/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 27 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 199 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(2(1(0(x1)))) -> 0(4(2(0(4(2(4(3(0(0(x1)))))))))) 2(1(0(1(x1)))) -> 3(4(1(2(0(0(0(4(2(1(x1)))))))))) 2(5(5(0(x1)))) -> 2(4(2(5(2(4(4(3(5(2(x1)))))))))) 2(5(5(5(x1)))) -> 2(2(2(0(0(3(4(1(2(4(x1)))))))))) 5(4(0(3(x1)))) -> 5(0(1(0(0(0(2(0(2(0(x1)))))))))) 1(5(4(2(1(x1))))) -> 1(1(0(2(4(2(2(2(4(1(x1)))))))))) 2(3(2(1(0(x1))))) -> 2(0(0(3(2(2(4(2(1(4(x1)))))))))) 2(5(5(5(2(x1))))) -> 4(5(4(5(3(3(0(3(2(0(x1)))))))))) 4(5(1(0(1(x1))))) -> 4(1(2(0(2(0(0(3(4(1(x1)))))))))) 5(0(4(4(2(x1))))) -> 1(2(0(2(0(0(4(0(4(2(x1)))))))))) 5(0(5(1(5(x1))))) -> 5(0(3(5(2(0(4(2(0(4(x1)))))))))) 5(1(1(1(5(x1))))) -> 3(5(3(3(1(4(0(4(2(4(x1)))))))))) 5(1(2(3(2(x1))))) -> 5(5(3(0(0(0(0(0(2(0(x1)))))))))) 5(1(4(3(0(x1))))) -> 3(5(2(2(4(1(3(0(2(0(x1)))))))))) 5(5(0(2(5(x1))))) -> 3(5(1(1(2(2(4(3(2(3(x1)))))))))) 5(5(5(5(0(x1))))) -> 3(5(4(3(3(3(0(4(2(2(x1)))))))))) 1(0(3(1(1(0(x1)))))) -> 2(2(0(4(2(3(5(3(4(0(x1)))))))))) 1(5(0(5(3(5(x1)))))) -> 2(5(2(0(4(5(2(4(4(5(x1)))))))))) 2(1(2(3(1(2(x1)))))) -> 2(0(1(2(3(4(2(0(2(0(x1)))))))))) 2(3(2(1(3(2(x1)))))) -> 2(3(0(4(3(0(4(2(0(0(x1)))))))))) 2(3(2(2(2(3(x1)))))) -> 1(0(4(1(3(2(0(3(4(2(x1)))))))))) 2(5(5(2(2(3(x1)))))) -> 2(1(2(2(1(4(0(0(3(0(x1)))))))))) 3(2(3(5(2(1(x1)))))) -> 3(3(2(4(3(3(2(1(3(3(x1)))))))))) 3(5(0(5(5(4(x1)))))) -> 5(1(4(3(0(0(4(4(4(3(x1)))))))))) 4(1(0(5(0(3(x1)))))) -> 2(4(1(4(4(0(1(3(4(3(x1)))))))))) 4(1(1(3(5(0(x1)))))) -> 4(2(2(0(5(2(0(4(3(0(x1)))))))))) 5(0(1(2(1(2(x1)))))) -> 0(2(4(2(1(4(4(0(2(0(x1)))))))))) 5(4(0(1(1(5(x1)))))) -> 5(0(2(2(0(1(3(3(3(5(x1)))))))))) 5(4(0(3(5(5(x1)))))) -> 3(1(0(0(2(0(0(3(1(5(x1)))))))))) 0(3(5(2(5(5(2(x1))))))) -> 4(3(2(4(0(4(4(3(1(2(x1)))))))))) 0(5(0(0(1(0(5(x1))))))) -> 2(0(2(3(1(5(2(4(2(4(x1)))))))))) 1(1(5(0(4(2(1(x1))))))) -> 2(5(2(0(1(4(3(0(0(4(x1)))))))))) 1(2(3(5(5(1(5(x1))))))) -> 1(4(3(0(3(0(4(2(5(4(x1)))))))))) 1(3(2(5(5(4(2(x1))))))) -> 1(1(4(4(1(2(5(2(0(0(x1)))))))))) 2(1(0(3(3(1(1(x1))))))) -> 1(1(4(0(0(0(1(4(3(1(x1)))))))))) 2(3(0(3(2(5(5(x1))))))) -> 1(4(3(0(2(4(5(4(4(4(x1)))))))))) 2(3(3(2(2(3(3(x1))))))) -> 2(1(4(2(4(2(5(2(0(3(x1)))))))))) 2(3(5(3(1(0(3(x1))))))) -> 4(5(2(0(2(1(4(1(2(0(x1)))))))))) 2(5(2(1(2(5(5(x1))))))) -> 4(0(4(3(3(1(1(1(1(5(x1)))))))))) 2(5(5(3(4(4(1(x1))))))) -> 1(0(2(2(1(4(2(2(4(1(x1)))))))))) 2(5(5(5(2(4(5(x1))))))) -> 1(3(0(0(0(5(1(4(1(5(x1)))))))))) 3(2(5(0(5(2(5(x1))))))) -> 3(3(3(1(3(5(1(3(4(5(x1)))))))))) 3(3(2(5(5(5(0(x1))))))) -> 1(3(4(4(1(4(0(1(1(0(x1)))))))))) 3(5(1(3(1(5(0(x1))))))) -> 5(3(1(0(2(0(4(3(0(2(x1)))))))))) 4(0(1(0(5(1(1(x1))))))) -> 2(0(0(0(1(4(0(1(1(3(x1)))))))))) 4(1(0(5(5(5(1(x1))))))) -> 2(0(2(3(5(4(3(5(0(1(x1)))))))))) 4(2(5(5(0(2(3(x1))))))) -> 4(2(3(0(1(3(4(3(3(2(x1)))))))))) 5(0(3(2(5(0(4(x1))))))) -> 1(2(4(1(1(1(4(0(2(4(x1)))))))))) 5(0(3(5(3(5(2(x1))))))) -> 5(0(1(5(3(2(4(4(5(2(x1)))))))))) 5(0(5(2(3(2(3(x1))))))) -> 5(3(0(3(0(1(4(3(0(0(x1)))))))))) 5(1(4(1(2(5(5(x1))))))) -> 0(2(4(1(4(0(1(0(0(3(x1)))))))))) 5(5(0(3(0(1(1(x1))))))) -> 2(0(0(2(4(4(4(4(2(5(x1)))))))))) 5(5(1(1(1(1(1(x1))))))) -> 3(5(0(0(0(4(1(4(0(1(x1)))))))))) 5(5(3(0(3(2(5(x1))))))) -> 0(2(0(4(4(1(4(3(2(5(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_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_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_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(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(2(1(0(x1)))) -> 0(4(2(0(4(2(4(3(0(0(x1)))))))))) 2(1(0(1(x1)))) -> 3(4(1(2(0(0(0(4(2(1(x1)))))))))) 2(5(5(0(x1)))) -> 2(4(2(5(2(4(4(3(5(2(x1)))))))))) 2(5(5(5(x1)))) -> 2(2(2(0(0(3(4(1(2(4(x1)))))))))) 5(4(0(3(x1)))) -> 5(0(1(0(0(0(2(0(2(0(x1)))))))))) 1(5(4(2(1(x1))))) -> 1(1(0(2(4(2(2(2(4(1(x1)))))))))) 2(3(2(1(0(x1))))) -> 2(0(0(3(2(2(4(2(1(4(x1)))))))))) 2(5(5(5(2(x1))))) -> 4(5(4(5(3(3(0(3(2(0(x1)))))))))) 4(5(1(0(1(x1))))) -> 4(1(2(0(2(0(0(3(4(1(x1)))))))))) 5(0(4(4(2(x1))))) -> 1(2(0(2(0(0(4(0(4(2(x1)))))))))) 5(0(5(1(5(x1))))) -> 5(0(3(5(2(0(4(2(0(4(x1)))))))))) 5(1(1(1(5(x1))))) -> 3(5(3(3(1(4(0(4(2(4(x1)))))))))) 5(1(2(3(2(x1))))) -> 5(5(3(0(0(0(0(0(2(0(x1)))))))))) 5(1(4(3(0(x1))))) -> 3(5(2(2(4(1(3(0(2(0(x1)))))))))) 5(5(0(2(5(x1))))) -> 3(5(1(1(2(2(4(3(2(3(x1)))))))))) 5(5(5(5(0(x1))))) -> 3(5(4(3(3(3(0(4(2(2(x1)))))))))) 1(0(3(1(1(0(x1)))))) -> 2(2(0(4(2(3(5(3(4(0(x1)))))))))) 1(5(0(5(3(5(x1)))))) -> 2(5(2(0(4(5(2(4(4(5(x1)))))))))) 2(1(2(3(1(2(x1)))))) -> 2(0(1(2(3(4(2(0(2(0(x1)))))))))) 2(3(2(1(3(2(x1)))))) -> 2(3(0(4(3(0(4(2(0(0(x1)))))))))) 2(3(2(2(2(3(x1)))))) -> 1(0(4(1(3(2(0(3(4(2(x1)))))))))) 2(5(5(2(2(3(x1)))))) -> 2(1(2(2(1(4(0(0(3(0(x1)))))))))) 3(2(3(5(2(1(x1)))))) -> 3(3(2(4(3(3(2(1(3(3(x1)))))))))) 3(5(0(5(5(4(x1)))))) -> 5(1(4(3(0(0(4(4(4(3(x1)))))))))) 4(1(0(5(0(3(x1)))))) -> 2(4(1(4(4(0(1(3(4(3(x1)))))))))) 4(1(1(3(5(0(x1)))))) -> 4(2(2(0(5(2(0(4(3(0(x1)))))))))) 5(0(1(2(1(2(x1)))))) -> 0(2(4(2(1(4(4(0(2(0(x1)))))))))) 5(4(0(1(1(5(x1)))))) -> 5(0(2(2(0(1(3(3(3(5(x1)))))))))) 5(4(0(3(5(5(x1)))))) -> 3(1(0(0(2(0(0(3(1(5(x1)))))))))) 0(3(5(2(5(5(2(x1))))))) -> 4(3(2(4(0(4(4(3(1(2(x1)))))))))) 0(5(0(0(1(0(5(x1))))))) -> 2(0(2(3(1(5(2(4(2(4(x1)))))))))) 1(1(5(0(4(2(1(x1))))))) -> 2(5(2(0(1(4(3(0(0(4(x1)))))))))) 1(2(3(5(5(1(5(x1))))))) -> 1(4(3(0(3(0(4(2(5(4(x1)))))))))) 1(3(2(5(5(4(2(x1))))))) -> 1(1(4(4(1(2(5(2(0(0(x1)))))))))) 2(1(0(3(3(1(1(x1))))))) -> 1(1(4(0(0(0(1(4(3(1(x1)))))))))) 2(3(0(3(2(5(5(x1))))))) -> 1(4(3(0(2(4(5(4(4(4(x1)))))))))) 2(3(3(2(2(3(3(x1))))))) -> 2(1(4(2(4(2(5(2(0(3(x1)))))))))) 2(3(5(3(1(0(3(x1))))))) -> 4(5(2(0(2(1(4(1(2(0(x1)))))))))) 2(5(2(1(2(5(5(x1))))))) -> 4(0(4(3(3(1(1(1(1(5(x1)))))))))) 2(5(5(3(4(4(1(x1))))))) -> 1(0(2(2(1(4(2(2(4(1(x1)))))))))) 2(5(5(5(2(4(5(x1))))))) -> 1(3(0(0(0(5(1(4(1(5(x1)))))))))) 3(2(5(0(5(2(5(x1))))))) -> 3(3(3(1(3(5(1(3(4(5(x1)))))))))) 3(3(2(5(5(5(0(x1))))))) -> 1(3(4(4(1(4(0(1(1(0(x1)))))))))) 3(5(1(3(1(5(0(x1))))))) -> 5(3(1(0(2(0(4(3(0(2(x1)))))))))) 4(0(1(0(5(1(1(x1))))))) -> 2(0(0(0(1(4(0(1(1(3(x1)))))))))) 4(1(0(5(5(5(1(x1))))))) -> 2(0(2(3(5(4(3(5(0(1(x1)))))))))) 4(2(5(5(0(2(3(x1))))))) -> 4(2(3(0(1(3(4(3(3(2(x1)))))))))) 5(0(3(2(5(0(4(x1))))))) -> 1(2(4(1(1(1(4(0(2(4(x1)))))))))) 5(0(3(5(3(5(2(x1))))))) -> 5(0(1(5(3(2(4(4(5(2(x1)))))))))) 5(0(5(2(3(2(3(x1))))))) -> 5(3(0(3(0(1(4(3(0(0(x1)))))))))) 5(1(4(1(2(5(5(x1))))))) -> 0(2(4(1(4(0(1(0(0(3(x1)))))))))) 5(5(0(3(0(1(1(x1))))))) -> 2(0(0(2(4(4(4(4(2(5(x1)))))))))) 5(5(1(1(1(1(1(x1))))))) -> 3(5(0(0(0(4(1(4(0(1(x1)))))))))) 5(5(3(0(3(2(5(x1))))))) -> 0(2(0(4(4(1(4(3(2(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_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_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(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(2(1(0(x1)))) -> 0(4(2(0(4(2(4(3(0(0(x1)))))))))) 2(1(0(1(x1)))) -> 3(4(1(2(0(0(0(4(2(1(x1)))))))))) 2(5(5(0(x1)))) -> 2(4(2(5(2(4(4(3(5(2(x1)))))))))) 2(5(5(5(x1)))) -> 2(2(2(0(0(3(4(1(2(4(x1)))))))))) 5(4(0(3(x1)))) -> 5(0(1(0(0(0(2(0(2(0(x1)))))))))) 1(5(4(2(1(x1))))) -> 1(1(0(2(4(2(2(2(4(1(x1)))))))))) 2(3(2(1(0(x1))))) -> 2(0(0(3(2(2(4(2(1(4(x1)))))))))) 2(5(5(5(2(x1))))) -> 4(5(4(5(3(3(0(3(2(0(x1)))))))))) 4(5(1(0(1(x1))))) -> 4(1(2(0(2(0(0(3(4(1(x1)))))))))) 5(0(4(4(2(x1))))) -> 1(2(0(2(0(0(4(0(4(2(x1)))))))))) 5(0(5(1(5(x1))))) -> 5(0(3(5(2(0(4(2(0(4(x1)))))))))) 5(1(1(1(5(x1))))) -> 3(5(3(3(1(4(0(4(2(4(x1)))))))))) 5(1(2(3(2(x1))))) -> 5(5(3(0(0(0(0(0(2(0(x1)))))))))) 5(1(4(3(0(x1))))) -> 3(5(2(2(4(1(3(0(2(0(x1)))))))))) 5(5(0(2(5(x1))))) -> 3(5(1(1(2(2(4(3(2(3(x1)))))))))) 5(5(5(5(0(x1))))) -> 3(5(4(3(3(3(0(4(2(2(x1)))))))))) 1(0(3(1(1(0(x1)))))) -> 2(2(0(4(2(3(5(3(4(0(x1)))))))))) 1(5(0(5(3(5(x1)))))) -> 2(5(2(0(4(5(2(4(4(5(x1)))))))))) 2(1(2(3(1(2(x1)))))) -> 2(0(1(2(3(4(2(0(2(0(x1)))))))))) 2(3(2(1(3(2(x1)))))) -> 2(3(0(4(3(0(4(2(0(0(x1)))))))))) 2(3(2(2(2(3(x1)))))) -> 1(0(4(1(3(2(0(3(4(2(x1)))))))))) 2(5(5(2(2(3(x1)))))) -> 2(1(2(2(1(4(0(0(3(0(x1)))))))))) 3(2(3(5(2(1(x1)))))) -> 3(3(2(4(3(3(2(1(3(3(x1)))))))))) 3(5(0(5(5(4(x1)))))) -> 5(1(4(3(0(0(4(4(4(3(x1)))))))))) 4(1(0(5(0(3(x1)))))) -> 2(4(1(4(4(0(1(3(4(3(x1)))))))))) 4(1(1(3(5(0(x1)))))) -> 4(2(2(0(5(2(0(4(3(0(x1)))))))))) 5(0(1(2(1(2(x1)))))) -> 0(2(4(2(1(4(4(0(2(0(x1)))))))))) 5(4(0(1(1(5(x1)))))) -> 5(0(2(2(0(1(3(3(3(5(x1)))))))))) 5(4(0(3(5(5(x1)))))) -> 3(1(0(0(2(0(0(3(1(5(x1)))))))))) 0(3(5(2(5(5(2(x1))))))) -> 4(3(2(4(0(4(4(3(1(2(x1)))))))))) 0(5(0(0(1(0(5(x1))))))) -> 2(0(2(3(1(5(2(4(2(4(x1)))))))))) 1(1(5(0(4(2(1(x1))))))) -> 2(5(2(0(1(4(3(0(0(4(x1)))))))))) 1(2(3(5(5(1(5(x1))))))) -> 1(4(3(0(3(0(4(2(5(4(x1)))))))))) 1(3(2(5(5(4(2(x1))))))) -> 1(1(4(4(1(2(5(2(0(0(x1)))))))))) 2(1(0(3(3(1(1(x1))))))) -> 1(1(4(0(0(0(1(4(3(1(x1)))))))))) 2(3(0(3(2(5(5(x1))))))) -> 1(4(3(0(2(4(5(4(4(4(x1)))))))))) 2(3(3(2(2(3(3(x1))))))) -> 2(1(4(2(4(2(5(2(0(3(x1)))))))))) 2(3(5(3(1(0(3(x1))))))) -> 4(5(2(0(2(1(4(1(2(0(x1)))))))))) 2(5(2(1(2(5(5(x1))))))) -> 4(0(4(3(3(1(1(1(1(5(x1)))))))))) 2(5(5(3(4(4(1(x1))))))) -> 1(0(2(2(1(4(2(2(4(1(x1)))))))))) 2(5(5(5(2(4(5(x1))))))) -> 1(3(0(0(0(5(1(4(1(5(x1)))))))))) 3(2(5(0(5(2(5(x1))))))) -> 3(3(3(1(3(5(1(3(4(5(x1)))))))))) 3(3(2(5(5(5(0(x1))))))) -> 1(3(4(4(1(4(0(1(1(0(x1)))))))))) 3(5(1(3(1(5(0(x1))))))) -> 5(3(1(0(2(0(4(3(0(2(x1)))))))))) 4(0(1(0(5(1(1(x1))))))) -> 2(0(0(0(1(4(0(1(1(3(x1)))))))))) 4(1(0(5(5(5(1(x1))))))) -> 2(0(2(3(5(4(3(5(0(1(x1)))))))))) 4(2(5(5(0(2(3(x1))))))) -> 4(2(3(0(1(3(4(3(3(2(x1)))))))))) 5(0(3(2(5(0(4(x1))))))) -> 1(2(4(1(1(1(4(0(2(4(x1)))))))))) 5(0(3(5(3(5(2(x1))))))) -> 5(0(1(5(3(2(4(4(5(2(x1)))))))))) 5(0(5(2(3(2(3(x1))))))) -> 5(3(0(3(0(1(4(3(0(0(x1)))))))))) 5(1(4(1(2(5(5(x1))))))) -> 0(2(4(1(4(0(1(0(0(3(x1)))))))))) 5(5(0(3(0(1(1(x1))))))) -> 2(0(0(2(4(4(4(4(2(5(x1)))))))))) 5(5(1(1(1(1(1(x1))))))) -> 3(5(0(0(0(4(1(4(0(1(x1)))))))))) 5(5(3(0(3(2(5(x1))))))) -> 0(2(0(4(4(1(4(3(2(5(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_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_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(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(2(1(0(x1)))) -> 0(4(2(0(4(2(4(3(0(0(x1)))))))))) 2(1(0(1(x1)))) -> 3(4(1(2(0(0(0(4(2(1(x1)))))))))) 2(5(5(0(x1)))) -> 2(4(2(5(2(4(4(3(5(2(x1)))))))))) 2(5(5(5(x1)))) -> 2(2(2(0(0(3(4(1(2(4(x1)))))))))) 5(4(0(3(x1)))) -> 5(0(1(0(0(0(2(0(2(0(x1)))))))))) 1(5(4(2(1(x1))))) -> 1(1(0(2(4(2(2(2(4(1(x1)))))))))) 2(3(2(1(0(x1))))) -> 2(0(0(3(2(2(4(2(1(4(x1)))))))))) 2(5(5(5(2(x1))))) -> 4(5(4(5(3(3(0(3(2(0(x1)))))))))) 4(5(1(0(1(x1))))) -> 4(1(2(0(2(0(0(3(4(1(x1)))))))))) 5(0(4(4(2(x1))))) -> 1(2(0(2(0(0(4(0(4(2(x1)))))))))) 5(0(5(1(5(x1))))) -> 5(0(3(5(2(0(4(2(0(4(x1)))))))))) 5(1(1(1(5(x1))))) -> 3(5(3(3(1(4(0(4(2(4(x1)))))))))) 5(1(2(3(2(x1))))) -> 5(5(3(0(0(0(0(0(2(0(x1)))))))))) 5(1(4(3(0(x1))))) -> 3(5(2(2(4(1(3(0(2(0(x1)))))))))) 5(5(0(2(5(x1))))) -> 3(5(1(1(2(2(4(3(2(3(x1)))))))))) 5(5(5(5(0(x1))))) -> 3(5(4(3(3(3(0(4(2(2(x1)))))))))) 1(0(3(1(1(0(x1)))))) -> 2(2(0(4(2(3(5(3(4(0(x1)))))))))) 1(5(0(5(3(5(x1)))))) -> 2(5(2(0(4(5(2(4(4(5(x1)))))))))) 2(1(2(3(1(2(x1)))))) -> 2(0(1(2(3(4(2(0(2(0(x1)))))))))) 2(3(2(1(3(2(x1)))))) -> 2(3(0(4(3(0(4(2(0(0(x1)))))))))) 2(3(2(2(2(3(x1)))))) -> 1(0(4(1(3(2(0(3(4(2(x1)))))))))) 2(5(5(2(2(3(x1)))))) -> 2(1(2(2(1(4(0(0(3(0(x1)))))))))) 3(2(3(5(2(1(x1)))))) -> 3(3(2(4(3(3(2(1(3(3(x1)))))))))) 3(5(0(5(5(4(x1)))))) -> 5(1(4(3(0(0(4(4(4(3(x1)))))))))) 4(1(0(5(0(3(x1)))))) -> 2(4(1(4(4(0(1(3(4(3(x1)))))))))) 4(1(1(3(5(0(x1)))))) -> 4(2(2(0(5(2(0(4(3(0(x1)))))))))) 5(0(1(2(1(2(x1)))))) -> 0(2(4(2(1(4(4(0(2(0(x1)))))))))) 5(4(0(1(1(5(x1)))))) -> 5(0(2(2(0(1(3(3(3(5(x1)))))))))) 5(4(0(3(5(5(x1)))))) -> 3(1(0(0(2(0(0(3(1(5(x1)))))))))) 0(3(5(2(5(5(2(x1))))))) -> 4(3(2(4(0(4(4(3(1(2(x1)))))))))) 0(5(0(0(1(0(5(x1))))))) -> 2(0(2(3(1(5(2(4(2(4(x1)))))))))) 1(1(5(0(4(2(1(x1))))))) -> 2(5(2(0(1(4(3(0(0(4(x1)))))))))) 1(2(3(5(5(1(5(x1))))))) -> 1(4(3(0(3(0(4(2(5(4(x1)))))))))) 1(3(2(5(5(4(2(x1))))))) -> 1(1(4(4(1(2(5(2(0(0(x1)))))))))) 2(1(0(3(3(1(1(x1))))))) -> 1(1(4(0(0(0(1(4(3(1(x1)))))))))) 2(3(0(3(2(5(5(x1))))))) -> 1(4(3(0(2(4(5(4(4(4(x1)))))))))) 2(3(3(2(2(3(3(x1))))))) -> 2(1(4(2(4(2(5(2(0(3(x1)))))))))) 2(3(5(3(1(0(3(x1))))))) -> 4(5(2(0(2(1(4(1(2(0(x1)))))))))) 2(5(2(1(2(5(5(x1))))))) -> 4(0(4(3(3(1(1(1(1(5(x1)))))))))) 2(5(5(3(4(4(1(x1))))))) -> 1(0(2(2(1(4(2(2(4(1(x1)))))))))) 2(5(5(5(2(4(5(x1))))))) -> 1(3(0(0(0(5(1(4(1(5(x1)))))))))) 3(2(5(0(5(2(5(x1))))))) -> 3(3(3(1(3(5(1(3(4(5(x1)))))))))) 3(3(2(5(5(5(0(x1))))))) -> 1(3(4(4(1(4(0(1(1(0(x1)))))))))) 3(5(1(3(1(5(0(x1))))))) -> 5(3(1(0(2(0(4(3(0(2(x1)))))))))) 4(0(1(0(5(1(1(x1))))))) -> 2(0(0(0(1(4(0(1(1(3(x1)))))))))) 4(1(0(5(5(5(1(x1))))))) -> 2(0(2(3(5(4(3(5(0(1(x1)))))))))) 4(2(5(5(0(2(3(x1))))))) -> 4(2(3(0(1(3(4(3(3(2(x1)))))))))) 5(0(3(2(5(0(4(x1))))))) -> 1(2(4(1(1(1(4(0(2(4(x1)))))))))) 5(0(3(5(3(5(2(x1))))))) -> 5(0(1(5(3(2(4(4(5(2(x1)))))))))) 5(0(5(2(3(2(3(x1))))))) -> 5(3(0(3(0(1(4(3(0(0(x1)))))))))) 5(1(4(1(2(5(5(x1))))))) -> 0(2(4(1(4(0(1(0(0(3(x1)))))))))) 5(5(0(3(0(1(1(x1))))))) -> 2(0(0(2(4(4(4(4(2(5(x1)))))))))) 5(5(1(1(1(1(1(x1))))))) -> 3(5(0(0(0(4(1(4(0(1(x1)))))))))) 5(5(3(0(3(2(5(x1))))))) -> 0(2(0(4(4(1(4(3(2(5(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_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_4(x_1) -> 4(encArg(x_1)) encode_3(x_1) -> 3(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 3. The certificate found is represented by the following graph. "[109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 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, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660] {(109,110,[0_1|0, 2_1|0, 5_1|0, 1_1|0, 4_1|0, 3_1|0, encArg_1|0, encode_0_1|0, encode_2_1|0, encode_1_1|0, encode_4_1|0, encode_3_1|0, encode_5_1|0]), (109,111,[0_1|1, 2_1|1, 5_1|1, 1_1|1, 4_1|1, 3_1|1]), (109,112,[0_1|2]), (109,121,[4_1|2]), (109,130,[2_1|2]), (109,139,[3_1|2]), (109,148,[1_1|2]), (109,157,[2_1|2]), (109,166,[2_1|2]), (109,175,[2_1|2]), (109,184,[4_1|2]), (109,193,[1_1|2]), (109,202,[2_1|2]), (109,211,[1_1|2]), (109,220,[4_1|2]), (109,229,[2_1|2]), (109,238,[2_1|2]), (109,247,[1_1|2]), (109,256,[1_1|2]), (109,265,[2_1|2]), (109,274,[4_1|2]), (109,283,[5_1|2]), (109,292,[3_1|2]), (109,301,[5_1|2]), (109,310,[1_1|2]), (109,319,[5_1|2]), (109,328,[5_1|2]), (109,337,[0_1|2]), (109,346,[1_1|2]), (109,355,[5_1|2]), (109,364,[3_1|2]), (109,373,[5_1|2]), (109,382,[3_1|2]), (109,391,[0_1|2]), (109,400,[3_1|2]), (109,409,[2_1|2]), (109,418,[3_1|2]), (109,427,[3_1|2]), (109,436,[0_1|2]), (109,445,[1_1|2]), (109,454,[2_1|2]), (109,463,[2_1|2]), (109,472,[2_1|2]), (109,481,[1_1|2]), (109,490,[1_1|2]), (109,499,[4_1|2]), (109,508,[2_1|2]), (109,517,[2_1|2]), (109,526,[4_1|2]), (109,535,[2_1|2]), (109,544,[4_1|2]), (109,553,[3_1|2]), (109,562,[3_1|2]), (109,571,[5_1|2]), (109,580,[5_1|2]), (109,589,[1_1|2]), (109,598,[3_1|3]), (110,110,[cons_0_1|0, cons_2_1|0, cons_5_1|0, cons_1_1|0, cons_4_1|0, cons_3_1|0]), (111,110,[encArg_1|1]), (111,111,[0_1|1, 2_1|1, 5_1|1, 1_1|1, 4_1|1, 3_1|1]), (111,112,[0_1|2]), (111,121,[4_1|2]), (111,130,[2_1|2]), (111,139,[3_1|2]), (111,148,[1_1|2]), (111,157,[2_1|2]), (111,166,[2_1|2]), (111,175,[2_1|2]), (111,184,[4_1|2]), (111,193,[1_1|2]), (111,202,[2_1|2]), (111,211,[1_1|2]), (111,220,[4_1|2]), (111,229,[2_1|2]), (111,238,[2_1|2]), (111,247,[1_1|2]), (111,256,[1_1|2]), (111,265,[2_1|2]), (111,274,[4_1|2]), (111,283,[5_1|2]), (111,292,[3_1|2]), (111,301,[5_1|2]), (111,310,[1_1|2]), (111,319,[5_1|2]), (111,328,[5_1|2]), (111,337,[0_1|2]), (111,346,[1_1|2]), (111,355,[5_1|2]), (111,364,[3_1|2]), (111,373,[5_1|2]), (111,382,[3_1|2]), (111,391,[0_1|2]), (111,400,[3_1|2]), (111,409,[2_1|2]), (111,418,[3_1|2]), (111,427,[3_1|2]), (111,436,[0_1|2]), (111,445,[1_1|2]), (111,454,[2_1|2]), (111,463,[2_1|2]), (111,472,[2_1|2]), (111,481,[1_1|2]), (111,490,[1_1|2]), (111,499,[4_1|2]), (111,508,[2_1|2]), (111,517,[2_1|2]), (111,526,[4_1|2]), (111,535,[2_1|2]), (111,544,[4_1|2]), (111,553,[3_1|2]), (111,562,[3_1|2]), (111,571,[5_1|2]), (111,580,[5_1|2]), (111,589,[1_1|2]), (111,598,[3_1|3]), (112,113,[4_1|2]), (113,114,[2_1|2]), (114,115,[0_1|2]), (115,116,[4_1|2]), (116,117,[2_1|2]), (117,118,[4_1|2]), (118,119,[3_1|2]), (119,120,[0_1|2]), (120,111,[0_1|2]), (120,112,[0_1|2]), (120,337,[0_1|2]), (120,391,[0_1|2]), (120,436,[0_1|2]), (120,212,[0_1|2]), (120,248,[0_1|2]), (120,121,[4_1|2]), (120,130,[2_1|2]), (121,122,[3_1|2]), (122,123,[2_1|2]), (123,124,[4_1|2]), (124,125,[0_1|2]), (125,126,[4_1|2]), (126,127,[4_1|2]), (127,128,[3_1|2]), (128,129,[1_1|2]), (128,481,[1_1|2]), (129,111,[2_1|2]), (129,130,[2_1|2]), (129,157,[2_1|2]), (129,166,[2_1|2]), (129,175,[2_1|2]), (129,202,[2_1|2]), (129,229,[2_1|2]), (129,238,[2_1|2]), (129,265,[2_1|2]), (129,409,[2_1|2]), (129,454,[2_1|2]), (129,463,[2_1|2]), (129,472,[2_1|2]), (129,508,[2_1|2]), (129,517,[2_1|2]), (129,535,[2_1|2]), (129,139,[3_1|2]), (129,148,[1_1|2]), (129,184,[4_1|2]), (129,193,[1_1|2]), (129,211,[1_1|2]), (129,220,[4_1|2]), (129,247,[1_1|2]), (129,256,[1_1|2]), (129,274,[4_1|2]), (130,131,[0_1|2]), (131,132,[2_1|2]), (132,133,[3_1|2]), (133,134,[1_1|2]), (134,135,[5_1|2]), (135,136,[2_1|2]), (136,137,[4_1|2]), (137,138,[2_1|2]), (138,111,[4_1|2]), (138,283,[4_1|2]), (138,301,[4_1|2]), (138,319,[4_1|2]), (138,328,[4_1|2]), (138,355,[4_1|2]), (138,373,[4_1|2]), (138,571,[4_1|2]), (138,580,[4_1|2]), (138,499,[4_1|2]), (138,508,[2_1|2]), (138,517,[2_1|2]), (138,526,[4_1|2]), (138,535,[2_1|2]), (138,544,[4_1|2]), (139,140,[4_1|2]), (140,141,[1_1|2]), (141,142,[2_1|2]), (142,143,[0_1|2]), (143,144,[0_1|2]), (144,145,[0_1|2]), (145,146,[4_1|2]), (146,147,[2_1|2]), (146,139,[3_1|2]), (146,148,[1_1|2]), (146,157,[2_1|2]), (147,111,[1_1|2]), (147,148,[1_1|2]), (147,193,[1_1|2]), (147,211,[1_1|2]), (147,247,[1_1|2]), (147,256,[1_1|2]), (147,310,[1_1|2]), (147,346,[1_1|2]), (147,445,[1_1|2]), (147,481,[1_1|2]), (147,490,[1_1|2]), (147,589,[1_1|2]), (147,454,[2_1|2]), (147,463,[2_1|2]), (147,472,[2_1|2]), (148,149,[1_1|2]), (149,150,[4_1|2]), (150,151,[0_1|2]), (151,152,[0_1|2]), (152,153,[0_1|2]), (153,154,[1_1|2]), (154,155,[4_1|2]), (155,156,[3_1|2]), (156,111,[1_1|2]), (156,148,[1_1|2]), (156,193,[1_1|2]), (156,211,[1_1|2]), (156,247,[1_1|2]), (156,256,[1_1|2]), (156,310,[1_1|2]), (156,346,[1_1|2]), (156,445,[1_1|2]), (156,481,[1_1|2]), (156,490,[1_1|2]), (156,589,[1_1|2]), (156,149,[1_1|2]), (156,446,[1_1|2]), (156,491,[1_1|2]), (156,454,[2_1|2]), (156,463,[2_1|2]), (156,472,[2_1|2]), (157,158,[0_1|2]), (158,159,[1_1|2]), (159,160,[2_1|2]), (160,161,[3_1|2]), (161,162,[4_1|2]), (162,163,[2_1|2]), (163,164,[0_1|2]), (164,165,[2_1|2]), (165,111,[0_1|2]), (165,130,[0_1|2, 2_1|2]), (165,157,[0_1|2]), (165,166,[0_1|2]), (165,175,[0_1|2]), (165,202,[0_1|2]), (165,229,[0_1|2]), (165,238,[0_1|2]), (165,265,[0_1|2]), 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(597,111,[0_1|2]), (597,112,[0_1|2]), (597,337,[0_1|2]), (597,391,[0_1|2]), (597,436,[0_1|2]), (597,284,[0_1|2]), (597,302,[0_1|2]), (597,320,[0_1|2]), (597,356,[0_1|2]), (597,121,[4_1|2]), (597,130,[2_1|2]), (598,599,[5_1|3]), (599,600,[2_1|3]), (600,601,[2_1|3]), (601,602,[4_1|3]), (602,603,[1_1|3]), (603,604,[3_1|3]), (604,605,[0_1|3]), (605,606,[2_1|3]), (606,575,[0_1|3]), (607,608,[5_1|3]), (608,609,[2_1|3]), (609,610,[2_1|3]), (610,611,[4_1|3]), (611,612,[1_1|3]), (612,613,[3_1|3]), (613,614,[0_1|3]), (614,615,[2_1|3]), (615,259,[0_1|3]), (615,484,[0_1|3]), (615,575,[0_1|3]), (616,617,[4_1|3]), (617,618,[2_1|3]), (618,619,[5_1|3]), (619,620,[2_1|3]), (620,621,[4_1|3]), (621,622,[4_1|3]), (622,623,[3_1|3]), (623,624,[5_1|3]), (624,284,[2_1|3]), (624,302,[2_1|3]), (624,320,[2_1|3]), (624,356,[2_1|3]), (625,626,[2_1|3]), (626,627,[2_1|3]), (627,628,[0_1|3]), (628,629,[0_1|3]), (629,630,[3_1|3]), (630,631,[4_1|3]), (631,632,[1_1|3]), (632,633,[2_1|3]), (633,374,[4_1|3]), (634,635,[2_1|3]), (635,636,[4_1|3]), (636,637,[2_1|3]), (637,638,[1_1|3]), (638,639,[4_1|3]), (639,640,[4_1|3]), (640,641,[0_1|3]), (641,642,[2_1|3]), (642,204,[0_1|3]), (643,644,[4_1|3]), (644,645,[2_1|3]), (645,646,[0_1|3]), (646,647,[4_1|3]), (647,648,[2_1|3]), (648,649,[4_1|3]), (649,650,[3_1|3]), (650,651,[0_1|3]), (651,212,[0_1|3]), (651,248,[0_1|3]), (651,286,[0_1|3]), (652,653,[0_1|3]), (653,654,[1_1|3]), (654,655,[0_1|3]), (655,656,[0_1|3]), (656,657,[0_1|3]), (657,658,[2_1|3]), (658,659,[0_1|3]), (659,660,[2_1|3]), (660,321,[0_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)