/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF).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), 511 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 50 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, 7725 ms] (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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> 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(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) 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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) 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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) 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 isNatIList(n__cons(N, IL)) ->^+ and(isNat(activate(N)), isNatIList(IL)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [IL / n__cons(N, IL)]. 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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) 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: and(tt, T) -> T isNatIList(IL) -> isNatList(activate(IL)) isNat(n__0) -> tt isNat(n__s(N)) -> isNat(activate(N)) isNat(n__length(L)) -> isNatList(activate(L)) isNatIList(n__zeros) -> tt isNatIList(n__cons(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) isNatList(n__nil) -> tt isNatList(n__cons(N, L)) -> and(isNat(activate(N)), isNatList(activate(L))) isNatList(n__take(N, IL)) -> and(isNat(activate(N)), isNatIList(activate(IL))) zeros -> cons(0, n__zeros) take(0, IL) -> uTake1(isNatIList(IL)) uTake1(tt) -> nil take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL)) uTake2(tt, M, N, IL) -> cons(activate(N), n__take(activate(M), activate(IL))) length(cons(N, L)) -> uLength(and(isNat(N), isNatList(activate(L))), activate(L)) uLength(tt, L) -> s(length(activate(L))) 0 -> n__0 s(X) -> n__s(X) length(X) -> n__length(X) zeros -> n__zeros cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil take(X1, X2) -> n__take(X1, X2) activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__length(X)) -> length(X) activate(n__zeros) -> zeros activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(n__take(X1, X2)) -> take(X1, X2) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(n__0) -> n__0 encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__length(x_1)) -> n__length(encArg(x_1)) encArg(n__zeros) -> n__zeros encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__nil) -> n__nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_isNatIList(x_1)) -> isNatIList(encArg(x_1)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encArg(cons_isNatList(x_1)) -> isNatList(encArg(x_1)) encArg(cons_zeros) -> zeros encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_uTake1(x_1)) -> uTake1(encArg(x_1)) encArg(cons_uTake2(x_1, x_2, x_3, x_4)) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_uLength(x_1, x_2)) -> uLength(encArg(x_1), encArg(x_2)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNatIList(x_1) -> isNatIList(encArg(x_1)) encode_isNatList(x_1) -> isNatList(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_isNat(x_1) -> isNat(encArg(x_1)) encode_n__0 -> n__0 encode_n__s(x_1) -> n__s(encArg(x_1)) encode_n__length(x_1) -> n__length(encArg(x_1)) encode_n__zeros -> n__zeros encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_uTake1(x_1) -> uTake1(encArg(x_1)) encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_uTake2(x_1, x_2, x_3, x_4) -> uTake2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_length(x_1) -> length(encArg(x_1)) encode_uLength(x_1, x_2) -> uLength(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence isNatIList(n__cons(N, n__zeros)) ->^+ and(isNat(activate(N)), isNatIList(n__cons(n__0, n__zeros))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [ ]. The result substitution is [N / n__0]. ---------------------------------------- (14) BOUNDS(INF, INF)