/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 865 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(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(length(X)) ->^+ a__length(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / length(X)]. 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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(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(n^1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__U21(tt) -> nil a__U31(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__isNatList(take(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) a__take(0, IL) -> a__U21(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2, X3, X4)) -> a__U31(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) 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__length(X) -> length(X) a__U21(X) -> U21(X) a__U31(X1, X2, X3, X4) -> U31(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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(nil) -> nil encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(length(x_1)) -> length(encArg(x_1)) encArg(isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(isNat(x_1)) -> isNat(encArg(x_1)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2)) encArg(U21(x_1)) -> U21(encArg(x_1)) encArg(U31(x_1, x_2, x_3, x_4)) -> U31(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__U21(x_1)) -> a__U21(encArg(x_1)) encArg(cons_a__U31(x_1, x_2, x_3, x_4)) -> a__U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_a__and(x_1, x_2)) -> a__and(encArg(x_1), encArg(x_2)) encArg(cons_a__isNat(x_1)) -> a__isNat(encArg(x_1)) encArg(cons_a__isNatIList(x_1)) -> a__isNatIList(encArg(x_1)) encArg(cons_a__isNatList(x_1)) -> a__isNatList(encArg(x_1)) 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_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) -> a__U21(encArg(x_1)) encode_nil -> nil encode_a__U31(x_1, x_2, x_3, x_4) -> a__U31(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_a__and(x_1, x_2) -> a__and(encArg(x_1), encArg(x_2)) encode_a__isNat(x_1) -> a__isNat(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_a__isNatList(x_1) -> a__isNatList(encArg(x_1)) encode_a__isNatIList(x_1) -> a__isNatIList(encArg(x_1)) encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_U11(x_1, x_2) -> U11(encArg(x_1), encArg(x_2)) encode_U21(x_1) -> U21(encArg(x_1)) encode_U31(x_1, x_2, x_3, x_4) -> U31(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST