(5(2(2(3(4(1(x1)))))))))) 3(1(0(0(x1)))) -> 3(5(5(3(5(0(0(0(0(0(x1)))))))))) 0(2(2(5(5(x1))))) -> 2(5(4(3(0(4(1(3(5(4(x1)))))))))) 2(3(3(2(4(x1))))) -> 2(1(2(5(0(1(0(5(5(4(x1)))))))))) 4(0(3(0(0(x1))))) -> 4(3(5(2(4(1(2(1(2(0(x1)))))))))) 4(5(2(5(2(x1))))) -> 5(2(1(1(1(0(2(5(0(2(x1)))))))))) 0(0(2(2(5(0(x1)))))) -> 0(2(5(3(4(0(1(2(5(0(x1)))))))))) 1(3(2(2(5(2(x1)))))) -> 1(3(5(2(4(5(2(3(4(4(x1)))))))))) 1(5(2(5(3(1(x1)))))) -> 2(5(4(1(1(0(2(5(3(2(x1)))))))))) 2(2(4(5(2(5(x1)))))) -> 2(2(4(3(5(4(5(5(2(5(x1)))))))))) 2(4(0(2(0(0(x1)))))) -> 2(0(0(2(5(4(1(1(2(1(x1)))))))))) 2(4(5(2(5(4(x1)))))) -> 2(4(1(0(0(4(4(3(5(4(x1)))))))))) 3(1(5(1(3(2(x1)))))) -> 1(1(5(3(5(1(2(4(1(5(x1)))))))))) 3(2(0(3(0(0(x1)))))) -> 3(4(1(3(4(0(2(0(4(1(x1)))))))))) 4(5(2(3(1(2(x1)))))) -> 4(4(4(0(1(2(4(2(0(2(x1)))))))))) 4(5(2(5(3(2(x1)))))) -> 4(1(3(4(3(2(4(4(0(2(x1)))))))))) 5(0(5(2(5(2(x1)))))) -> 5(4(2(4(4(3(4(5(0(0(x1)))))))))) 5(2(5(2(2(5(x1)))))) -> 4(4(5(5(1(0(5(2(1(2(x1)))))))))) 0(0(5(2(4(2(5(x1))))))) -> 0(2(0(4(2(1(2(1(3(3(x1)))))))))) 0(5(2(0(3(2(2(x1))))))) -> 2(1(2(4(3(5(2(2(2(2(x1)))))))))) 0(5(2(0(4(3(0(x1))))))) -> 1(1(1(2(4(3(3(4(5(4(x1)))))))))) 1(0(3(2(1(3(2(x1))))))) -> 1(1(1(1(5(5(3(5(1(5(x1)))))))))) 2(0(1(4(5(2(0(x1))))))) -> 2(2(1(0(0(0(3(3(4(4(x1)))))))))) 2(1(5(5(0(1(1(x1))))))) -> 0(1(2(3(4(4(3(3(4(0(x1)))))))))) 2(4(1(3(0(5(2(x1))))))) -> 2(4(4(1(5(3(4(4(4(0(x1)))))))))) 2(4(4(3(1(0(0(x1))))))) -> 2(4(2(1(3(5(4(4(0(1(x1)))))))))) 3(1(4(2(3(0(0(x1))))))) -> 1(0(3(4(3(3(4(2(4(1(x1)))))))))) 3(2(1(4(2(1(4(x1))))))) -> 3(5(4(2(3(0(1(1(5(5(x1)))))))))) 3(2(2(2(2(4(2(x1))))))) -> 3(1(3(0(1(1(4(2(1(0(x1)))))))))) 3(2(5(0(0(3(0(x1))))))) -> 4(3(4(1(2(2(0(2(5(4(x1)))))))))) 3(3(5(2(2(5(2(x1))))))) -> 3(0(1(0(5(1(5(5(4(2(x1)))))))))) 4(2(1(5(1(5(0(x1))))))) -> 4(4(5(4(0(2(3(4(5(0(x1)))))))))) 5(0(0(3(1(5(0(x1))))))) -> 4(0(0(4(0(5(4(4(1(0(x1)))))))))) 5(0(0(4(4(5(0(x1))))))) -> 4(2(1(1(3(4(4(0(5(0(x1)))))))))) 5(0(3(0(5(0(1(x1))))))) -> 5(1(3(3(4(5(3(0(4(1(x1)))))))))) 5(0(3(2(0(5(0(x1))))))) -> 5(1(2(4(0(0(1(5(1(0(x1)))))))))) 5(0(3(3(2(2(0(x1))))))) -> 4(2(2(2(2(5(3(5(1(0(x1)))))))))) 5(1(4(2(0(5(2(x1))))))) -> 4(0(5(1(0(4(4(0(4(2(x1)))))))))) 5(3(3(2(1(5(5(x1))))))) -> 1(3(5(0(2(1(2(0(2(3(x1)))))))))) 5(5(0(3(2(0(3(x1))))))) -> 0(4(4(0(2(5(4(1(0(3(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(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: 3(3(0(x1))) -> 3(0(2(1(0(0(1(2(1(0(x1)))))))))) 5(1(5(x1))) -> 4(4(1(0(0(3(4(1(0(2(x1)))))))))) 0(5(0(0(x1)))) -> 2(3(0(3(5(2(2(3(4(1(x1)))))))))) 3(1(0(0(x1)))) -> 3(5(5(3(5(0(0(0(0(0(x1)))))))))) 0(2(2(5(5(x1))))) -> 2(5(4(3(0(4(1(3(5(4(x1)))))))))) 2(3(3(2(4(x1))))) -> 2(1(2(5(0(1(0(5(5(4(x1)))))))))) 4(0(3(0(0(x1))))) -> 4(3(5(2(4(1(2(1(2(0(x1)))))))))) 4(5(2(5(2(x1))))) -> 5(2(1(1(1(0(2(5(0(2(x1)))))))))) 0(0(2(2(5(0(x1)))))) -> 0(2(5(3(4(0(1(2(5(0(x1)))))))))) 1(3(2(2(5(2(x1)))))) -> 1(3(5(2(4(5(2(3(4(4(x1)))))))))) 1(5(2(5(3(1(x1)))))) -> 2(5(4(1(1(0(2(5(3(2(x1)))))))))) 2(2(4(5(2(5(x1)))))) -> 2(2(4(3(5(4(5(5(2(5(x1)))))))))) 2(4(0(2(0(0(x1)))))) -> 2(0(0(2(5(4(1(1(2(1(x1)))))))))) 2(4(5(2(5(4(x1)))))) -> 2(4(1(0(0(4(4(3(5(4(x1)))))))))) 3(1(5(1(3(2(x1)))))) -> 1(1(5(3(5(1(2(4(1(5(x1)))))))))) 3(2(0(3(0(0(x1)))))) -> 3(4(1(3(4(0(2(0(4(1(x1)))))))))) 4(5(2(3(1(2(x1)))))) -> 4(4(4(0(1(2(4(2(0(2(x1)))))))))) 4(5(2(5(3(2(x1)))))) -> 4(1(3(4(3(2(4(4(0(2(x1)))))))))) 5(0(5(2(5(2(x1)))))) -> 5(4(2(4(4(3(4(5(0(0(x1)))))))))) 5(2(5(2(2(5(x1)))))) -> 4(4(5(5(1(0(5(2(1(2(x1)))))))))) 0(0(5(2(4(2(5(x1))))))) -> 0(2(0(4(2(1(2(1(3(3(x1)))))))))) 0(5(2(0(3(2(2(x1))))))) -> 2(1(2(4(3(5(2(2(2(2(x1)))))))))) 0(5(2(0(4(3(0(x1))))))) -> 1(1(1(2(4(3(3(4(5(4(x1)))))))))) 1(0(3(2(1(3(2(x1))))))) -> 1(1(1(1(5(5(3(5(1(5(x1)))))))))) 2(0(1(4(5(2(0(x1))))))) -> 2(2(1(0(0(0(3(3(4(4(x1)))))))))) 2(1(5(5(0(1(1(x1))))))) -> 0(1(2(3(4(4(3(3(4(0(x1)))))))))) 2(4(1(3(0(5(2(x1))))))) -> 2(4(4(1(5(3(4(4(4(0(x1)))))))))) 2(4(4(3(1(0(0(x1))))))) -> 2(4(2(1(3(5(4(4(0(1(x1)))))))))) 3(1(4(2(3(0(0(x1))))))) -> 1(0(3(4(3(3(4(2(4(1(x1)))))))))) 3(2(1(4(2(1(4(x1))))))) -> 3(5(4(2(3(0(1(1(5(5(x1)))))))))) 3(2(2(2(2(4(2(x1))))))) -> 3(1(3(0(1(1(4(2(1(0(x1)))))))))) 3(2(5(0(0(3(0(x1))))))) -> 4(3(4(1(2(2(0(2(5(4(x1)))))))))) 3(3(5(2(2(5(2(x1))))))) -> 3(0(1(0(5(1(5(5(4(2(x1)))))))))) 4(2(1(5(1(5(0(x1))))))) -> 4(4(5(4(0(2(3(4(5(0(x1)))))))))) 5(0(0(3(1(5(0(x1))))))) -> 4(0(0(4(0(5(4(4(1(0(x1)))))))))) 5(0(0(4(4(5(0(x1))))))) -> 4(2(1(1(3(4(4(0(5(0(x1)))))))))) 5(0(3(0(5(0(1(x1))))))) -> 5(1(3(3(4(5(3(0(4(1(x1)))))))))) 5(0(3(2(0(5(0(x1))))))) -> 5(1(2(4(0(0(1(5(1(0(x1)))))))))) 5(0(3(3(2(2(0(x1))))))) -> 4(2(2(2(2(5(3(5(1(0(x1)))))))))) 5(1(4(2(0(5(2(x1))))))) -> 4(0(5(1(0(4(4(0(4(2(x1)))))))))) 5(3(3(2(1(5(5(x1))))))) -> 1(3(5(0(2(1(2(0(2(3(x1)))))))))) 5(5(0(3(2(0(3(x1))))))) -> 0(4(4(0(2(5(4(1(0(3(x1)))))))))) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) encode_4(x_1) -> 4(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 3. The certificate found is represented by the following graph. "[151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576] {(151,152,[3_1|0, 5_1|0, 0_1|0, 2_1|0, 4_1|0, 1_1|0, encArg_1|0, encode_3_1|0, encode_0_1|0, encode_2_1|0, encode_1_1|0, encode_5_1|0, encode_4_1|0]), (151,153,[3_1|1, 5_1|1, 0_1|1, 2_1|1, 4_1|1, 1_1|1]), (151,154,[3_1|2]), (151,163,[3_1|2]), (151,172,[3_1|2]), (151,181,[1_1|2]), (151,190,[1_1|2]), (151,199,[3_1|2]), (151,208,[3_1|2]), (151,217,[3_1|2]), (151,226,[4_1|2]), (151,235,[4_1|2]), (151,244,[4_1|2]), (151,253,[5_1|2]), (151,262,[4_1|2]), (151,271,[4_1|2]), (151,280,[5_1|2]), (151,289,[5_1|2]), (151,298,[4_1|2]), (151,307,[4_1|2]), (151,316,[1_1|2]), (151,325,[0_1|2]), (151,334,[2_1|2]), (151,343,[2_1|2]), (151,352,[1_1|2]), (151,361,[2_1|2]), (151,370,[0_1|2]), (151,379,[0_1|2]), (151,388,[2_1|2]), (151,397,[2_1|2]), (151,406,[2_1|2]), (151,415,[2_1|2]), (151,424,[2_1|2]), (151,433,[2_1|2]), (151,442,[2_1|2]), (151,451,[0_1|2]), 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(387,208,[3_1|2]), (387,217,[3_1|2]), (387,226,[4_1|2]), (387,550,[3_1|3]), (388,389,[1_1|2]), (389,390,[2_1|2]), (390,391,[5_1|2]), (391,392,[0_1|2]), (392,393,[1_1|2]), (393,394,[0_1|2]), (394,395,[5_1|2]), (395,396,[5_1|2]), (396,153,[4_1|2]), (396,226,[4_1|2]), (396,235,[4_1|2]), (396,244,[4_1|2]), (396,262,[4_1|2]), (396,271,[4_1|2]), (396,298,[4_1|2]), (396,307,[4_1|2]), (396,460,[4_1|2]), (396,478,[4_1|2]), (396,487,[4_1|2]), (396,496,[4_1|2]), (396,416,[4_1|2]), (396,425,[4_1|2]), (396,434,[4_1|2]), (396,469,[5_1|2]), (397,398,[2_1|2]), (398,399,[4_1|2]), (399,400,[3_1|2]), (400,401,[5_1|2]), (401,402,[4_1|2]), (402,403,[5_1|2]), (403,404,[5_1|2]), (403,307,[4_1|2]), (404,405,[2_1|2]), (405,153,[5_1|2]), (405,253,[5_1|2]), (405,280,[5_1|2]), (405,289,[5_1|2]), (405,469,[5_1|2]), (405,362,[5_1|2]), (405,515,[5_1|2]), (405,235,[4_1|2]), (405,244,[4_1|2]), (405,262,[4_1|2]), (405,271,[4_1|2]), (405,298,[4_1|2]), (405,307,[4_1|2]), (405,316,[1_1|2]), (405,325,[0_1|2]), (406,407,[0_1|2]), (407,408,[0_1|2]), (408,409,[2_1|2]), (409,410,[5_1|2]), (410,411,[4_1|2]), (411,412,[1_1|2]), (412,413,[1_1|2]), (413,414,[2_1|2]), (413,451,[0_1|2]), (414,153,[1_1|2]), (414,325,[1_1|2]), (414,370,[1_1|2]), (414,379,[1_1|2]), (414,451,[1_1|2]), (414,408,[1_1|2]), (414,505,[1_1|2]), (414,514,[2_1|2]), (414,523,[1_1|2]), (415,416,[4_1|2]), (416,417,[1_1|2]), (417,418,[0_1|2]), (418,419,[0_1|2]), (419,420,[4_1|2]), (420,421,[4_1|2]), (421,422,[3_1|2]), (422,423,[5_1|2]), (423,153,[4_1|2]), (423,226,[4_1|2]), (423,235,[4_1|2]), (423,244,[4_1|2]), (423,262,[4_1|2]), (423,271,[4_1|2]), (423,298,[4_1|2]), (423,307,[4_1|2]), (423,460,[4_1|2]), (423,478,[4_1|2]), (423,487,[4_1|2]), (423,496,[4_1|2]), (423,254,[4_1|2]), (423,363,[4_1|2]), (423,516,[4_1|2]), (423,469,[5_1|2]), (424,425,[4_1|2]), (425,426,[4_1|2]), (426,427,[1_1|2]), (427,428,[5_1|2]), (428,429,[3_1|2]), (429,430,[4_1|2]), (430,431,[4_1|2]), (431,432,[4_1|2]), (431,460,[4_1|2]), (432,153,[0_1|2]), (432,334,[0_1|2, 2_1|2]), (432,343,[0_1|2, 2_1|2]), (432,361,[0_1|2, 2_1|2]), (432,388,[0_1|2]), (432,397,[0_1|2]), (432,406,[0_1|2]), (432,415,[0_1|2]), (432,424,[0_1|2]), (432,433,[0_1|2]), (432,442,[0_1|2]), (432,514,[0_1|2]), (432,470,[0_1|2]), (432,352,[1_1|2]), (432,370,[0_1|2]), (432,379,[0_1|2]), (433,434,[4_1|2]), (434,435,[2_1|2]), (435,436,[1_1|2]), (436,437,[3_1|2]), (437,438,[5_1|2]), (438,439,[4_1|2]), (439,440,[4_1|2]), (440,441,[0_1|2]), (441,153,[1_1|2]), (441,325,[1_1|2]), (441,370,[1_1|2]), (441,379,[1_1|2]), (441,451,[1_1|2]), (441,505,[1_1|2]), (441,514,[2_1|2]), (441,523,[1_1|2]), (442,443,[2_1|2]), (443,444,[1_1|2]), (444,445,[0_1|2]), (445,446,[0_1|2]), (446,447,[0_1|2]), (447,448,[3_1|2]), (448,449,[3_1|2]), (449,450,[4_1|2]), (450,153,[4_1|2]), (450,325,[4_1|2]), (450,370,[4_1|2]), (450,379,[4_1|2]), (450,451,[4_1|2]), (450,407,[4_1|2]), (450,460,[4_1|2]), (450,469,[5_1|2]), (450,478,[4_1|2]), (450,487,[4_1|2]), (450,496,[4_1|2]), (451,452,[1_1|2]), (452,453,[2_1|2]), (453,454,[3_1|2]), (454,455,[4_1|2]), (455,456,[4_1|2]), (456,457,[3_1|2]), (457,458,[3_1|2]), (458,459,[4_1|2]), (458,460,[4_1|2]), (459,153,[0_1|2]), (459,181,[0_1|2]), (459,190,[0_1|2]), (459,316,[0_1|2]), (459,352,[0_1|2, 1_1|2]), (459,505,[0_1|2]), (459,523,[0_1|2]), (459,182,[0_1|2]), (459,353,[0_1|2]), (459,524,[0_1|2]), (459,334,[2_1|2]), (459,343,[2_1|2]), (459,361,[2_1|2]), (459,370,[0_1|2]), (459,379,[0_1|2]), (460,461,[3_1|2]), (461,462,[5_1|2]), (462,463,[2_1|2]), (463,464,[4_1|2]), (464,465,[1_1|2]), (465,466,[2_1|2]), (466,467,[1_1|2]), (467,468,[2_1|2]), (467,442,[2_1|2]), (468,153,[0_1|2]), (468,325,[0_1|2]), (468,370,[0_1|2]), (468,379,[0_1|2]), (468,451,[0_1|2]), (468,334,[2_1|2]), (468,343,[2_1|2]), (468,352,[1_1|2]), (468,361,[2_1|2]), (469,470,[2_1|2]), (470,471,[1_1|2]), (471,472,[1_1|2]), (472,473,[1_1|2]), (473,474,[0_1|2]), (474,475,[2_1|2]), (475,476,[5_1|2]), (476,477,[0_1|2]), (476,361,[2_1|2]), (477,153,[2_1|2]), (477,334,[2_1|2]), (477,343,[2_1|2]), (477,361,[2_1|2]), (477,388,[2_1|2]), (477,397,[2_1|2]), (477,406,[2_1|2]), (477,415,[2_1|2]), (477,424,[2_1|2]), (477,433,[2_1|2]), (477,442,[2_1|2]), (477,514,[2_1|2]), (477,470,[2_1|2]), (477,451,[0_1|2]), (478,479,[1_1|2]), (479,480,[3_1|2]), (480,481,[4_1|2]), (481,482,[3_1|2]), (482,483,[2_1|2]), (483,484,[4_1|2]), (484,485,[4_1|2]), (485,486,[0_1|2]), (485,361,[2_1|2]), (486,153,[2_1|2]), (486,334,[2_1|2]), (486,343,[2_1|2]), (486,361,[2_1|2]), (486,388,[2_1|2]), (486,397,[2_1|2]), (486,406,[2_1|2]), (486,415,[2_1|2]), (486,424,[2_1|2]), (486,433,[2_1|2]), (486,442,[2_1|2]), (486,514,[2_1|2]), (486,451,[0_1|2]), (487,488,[4_1|2]), (488,489,[4_1|2]), (489,490,[0_1|2]), (490,491,[1_1|2]), (491,492,[2_1|2]), (492,493,[4_1|2]), (493,494,[2_1|2]), (494,495,[0_1|2]), (494,361,[2_1|2]), (495,153,[2_1|2]), (495,334,[2_1|2]), (495,343,[2_1|2]), (495,361,[2_1|2]), (495,388,[2_1|2]), (495,397,[2_1|2]), (495,406,[2_1|2]), (495,415,[2_1|2]), (495,424,[2_1|2]), (495,433,[2_1|2]), (495,442,[2_1|2]), (495,514,[2_1|2]), (495,451,[0_1|2]), (496,497,[4_1|2]), (497,498,[5_1|2]), (498,499,[4_1|2]), (499,500,[0_1|2]), (500,501,[2_1|2]), (501,502,[3_1|2]), (502,503,[4_1|2]), (503,504,[5_1|2]), (503,253,[5_1|2]), (503,262,[4_1|2]), (503,271,[4_1|2]), (503,280,[5_1|2]), (503,289,[5_1|2]), (503,298,[4_1|2]), (504,153,[0_1|2]), (504,325,[0_1|2]), (504,370,[0_1|2]), (504,379,[0_1|2]), (504,451,[0_1|2]), (504,334,[2_1|2]), (504,343,[2_1|2]), (504,352,[1_1|2]), (504,361,[2_1|2]), (505,506,[3_1|2]), (506,507,[5_1|2]), (507,508,[2_1|2]), (508,509,[4_1|2]), (509,510,[5_1|2]), (510,511,[2_1|2]), (511,512,[3_1|2]), (512,513,[4_1|2]), (513,153,[4_1|2]), (513,334,[4_1|2]), (513,343,[4_1|2]), (513,361,[4_1|2]), (513,388,[4_1|2]), (513,397,[4_1|2]), (513,406,[4_1|2]), (513,415,[4_1|2]), (513,424,[4_1|2]), (513,433,[4_1|2]), (513,442,[4_1|2]), (513,514,[4_1|2]), (513,470,[4_1|2]), (513,460,[4_1|2]), (513,469,[5_1|2]), (513,478,[4_1|2]), (513,487,[4_1|2]), (513,496,[4_1|2]), (514,515,[5_1|2]), (515,516,[4_1|2]), (516,517,[1_1|2]), (517,518,[1_1|2]), (518,519,[0_1|2]), (519,520,[2_1|2]), (520,521,[5_1|2]), (521,522,[3_1|2]), (521,199,[3_1|2]), (521,208,[3_1|2]), (521,217,[3_1|2]), (521,226,[4_1|2]), (522,153,[2_1|2]), (522,181,[2_1|2]), (522,190,[2_1|2]), (522,316,[2_1|2]), (522,352,[2_1|2]), (522,505,[2_1|2]), (522,523,[2_1|2]), (522,218,[2_1|2]), (522,388,[2_1|2]), (522,397,[2_1|2]), (522,406,[2_1|2]), (522,415,[2_1|2]), (522,424,[2_1|2]), (522,433,[2_1|2]), (522,442,[2_1|2]), (522,451,[0_1|2]), (523,524,[1_1|2]), (524,525,[1_1|2]), (525,526,[1_1|2]), (526,527,[5_1|2]), (527,528,[5_1|2]), (528,529,[3_1|2]), (529,530,[5_1|2]), (529,568,[4_1|3]), (530,531,[1_1|2]), (530,514,[2_1|2]), (531,153,[5_1|2]), (531,334,[5_1|2]), (531,343,[5_1|2]), (531,361,[5_1|2]), (531,388,[5_1|2]), (531,397,[5_1|2]), (531,406,[5_1|2]), (531,415,[5_1|2]), (531,424,[5_1|2]), (531,433,[5_1|2]), (531,442,[5_1|2]), (531,514,[5_1|2]), (531,235,[4_1|2]), (531,244,[4_1|2]), (531,253,[5_1|2]), (531,262,[4_1|2]), (531,271,[4_1|2]), (531,280,[5_1|2]), (531,289,[5_1|2]), (531,298,[4_1|2]), (531,307,[4_1|2]), (531,316,[1_1|2]), (531,325,[0_1|2]), (532,533,[4_1|3]), (533,534,[1_1|3]), (534,535,[0_1|3]), (535,536,[0_1|3]), (536,537,[3_1|3]), (537,538,[4_1|3]), (538,539,[1_1|3]), (539,540,[0_1|3]), (540,169,[2_1|3]), (541,542,[3_1|3]), (542,543,[0_1|3]), (543,544,[3_1|3]), (544,545,[5_1|3]), (545,546,[2_1|3]), (546,547,[2_1|3]), (547,548,[3_1|3]), (548,549,[4_1|3]), (549,325,[1_1|3]), (549,370,[1_1|3]), (549,379,[1_1|3]), (549,451,[1_1|3]), (550,551,[0_1|3]), (551,552,[2_1|3]), (552,553,[1_1|3]), (553,554,[0_1|3]), (554,555,[0_1|3]), (555,556,[1_1|3]), (556,557,[2_1|3]), (557,558,[1_1|3]), (558,155,[0_1|3]), (558,164,[0_1|3]), (559,560,[0_1|3]), (560,561,[2_1|3]), (561,562,[1_1|3]), (562,563,[0_1|3]), (563,564,[0_1|3]), (564,565,[1_1|3]), (565,566,[2_1|3]), (566,567,[1_1|3]), (567,325,[0_1|3]), (567,370,[0_1|3]), (567,379,[0_1|3]), (567,451,[0_1|3]), (567,155,[0_1|3]), (567,164,[0_1|3]), (567,551,[0_1|3]), (568,569,[4_1|3]), (569,570,[1_1|3]), (570,571,[0_1|3]), (571,572,[0_1|3]), (572,573,[3_1|3]), (573,574,[4_1|3]), (574,575,[1_1|3]), (575,576,[0_1|3]), (575,361,[2_1|2]), (576,153,[2_1|3]), (576,334,[2_1|3]), (576,343,[2_1|3]), (576,361,[2_1|3]), (576,388,[2_1|3, 2_1|2]), (576,397,[2_1|3, 2_1|2]), (576,406,[2_1|3, 2_1|2]), (576,415,[2_1|3, 2_1|2]), (576,424,[2_1|3, 2_1|2]), (576,433,[2_1|3, 2_1|2]), (576,442,[2_1|3, 2_1|2]), (576,514,[2_1|3]), (576,253,[2_1|3]), (576,280,[2_1|3]), (576,289,[2_1|3]), (576,451,[0_1|2])}" ---------------------------------------- (8) BOUNDS(1, n^1)