/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 641 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 4 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection (13) InfiniteLowerBoundProof [FINISHED, 37.9 s] (14) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) 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(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(U21(X1, X2, X3, X4)) ->^+ a__U21(mark(X1), X2, X3, X4) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / U21(X1, X2, X3, X4)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N) a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N) a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) a__take(0, IL) -> nil a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4) mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4) mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4) a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4) a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(zeros) -> zeros encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U12(x_1, x_2)) -> U12(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(U21(x_1, x_2, x_3, x_4)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U22(x_1, x_2, x_3, x_4)) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(U23(x_1, x_2, x_3, x_4)) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__zeros) -> a__zeros encArg(cons_a__U11(x_1, x_2)) -> a__U11(encArg(x_1), encArg(x_2)) encArg(cons_a__U12(x_1, x_2)) -> a__U12(encArg(x_1), encArg(x_2)) encArg(cons_a__U21(x_1, x_2, x_3, x_4)) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U22(x_1, x_2, x_3, x_4)) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__U23(x_1, x_2, x_3, x_4)) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__zeros -> a__zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_zeros -> zeros encode_a__U11(x_1, x_2) -> a__U11(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_a__U12(x_1, x_2) -> a__U12(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__U21(x_1, x_2, x_3, x_4) -> a__U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U22(x_1, x_2, x_3, x_4) -> a__U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_a__U23(x_1, x_2, x_3, x_4) -> a__U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U12(x_1, x_2) -> U12(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_U21(x_1, x_2, x_3, x_4) -> U21(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U22(x_1, x_2, x_3, x_4) -> U22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_U23(x_1, x_2, x_3, x_4) -> U23(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence a__U11(tt, zeros) ->^+ s(a__U11(tt, zeros)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [ ]. The result substitution is [ ]. ---------------------------------------- (14) BOUNDS(INF, INF)