/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).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), 49 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 248 ms] (8) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(1(x1)))) -> 0(1(1(1(0(0(1(2(2(2(x1)))))))))) 0(3(4(4(x1)))) -> 0(0(0(3(1(1(4(2(2(0(x1)))))))))) 1(5(1(5(4(x1))))) -> 1(0(0(2(2(0(5(2(2(4(x1)))))))))) 3(4(5(3(4(x1))))) -> 3(5(0(2(1(1(1(2(1(4(x1)))))))))) 4(3(4(3(4(x1))))) -> 2(1(1(1(4(1(2(4(2(0(x1)))))))))) 5(0(1(0(1(x1))))) -> 5(0(2(0(1(1(4(2(1(2(x1)))))))))) 1(0(1(0(1(4(x1)))))) -> 1(0(2(4(5(4(2(4(2(4(x1)))))))))) 1(1(3(4(1(5(x1)))))) -> 1(1(0(0(2(2(5(2(0(0(x1)))))))))) 1(3(1(1(3(3(x1)))))) -> 1(2(4(2(0(2(1(0(2(5(x1)))))))))) 1(3(1(5(2(3(x1)))))) -> 1(4(3(2(1(0(0(2(4(3(x1)))))))))) 1(5(0(5(5(3(x1)))))) -> 2(1(1(1(1(2(1(3(5(3(x1)))))))))) 2(1(3(1(5(5(x1)))))) -> 1(1(2(2(0(5(0(0(2(2(x1)))))))))) 2(2(4(3(4(5(x1)))))) -> 2(0(1(4(0(0(2(0(0(0(x1)))))))))) 2(3(1(0(3(4(x1)))))) -> 0(0(1(1(5(2(4(1(1(4(x1)))))))))) 2(3(3(4(1(5(x1)))))) -> 0(0(2(4(4(2(0(4(1(3(x1)))))))))) 2(5(0(5(5(1(x1)))))) -> 3(0(1(4(4(0(0(0(0(1(x1)))))))))) 3(0(4(3(3(4(x1)))))) -> 2(3(0(3(5(1(2(4(2(4(x1)))))))))) 3(2(3(3(0(4(x1)))))) -> 5(4(2(2(0(0(4(2(4(4(x1)))))))))) 3(3(5(4(3(4(x1)))))) -> 0(5(1(1(0(4(0(2(4(4(x1)))))))))) 3(5(4(2(1(0(x1)))))) -> 2(5(0(0(0(0(0(4(4(0(x1)))))))))) 4(1(3(4(3(1(x1)))))) -> 4(2(5(0(1(0(0(0(4(4(x1)))))))))) 4(2(1(0(2(5(x1)))))) -> 4(2(5(4(2(2(2(4(2(5(x1)))))))))) 4(3(3(5(1(1(x1)))))) -> 4(4(2(4(4(2(5(0(1(2(x1)))))))))) 5(4(0(1(3(0(x1)))))) -> 0(2(4(2(2(1(2(0(0(0(x1)))))))))) 0(4(1(5(5(3(5(x1))))))) -> 0(2(2(0(1(2(5(2(5(0(x1)))))))))) 0(4(4(3(4(1(3(x1))))))) -> 0(4(2(4(1(3(2(0(2(2(x1)))))))))) 1(3(3(4(5(2(5(x1))))))) -> 1(1(1(4(5(1(2(5(2(4(x1)))))))))) 1(5(2(5(1(5(2(x1))))))) -> 1(2(3(0(2(3(0(1(0(2(x1)))))))))) 1(5(3(2(4(5(4(x1))))))) -> 1(1(0(3(0(0(0(0(3(4(x1)))))))))) 1(5(5(5(3(2(1(x1))))))) -> 1(4(1(4(2(1(3(0(1(1(x1)))))))))) 2(1(2(3(1(3(3(x1))))))) -> 2(1(4(1(5(1(1(1(1(1(x1)))))))))) 3(0(2(1(3(2(1(x1))))))) -> 1(2(0(0(3(3(4(2(2(1(x1)))))))))) 3(0(4(4(3(4(5(x1))))))) -> 1(2(2(4(3(2(2(2(0(4(x1)))))))))) 3(1(3(1(5(4(1(x1))))))) -> 0(1(2(2(5(5(5(4(2(0(x1)))))))))) 3(2(5(2(1(3(4(x1))))))) -> 0(2(1(3(1(2(1(4(2(2(x1)))))))))) 3(3(1(3(1(3(3(x1))))))) -> 1(2(1(3(0(5(5(1(2(1(x1)))))))))) 3(3(1(5(0(3(4(x1))))))) -> 2(0(0(5(1(2(1(4(1(4(x1)))))))))) 3(3(3(5(2(4(5(x1))))))) -> 3(1(0(0(1(4(2(2(0(5(x1)))))))))) 3(4(3(3(4(3(5(x1))))))) -> 5(1(1(1(1(1(4(2(3(3(x1)))))))))) 3(4(3(4(4(3(2(x1))))))) -> 2(5(4(5(3(4(2(4(4(0(x1)))))))))) 4(1(3(4(1(0(2(x1))))))) -> 4(4(1(5(1(2(1(4(2(2(x1)))))))))) 4(5(0(4(1(3(1(x1))))))) -> 1(1(1(2(0(0(4(4(5(1(x1)))))))))) 4(5(0(5(3(2(1(x1))))))) -> 1(1(4(1(3(0(2(4(2(1(x1)))))))))) 4(5(0(5(3(4(5(x1))))))) -> 1(1(1(0(5(4(0(2(4(5(x1)))))))))) 4(5(3(1(4(4(3(x1))))))) -> 4(5(5(5(4(2(4(4(2(3(x1)))))))))) 4(5(3(4(1(4(5(x1))))))) -> 4(5(4(0(2(0(1(2(1(0(x1)))))))))) 4(5(4(3(4(1(0(x1))))))) -> 1(4(1(1(2(4(0(1(2(0(x1)))))))))) 5(3(2(1(5(3(4(x1))))))) -> 5(1(2(1(3(3(5(1(2(4(x1)))))))))) 5(3(4(3(1(3(3(x1))))))) -> 5(1(1(4(2(4(3(3(4(3(x1)))))))))) 5(3(4(4(3(1(2(x1))))))) -> 5(1(0(5(0(3(5(1(1(1(x1)))))))))) 5(4(3(2(3(1(3(x1))))))) -> 0(0(0(2(3(0(2(5(4(3(x1)))))))))) 5(4(3(4(3(1(5(x1))))))) -> 0(2(0(0(0(0(4(1(5(0(x1)))))))))) 5(5(3(3(3(5(4(x1))))))) -> 0(3(5(2(2(1(0(4(2(2(x1)))))))))) 5(5(4(5(3(5(5(x1))))))) -> 5(2(4(2(2(2(4(1(5(2(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_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(1(x1)))) -> 0(1(1(1(0(0(1(2(2(2(x1)))))))))) 0(3(4(4(x1)))) -> 0(0(0(3(1(1(4(2(2(0(x1)))))))))) 1(5(1(5(4(x1))))) -> 1(0(0(2(2(0(5(2(2(4(x1)))))))))) 3(4(5(3(4(x1))))) -> 3(5(0(2(1(1(1(2(1(4(x1)))))))))) 4(3(4(3(4(x1))))) -> 2(1(1(1(4(1(2(4(2(0(x1)))))))))) 5(0(1(0(1(x1))))) -> 5(0(2(0(1(1(4(2(1(2(x1)))))))))) 1(0(1(0(1(4(x1)))))) -> 1(0(2(4(5(4(2(4(2(4(x1)))))))))) 1(1(3(4(1(5(x1)))))) -> 1(1(0(0(2(2(5(2(0(0(x1)))))))))) 1(3(1(1(3(3(x1)))))) -> 1(2(4(2(0(2(1(0(2(5(x1)))))))))) 1(3(1(5(2(3(x1)))))) -> 1(4(3(2(1(0(0(2(4(3(x1)))))))))) 1(5(0(5(5(3(x1)))))) -> 2(1(1(1(1(2(1(3(5(3(x1)))))))))) 2(1(3(1(5(5(x1)))))) -> 1(1(2(2(0(5(0(0(2(2(x1)))))))))) 2(2(4(3(4(5(x1)))))) -> 2(0(1(4(0(0(2(0(0(0(x1)))))))))) 2(3(1(0(3(4(x1)))))) -> 0(0(1(1(5(2(4(1(1(4(x1)))))))))) 2(3(3(4(1(5(x1)))))) -> 0(0(2(4(4(2(0(4(1(3(x1)))))))))) 2(5(0(5(5(1(x1)))))) -> 3(0(1(4(4(0(0(0(0(1(x1)))))))))) 3(0(4(3(3(4(x1)))))) -> 2(3(0(3(5(1(2(4(2(4(x1)))))))))) 3(2(3(3(0(4(x1)))))) -> 5(4(2(2(0(0(4(2(4(4(x1)))))))))) 3(3(5(4(3(4(x1)))))) -> 0(5(1(1(0(4(0(2(4(4(x1)))))))))) 3(5(4(2(1(0(x1)))))) -> 2(5(0(0(0(0(0(4(4(0(x1)))))))))) 4(1(3(4(3(1(x1)))))) -> 4(2(5(0(1(0(0(0(4(4(x1)))))))))) 4(2(1(0(2(5(x1)))))) -> 4(2(5(4(2(2(2(4(2(5(x1)))))))))) 4(3(3(5(1(1(x1)))))) -> 4(4(2(4(4(2(5(0(1(2(x1)))))))))) 5(4(0(1(3(0(x1)))))) -> 0(2(4(2(2(1(2(0(0(0(x1)))))))))) 0(4(1(5(5(3(5(x1))))))) -> 0(2(2(0(1(2(5(2(5(0(x1)))))))))) 0(4(4(3(4(1(3(x1))))))) -> 0(4(2(4(1(3(2(0(2(2(x1)))))))))) 1(3(3(4(5(2(5(x1))))))) -> 1(1(1(4(5(1(2(5(2(4(x1)))))))))) 1(5(2(5(1(5(2(x1))))))) -> 1(2(3(0(2(3(0(1(0(2(x1)))))))))) 1(5(3(2(4(5(4(x1))))))) -> 1(1(0(3(0(0(0(0(3(4(x1)))))))))) 1(5(5(5(3(2(1(x1))))))) -> 1(4(1(4(2(1(3(0(1(1(x1)))))))))) 2(1(2(3(1(3(3(x1))))))) -> 2(1(4(1(5(1(1(1(1(1(x1)))))))))) 3(0(2(1(3(2(1(x1))))))) -> 1(2(0(0(3(3(4(2(2(1(x1)))))))))) 3(0(4(4(3(4(5(x1))))))) -> 1(2(2(4(3(2(2(2(0(4(x1)))))))))) 3(1(3(1(5(4(1(x1))))))) -> 0(1(2(2(5(5(5(4(2(0(x1)))))))))) 3(2(5(2(1(3(4(x1))))))) -> 0(2(1(3(1(2(1(4(2(2(x1)))))))))) 3(3(1(3(1(3(3(x1))))))) -> 1(2(1(3(0(5(5(1(2(1(x1)))))))))) 3(3(1(5(0(3(4(x1))))))) -> 2(0(0(5(1(2(1(4(1(4(x1)))))))))) 3(3(3(5(2(4(5(x1))))))) -> 3(1(0(0(1(4(2(2(0(5(x1)))))))))) 3(4(3(3(4(3(5(x1))))))) -> 5(1(1(1(1(1(4(2(3(3(x1)))))))))) 3(4(3(4(4(3(2(x1))))))) -> 2(5(4(5(3(4(2(4(4(0(x1)))))))))) 4(1(3(4(1(0(2(x1))))))) -> 4(4(1(5(1(2(1(4(2(2(x1)))))))))) 4(5(0(4(1(3(1(x1))))))) -> 1(1(1(2(0(0(4(4(5(1(x1)))))))))) 4(5(0(5(3(2(1(x1))))))) -> 1(1(4(1(3(0(2(4(2(1(x1)))))))))) 4(5(0(5(3(4(5(x1))))))) -> 1(1(1(0(5(4(0(2(4(5(x1)))))))))) 4(5(3(1(4(4(3(x1))))))) -> 4(5(5(5(4(2(4(4(2(3(x1)))))))))) 4(5(3(4(1(4(5(x1))))))) -> 4(5(4(0(2(0(1(2(1(0(x1)))))))))) 4(5(4(3(4(1(0(x1))))))) -> 1(4(1(1(2(4(0(1(2(0(x1)))))))))) 5(3(2(1(5(3(4(x1))))))) -> 5(1(2(1(3(3(5(1(2(4(x1)))))))))) 5(3(4(3(1(3(3(x1))))))) -> 5(1(1(4(2(4(3(3(4(3(x1)))))))))) 5(3(4(4(3(1(2(x1))))))) -> 5(1(0(5(0(3(5(1(1(1(x1)))))))))) 5(4(3(2(3(1(3(x1))))))) -> 0(0(0(2(3(0(2(5(4(3(x1)))))))))) 5(4(3(4(3(1(5(x1))))))) -> 0(2(0(0(0(0(4(1(5(0(x1)))))))))) 5(5(3(3(3(5(4(x1))))))) -> 0(3(5(2(2(1(0(4(2(2(x1)))))))))) 5(5(4(5(3(5(5(x1))))))) -> 5(2(4(2(2(2(4(1(5(2(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(1(x1)))) -> 0(1(1(1(0(0(1(2(2(2(x1)))))))))) 0(3(4(4(x1)))) -> 0(0(0(3(1(1(4(2(2(0(x1)))))))))) 1(5(1(5(4(x1))))) -> 1(0(0(2(2(0(5(2(2(4(x1)))))))))) 3(4(5(3(4(x1))))) -> 3(5(0(2(1(1(1(2(1(4(x1)))))))))) 4(3(4(3(4(x1))))) -> 2(1(1(1(4(1(2(4(2(0(x1)))))))))) 5(0(1(0(1(x1))))) -> 5(0(2(0(1(1(4(2(1(2(x1)))))))))) 1(0(1(0(1(4(x1)))))) -> 1(0(2(4(5(4(2(4(2(4(x1)))))))))) 1(1(3(4(1(5(x1)))))) -> 1(1(0(0(2(2(5(2(0(0(x1)))))))))) 1(3(1(1(3(3(x1)))))) -> 1(2(4(2(0(2(1(0(2(5(x1)))))))))) 1(3(1(5(2(3(x1)))))) -> 1(4(3(2(1(0(0(2(4(3(x1)))))))))) 1(5(0(5(5(3(x1)))))) -> 2(1(1(1(1(2(1(3(5(3(x1)))))))))) 2(1(3(1(5(5(x1)))))) -> 1(1(2(2(0(5(0(0(2(2(x1)))))))))) 2(2(4(3(4(5(x1)))))) -> 2(0(1(4(0(0(2(0(0(0(x1)))))))))) 2(3(1(0(3(4(x1)))))) -> 0(0(1(1(5(2(4(1(1(4(x1)))))))))) 2(3(3(4(1(5(x1)))))) -> 0(0(2(4(4(2(0(4(1(3(x1)))))))))) 2(5(0(5(5(1(x1)))))) -> 3(0(1(4(4(0(0(0(0(1(x1)))))))))) 3(0(4(3(3(4(x1)))))) -> 2(3(0(3(5(1(2(4(2(4(x1)))))))))) 3(2(3(3(0(4(x1)))))) -> 5(4(2(2(0(0(4(2(4(4(x1)))))))))) 3(3(5(4(3(4(x1)))))) -> 0(5(1(1(0(4(0(2(4(4(x1)))))))))) 3(5(4(2(1(0(x1)))))) -> 2(5(0(0(0(0(0(4(4(0(x1)))))))))) 4(1(3(4(3(1(x1)))))) -> 4(2(5(0(1(0(0(0(4(4(x1)))))))))) 4(2(1(0(2(5(x1)))))) -> 4(2(5(4(2(2(2(4(2(5(x1)))))))))) 4(3(3(5(1(1(x1)))))) -> 4(4(2(4(4(2(5(0(1(2(x1)))))))))) 5(4(0(1(3(0(x1)))))) -> 0(2(4(2(2(1(2(0(0(0(x1)))))))))) 0(4(1(5(5(3(5(x1))))))) -> 0(2(2(0(1(2(5(2(5(0(x1)))))))))) 0(4(4(3(4(1(3(x1))))))) -> 0(4(2(4(1(3(2(0(2(2(x1)))))))))) 1(3(3(4(5(2(5(x1))))))) -> 1(1(1(4(5(1(2(5(2(4(x1)))))))))) 1(5(2(5(1(5(2(x1))))))) -> 1(2(3(0(2(3(0(1(0(2(x1)))))))))) 1(5(3(2(4(5(4(x1))))))) -> 1(1(0(3(0(0(0(0(3(4(x1)))))))))) 1(5(5(5(3(2(1(x1))))))) -> 1(4(1(4(2(1(3(0(1(1(x1)))))))))) 2(1(2(3(1(3(3(x1))))))) -> 2(1(4(1(5(1(1(1(1(1(x1)))))))))) 3(0(2(1(3(2(1(x1))))))) -> 1(2(0(0(3(3(4(2(2(1(x1)))))))))) 3(0(4(4(3(4(5(x1))))))) -> 1(2(2(4(3(2(2(2(0(4(x1)))))))))) 3(1(3(1(5(4(1(x1))))))) -> 0(1(2(2(5(5(5(4(2(0(x1)))))))))) 3(2(5(2(1(3(4(x1))))))) -> 0(2(1(3(1(2(1(4(2(2(x1)))))))))) 3(3(1(3(1(3(3(x1))))))) -> 1(2(1(3(0(5(5(1(2(1(x1)))))))))) 3(3(1(5(0(3(4(x1))))))) -> 2(0(0(5(1(2(1(4(1(4(x1)))))))))) 3(3(3(5(2(4(5(x1))))))) -> 3(1(0(0(1(4(2(2(0(5(x1)))))))))) 3(4(3(3(4(3(5(x1))))))) -> 5(1(1(1(1(1(4(2(3(3(x1)))))))))) 3(4(3(4(4(3(2(x1))))))) -> 2(5(4(5(3(4(2(4(4(0(x1)))))))))) 4(1(3(4(1(0(2(x1))))))) -> 4(4(1(5(1(2(1(4(2(2(x1)))))))))) 4(5(0(4(1(3(1(x1))))))) -> 1(1(1(2(0(0(4(4(5(1(x1)))))))))) 4(5(0(5(3(2(1(x1))))))) -> 1(1(4(1(3(0(2(4(2(1(x1)))))))))) 4(5(0(5(3(4(5(x1))))))) -> 1(1(1(0(5(4(0(2(4(5(x1)))))))))) 4(5(3(1(4(4(3(x1))))))) -> 4(5(5(5(4(2(4(4(2(3(x1)))))))))) 4(5(3(4(1(4(5(x1))))))) -> 4(5(4(0(2(0(1(2(1(0(x1)))))))))) 4(5(4(3(4(1(0(x1))))))) -> 1(4(1(1(2(4(0(1(2(0(x1)))))))))) 5(3(2(1(5(3(4(x1))))))) -> 5(1(2(1(3(3(5(1(2(4(x1)))))))))) 5(3(4(3(1(3(3(x1))))))) -> 5(1(1(4(2(4(3(3(4(3(x1)))))))))) 5(3(4(4(3(1(2(x1))))))) -> 5(1(0(5(0(3(5(1(1(1(x1)))))))))) 5(4(3(2(3(1(3(x1))))))) -> 0(0(0(2(3(0(2(5(4(3(x1)))))))))) 5(4(3(4(3(1(5(x1))))))) -> 0(2(0(0(0(0(4(1(5(0(x1)))))))))) 5(5(3(3(3(5(4(x1))))))) -> 0(3(5(2(2(1(0(4(2(2(x1)))))))))) 5(5(4(5(3(5(5(x1))))))) -> 5(2(4(2(2(2(4(1(5(2(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_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(1(0(1(x1)))) -> 0(1(1(1(0(0(1(2(2(2(x1)))))))))) 0(3(4(4(x1)))) -> 0(0(0(3(1(1(4(2(2(0(x1)))))))))) 1(5(1(5(4(x1))))) -> 1(0(0(2(2(0(5(2(2(4(x1)))))))))) 3(4(5(3(4(x1))))) -> 3(5(0(2(1(1(1(2(1(4(x1)))))))))) 4(3(4(3(4(x1))))) -> 2(1(1(1(4(1(2(4(2(0(x1)))))))))) 5(0(1(0(1(x1))))) -> 5(0(2(0(1(1(4(2(1(2(x1)))))))))) 1(0(1(0(1(4(x1)))))) -> 1(0(2(4(5(4(2(4(2(4(x1)))))))))) 1(1(3(4(1(5(x1)))))) -> 1(1(0(0(2(2(5(2(0(0(x1)))))))))) 1(3(1(1(3(3(x1)))))) -> 1(2(4(2(0(2(1(0(2(5(x1)))))))))) 1(3(1(5(2(3(x1)))))) -> 1(4(3(2(1(0(0(2(4(3(x1)))))))))) 1(5(0(5(5(3(x1)))))) -> 2(1(1(1(1(2(1(3(5(3(x1)))))))))) 2(1(3(1(5(5(x1)))))) -> 1(1(2(2(0(5(0(0(2(2(x1)))))))))) 2(2(4(3(4(5(x1)))))) -> 2(0(1(4(0(0(2(0(0(0(x1)))))))))) 2(3(1(0(3(4(x1)))))) -> 0(0(1(1(5(2(4(1(1(4(x1)))))))))) 2(3(3(4(1(5(x1)))))) -> 0(0(2(4(4(2(0(4(1(3(x1)))))))))) 2(5(0(5(5(1(x1)))))) -> 3(0(1(4(4(0(0(0(0(1(x1)))))))))) 3(0(4(3(3(4(x1)))))) -> 2(3(0(3(5(1(2(4(2(4(x1)))))))))) 3(2(3(3(0(4(x1)))))) -> 5(4(2(2(0(0(4(2(4(4(x1)))))))))) 3(3(5(4(3(4(x1)))))) -> 0(5(1(1(0(4(0(2(4(4(x1)))))))))) 3(5(4(2(1(0(x1)))))) -> 2(5(0(0(0(0(0(4(4(0(x1)))))))))) 4(1(3(4(3(1(x1)))))) -> 4(2(5(0(1(0(0(0(4(4(x1)))))))))) 4(2(1(0(2(5(x1)))))) -> 4(2(5(4(2(2(2(4(2(5(x1)))))))))) 4(3(3(5(1(1(x1)))))) -> 4(4(2(4(4(2(5(0(1(2(x1)))))))))) 5(4(0(1(3(0(x1)))))) -> 0(2(4(2(2(1(2(0(0(0(x1)))))))))) 0(4(1(5(5(3(5(x1))))))) -> 0(2(2(0(1(2(5(2(5(0(x1)))))))))) 0(4(4(3(4(1(3(x1))))))) -> 0(4(2(4(1(3(2(0(2(2(x1)))))))))) 1(3(3(4(5(2(5(x1))))))) -> 1(1(1(4(5(1(2(5(2(4(x1)))))))))) 1(5(2(5(1(5(2(x1))))))) -> 1(2(3(0(2(3(0(1(0(2(x1)))))))))) 1(5(3(2(4(5(4(x1))))))) -> 1(1(0(3(0(0(0(0(3(4(x1)))))))))) 1(5(5(5(3(2(1(x1))))))) -> 1(4(1(4(2(1(3(0(1(1(x1)))))))))) 2(1(2(3(1(3(3(x1))))))) -> 2(1(4(1(5(1(1(1(1(1(x1)))))))))) 3(0(2(1(3(2(1(x1))))))) -> 1(2(0(0(3(3(4(2(2(1(x1)))))))))) 3(0(4(4(3(4(5(x1))))))) -> 1(2(2(4(3(2(2(2(0(4(x1)))))))))) 3(1(3(1(5(4(1(x1))))))) -> 0(1(2(2(5(5(5(4(2(0(x1)))))))))) 3(2(5(2(1(3(4(x1))))))) -> 0(2(1(3(1(2(1(4(2(2(x1)))))))))) 3(3(1(3(1(3(3(x1))))))) -> 1(2(1(3(0(5(5(1(2(1(x1)))))))))) 3(3(1(5(0(3(4(x1))))))) -> 2(0(0(5(1(2(1(4(1(4(x1)))))))))) 3(3(3(5(2(4(5(x1))))))) -> 3(1(0(0(1(4(2(2(0(5(x1)))))))))) 3(4(3(3(4(3(5(x1))))))) -> 5(1(1(1(1(1(4(2(3(3(x1)))))))))) 3(4(3(4(4(3(2(x1))))))) -> 2(5(4(5(3(4(2(4(4(0(x1)))))))))) 4(1(3(4(1(0(2(x1))))))) -> 4(4(1(5(1(2(1(4(2(2(x1)))))))))) 4(5(0(4(1(3(1(x1))))))) -> 1(1(1(2(0(0(4(4(5(1(x1)))))))))) 4(5(0(5(3(2(1(x1))))))) -> 1(1(4(1(3(0(2(4(2(1(x1)))))))))) 4(5(0(5(3(4(5(x1))))))) -> 1(1(1(0(5(4(0(2(4(5(x1)))))))))) 4(5(3(1(4(4(3(x1))))))) -> 4(5(5(5(4(2(4(4(2(3(x1)))))))))) 4(5(3(4(1(4(5(x1))))))) -> 4(5(4(0(2(0(1(2(1(0(x1)))))))))) 4(5(4(3(4(1(0(x1))))))) -> 1(4(1(1(2(4(0(1(2(0(x1)))))))))) 5(3(2(1(5(3(4(x1))))))) -> 5(1(2(1(3(3(5(1(2(4(x1)))))))))) 5(3(4(3(1(3(3(x1))))))) -> 5(1(1(4(2(4(3(3(4(3(x1)))))))))) 5(3(4(4(3(1(2(x1))))))) -> 5(1(0(5(0(3(5(1(1(1(x1)))))))))) 5(4(3(2(3(1(3(x1))))))) -> 0(0(0(2(3(0(2(5(4(3(x1)))))))))) 5(4(3(4(3(1(5(x1))))))) -> 0(2(0(0(0(0(4(1(5(0(x1)))))))))) 5(5(3(3(3(5(4(x1))))))) -> 0(3(5(2(2(1(0(4(2(2(x1)))))))))) 5(5(4(5(3(5(5(x1))))))) -> 5(2(4(2(2(2(4(1(5(2(x1)))))))))) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encArg(cons_4(x_1)) -> 4(encArg(x_1)) encArg(cons_5(x_1)) -> 5(encArg(x_1)) encArg(cons_2(x_1)) -> 2(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. The certificate found is represented by the following graph. "[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] {(135,136,[0_1|0, 1_1|0, 3_1|0, 4_1|0, 5_1|0, 2_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]), (135,137,[0_1|1, 1_1|1, 3_1|1, 4_1|1, 5_1|1, 2_1|1]), (135,138,[0_1|2]), (135,147,[0_1|2]), (135,156,[0_1|2]), (135,165,[0_1|2]), (135,174,[1_1|2]), (135,183,[2_1|2]), (135,192,[1_1|2]), (135,201,[1_1|2]), (135,210,[1_1|2]), (135,219,[1_1|2]), (135,228,[1_1|2]), (135,237,[1_1|2]), (135,246,[1_1|2]), (135,255,[1_1|2]), (135,264,[3_1|2]), (135,273,[5_1|2]), (135,282,[2_1|2]), (135,291,[2_1|2]), (135,300,[1_1|2]), (135,309,[1_1|2]), (135,318,[5_1|2]), (135,327,[0_1|2]), (135,336,[0_1|2]), (135,345,[1_1|2]), (135,354,[2_1|2]), (135,363,[3_1|2]), (135,372,[2_1|2]), (135,381,[0_1|2]), (135,390,[2_1|2]), (135,399,[4_1|2]), (135,408,[4_1|2]), (135,417,[4_1|2]), (135,426,[4_1|2]), (135,435,[1_1|2]), (135,444,[1_1|2]), (135,453,[1_1|2]), (135,462,[4_1|2]), (135,471,[4_1|2]), (135,480,[1_1|2]), (135,489,[5_1|2]), (135,498,[0_1|2]), (135,507,[0_1|2]), (135,516,[0_1|2]), (135,525,[5_1|2]), (135,534,[5_1|2]), (135,543,[5_1|2]), (135,552,[0_1|2]), (135,561,[5_1|2]), (135,570,[1_1|2]), (135,579,[2_1|2]), (135,588,[2_1|2]), (135,597,[0_1|2]), (135,606,[0_1|2]), (135,615,[3_1|2]), (136,136,[cons_0_1|0, cons_1_1|0, cons_3_1|0, cons_4_1|0, cons_5_1|0, cons_2_1|0]), (137,136,[encArg_1|1]), (137,137,[0_1|1, 1_1|1, 3_1|1, 4_1|1, 5_1|1, 2_1|1]), (137,138,[0_1|2]), (137,147,[0_1|2]), (137,156,[0_1|2]), (137,165,[0_1|2]), (137,174,[1_1|2]), (137,183,[2_1|2]), (137,192,[1_1|2]), (137,201,[1_1|2]), (137,210,[1_1|2]), (137,219,[1_1|2]), (137,228,[1_1|2]), (137,237,[1_1|2]), (137,246,[1_1|2]), (137,255,[1_1|2]), (137,264,[3_1|2]), (137,273,[5_1|2]), (137,282,[2_1|2]), (137,291,[2_1|2]), (137,300,[1_1|2]), (137,309,[1_1|2]), (137,318,[5_1|2]), (137,327,[0_1|2]), (137,336,[0_1|2]), (137,345,[1_1|2]), (137,354,[2_1|2]), (137,363,[3_1|2]), (137,372,[2_1|2]), (137,381,[0_1|2]), (137,390,[2_1|2]), (137,399,[4_1|2]), (137,408,[4_1|2]), (137,417,[4_1|2]), (137,426,[4_1|2]), (137,435,[1_1|2]), (137,444,[1_1|2]), (137,453,[1_1|2]), (137,462,[4_1|2]), (137,471,[4_1|2]), (137,480,[1_1|2]), (137,489,[5_1|2]), (137,498,[0_1|2]), (137,507,[0_1|2]), (137,516,[0_1|2]), (137,525,[5_1|2]), (137,534,[5_1|2]), (137,543,[5_1|2]), (137,552,[0_1|2]), (137,561,[5_1|2]), (137,570,[1_1|2]), (137,579,[2_1|2]), (137,588,[2_1|2]), (137,597,[0_1|2]), (137,606,[0_1|2]), 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(578,570,[1_1|2]), (578,579,[2_1|2]), (578,588,[2_1|2]), (578,597,[0_1|2]), (578,606,[0_1|2]), (578,615,[3_1|2]), (579,580,[1_1|2]), (580,581,[4_1|2]), (581,582,[1_1|2]), (582,583,[5_1|2]), (583,584,[1_1|2]), (584,585,[1_1|2]), (585,586,[1_1|2]), (586,587,[1_1|2]), (586,228,[1_1|2]), (587,137,[1_1|2]), (587,264,[1_1|2]), (587,363,[1_1|2]), (587,615,[1_1|2]), (587,174,[1_1|2]), (587,183,[2_1|2]), (587,192,[1_1|2]), (587,201,[1_1|2]), (587,210,[1_1|2]), (587,219,[1_1|2]), (587,228,[1_1|2]), (587,237,[1_1|2]), (587,246,[1_1|2]), (587,255,[1_1|2]), (588,589,[0_1|2]), (589,590,[1_1|2]), (590,591,[4_1|2]), (591,592,[0_1|2]), (592,593,[0_1|2]), (593,594,[2_1|2]), (594,595,[0_1|2]), (595,596,[0_1|2]), (596,137,[0_1|2]), (596,273,[0_1|2]), (596,318,[0_1|2]), (596,489,[0_1|2]), (596,525,[0_1|2]), (596,534,[0_1|2]), (596,543,[0_1|2]), (596,561,[0_1|2]), (596,463,[0_1|2]), (596,472,[0_1|2]), (596,138,[0_1|2]), (596,147,[0_1|2]), (596,156,[0_1|2]), (596,165,[0_1|2]), (597,598,[0_1|2]), (598,599,[1_1|2]), (599,600,[1_1|2]), (600,601,[5_1|2]), (601,602,[2_1|2]), (602,603,[4_1|2]), (603,604,[1_1|2]), (604,605,[1_1|2]), (605,137,[4_1|2]), (605,399,[4_1|2]), (605,408,[4_1|2]), (605,417,[4_1|2]), (605,426,[4_1|2]), (605,462,[4_1|2]), (605,471,[4_1|2]), (605,390,[2_1|2]), (605,435,[1_1|2]), (605,444,[1_1|2]), (605,453,[1_1|2]), (605,480,[1_1|2]), (606,607,[0_1|2]), (607,608,[2_1|2]), (608,609,[4_1|2]), (609,610,[4_1|2]), (610,611,[2_1|2]), (611,612,[0_1|2]), (612,613,[4_1|2]), (612,408,[4_1|2]), (612,417,[4_1|2]), (613,614,[1_1|2]), (613,237,[1_1|2]), (613,246,[1_1|2]), (613,255,[1_1|2]), (614,137,[3_1|2]), (614,273,[3_1|2, 5_1|2]), (614,318,[3_1|2, 5_1|2]), (614,489,[3_1|2]), (614,525,[3_1|2]), (614,534,[3_1|2]), (614,543,[3_1|2]), (614,561,[3_1|2]), (614,264,[3_1|2]), (614,282,[2_1|2]), (614,291,[2_1|2]), (614,300,[1_1|2]), (614,309,[1_1|2]), (614,327,[0_1|2]), (614,336,[0_1|2]), (614,345,[1_1|2]), (614,354,[2_1|2]), (614,363,[3_1|2]), (614,372,[2_1|2]), (614,381,[0_1|2]), (615,616,[0_1|2]), (616,617,[1_1|2]), (617,618,[4_1|2]), (618,619,[4_1|2]), (619,620,[0_1|2]), (620,621,[0_1|2]), (621,622,[0_1|2]), (622,623,[0_1|2]), (622,138,[0_1|2]), (622,633,[0_1|3]), (623,137,[1_1|2]), (623,174,[1_1|2]), (623,192,[1_1|2]), (623,201,[1_1|2]), (623,210,[1_1|2]), (623,219,[1_1|2]), (623,228,[1_1|2]), (623,237,[1_1|2]), (623,246,[1_1|2]), (623,255,[1_1|2]), (623,300,[1_1|2]), (623,309,[1_1|2]), (623,345,[1_1|2]), (623,435,[1_1|2]), (623,444,[1_1|2]), (623,453,[1_1|2]), (623,480,[1_1|2]), (623,570,[1_1|2]), (623,274,[1_1|2]), (623,526,[1_1|2]), (623,535,[1_1|2]), (623,544,[1_1|2]), (623,183,[2_1|2]), (624,625,[0_1|3]), (625,626,[0_1|3]), (626,627,[3_1|3]), (627,628,[1_1|3]), (628,629,[1_1|3]), (629,630,[4_1|3]), (630,631,[2_1|3]), (631,632,[2_1|3]), (632,399,[0_1|3]), (632,408,[0_1|3]), (632,417,[0_1|3]), (632,426,[0_1|3]), (632,462,[0_1|3]), (632,471,[0_1|3]), (632,400,[0_1|3]), (632,418,[0_1|3]), (633,634,[1_1|3]), (634,635,[1_1|3]), (635,636,[1_1|3]), (636,637,[0_1|3]), (637,638,[0_1|3]), (638,639,[1_1|3]), (639,640,[2_1|3]), (640,641,[2_1|3]), (641,139,[2_1|3]), (641,382,[2_1|3]), (642,643,[1_1|3]), (643,644,[1_1|3]), (644,645,[1_1|3]), (645,646,[0_1|3]), (646,647,[0_1|3]), (647,648,[1_1|3]), (648,649,[2_1|3]), (649,650,[2_1|3]), (650,570,[2_1|3])}" ---------------------------------------- (8) BOUNDS(1, n^1)