MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { a__zeros^#() -> c_1() , a__zeros^#() -> c_2() , a__U11^#(X1, X2) -> c_3() , a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__U12^#(X) -> c_5() , a__U12^#(tt()) -> c_6() , a__isNatList^#(X) -> c_7() , a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(nil()) -> c_9() , a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(X1, X2, X3) -> c_33() , a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(X1, X2) -> c_86() , a__and^#(tt(), X) -> c_87(mark^#(X)) , a__isNatKind^#(X) -> c_93() , a__isNatKind^#(0()) -> c_94() , a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , a__U61^#(X1, X2, X3) -> c_39() , a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(X1, X2) -> c_11() , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__U22^#(X) -> c_13() , a__U22^#(tt()) -> c_14() , a__isNat^#(X) -> c_15() , a__isNat^#(0()) -> c_16() , a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(X) -> c_88() , a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(zeros()) -> c_90() , a__isNatIListKind^#(nil()) -> c_91() , a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(X1, X2) -> c_19() , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U32^#(X) -> c_21() , a__U32^#(tt()) -> c_22() , a__U41^#(X1, X2, X3) -> c_23() , a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(X1, X2) -> c_25() , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U43^#(X) -> c_27() , a__U43^#(tt()) -> c_28() , a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(X) -> c_30() , a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIList^#(zeros()) -> c_32() , a__U52^#(X1, X2) -> c_35() , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U53^#(X) -> c_37() , a__U53^#(tt()) -> c_38() , a__U62^#(X1, X2) -> c_41() , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U63^#(X) -> c_43() , a__U63^#(tt()) -> c_44() , a__U71^#(X1, X2) -> c_45() , a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , a__length^#(X) -> c_47() , a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , a__length^#(nil()) -> c_49() , mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , mark^#(0()) -> c_51() , mark^#(zeros()) -> c_52(a__zeros^#()) , mark^#(tt()) -> c_53() , mark^#(s(X)) -> c_54(mark^#(X)) , mark^#(nil()) -> c_55() , mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(X1, X2) -> c_97() , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U81^#(X) -> c_82() , a__U81^#(tt()) -> c_83() , a__U91^#(X1, X2, X3, X4) -> c_84() , a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__zeros^#() -> c_1() , a__zeros^#() -> c_2() , a__U11^#(X1, X2) -> c_3() , a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__U12^#(X) -> c_5() , a__U12^#(tt()) -> c_6() , a__isNatList^#(X) -> c_7() , a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(nil()) -> c_9() , a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(X1, X2, X3) -> c_33() , a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(X1, X2) -> c_86() , a__and^#(tt(), X) -> c_87(mark^#(X)) , a__isNatKind^#(X) -> c_93() , a__isNatKind^#(0()) -> c_94() , a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , a__U61^#(X1, X2, X3) -> c_39() , a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(X1, X2) -> c_11() , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__U22^#(X) -> c_13() , a__U22^#(tt()) -> c_14() , a__isNat^#(X) -> c_15() , a__isNat^#(0()) -> c_16() , a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(X) -> c_88() , a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(zeros()) -> c_90() , a__isNatIListKind^#(nil()) -> c_91() , a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(X1, X2) -> c_19() , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U32^#(X) -> c_21() , a__U32^#(tt()) -> c_22() , a__U41^#(X1, X2, X3) -> c_23() , a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(X1, X2) -> c_25() , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U43^#(X) -> c_27() , a__U43^#(tt()) -> c_28() , a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(X) -> c_30() , a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIList^#(zeros()) -> c_32() , a__U52^#(X1, X2) -> c_35() , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U53^#(X) -> c_37() , a__U53^#(tt()) -> c_38() , a__U62^#(X1, X2) -> c_41() , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U63^#(X) -> c_43() , a__U63^#(tt()) -> c_44() , a__U71^#(X1, X2) -> c_45() , a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , a__length^#(X) -> c_47() , a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , a__length^#(nil()) -> c_49() , mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , mark^#(0()) -> c_51() , mark^#(zeros()) -> c_52(a__zeros^#()) , mark^#(tt()) -> c_53() , mark^#(s(X)) -> c_54(mark^#(X)) , mark^#(nil()) -> c_55() , mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(X1, X2) -> c_97() , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U81^#(X) -> c_82() , a__U81^#(tt()) -> c_83() , a__U91^#(X1, X2, X3, X4) -> c_84() , a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } Weak Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,5,6,7,9,11,13,15,16,19,21,23,24,25,26,29,31,32,34,36,37,38,40,42,43,45,47,48,50,51,52,54,55,56,58,60,62,64,66,93,96,97,98} by applications of Pre({1,2,3,5,6,7,9,11,13,15,16,19,21,23,24,25,26,29,31,32,34,36,37,38,40,42,43,45,47,48,50,51,52,54,55,56,58,60,62,64,66,93,96,97,98}) = {4,8,10,12,14,17,18,20,22,27,28,30,33,35,39,41,44,46,49,53,57,59,61,63,65,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,94,95,99}. Here rules are labeled as follows: DPs: { 1: a__zeros^#() -> c_1() , 2: a__zeros^#() -> c_2() , 3: a__U11^#(X1, X2) -> c_3() , 4: a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , 5: a__U12^#(X) -> c_5() , 6: a__U12^#(tt()) -> c_6() , 7: a__isNatList^#(X) -> c_7() , 8: a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 9: a__isNatList^#(nil()) -> c_9() , 10: a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 11: a__U51^#(X1, X2, X3) -> c_33() , 12: a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , 13: a__and^#(X1, X2) -> c_86() , 14: a__and^#(tt(), X) -> c_87(mark^#(X)) , 15: a__isNatKind^#(X) -> c_93() , 16: a__isNatKind^#(0()) -> c_94() , 17: a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , 18: a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , 19: a__U61^#(X1, X2, X3) -> c_39() , 20: a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , 21: a__U21^#(X1, X2) -> c_11() , 22: a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , 23: a__U22^#(X) -> c_13() , 24: a__U22^#(tt()) -> c_14() , 25: a__isNat^#(X) -> c_15() , 26: a__isNat^#(0()) -> c_16() , 27: a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , 28: a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , 29: a__isNatIListKind^#(X) -> c_88() , 30: a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 31: a__isNatIListKind^#(zeros()) -> c_90() , 32: a__isNatIListKind^#(nil()) -> c_91() , 33: a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 34: a__U31^#(X1, X2) -> c_19() , 35: a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , 36: a__U32^#(X) -> c_21() , 37: a__U32^#(tt()) -> c_22() , 38: a__U41^#(X1, X2, X3) -> c_23() , 39: a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , 40: a__U42^#(X1, X2) -> c_25() , 41: a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , 42: a__U43^#(X) -> c_27() , 43: a__U43^#(tt()) -> c_28() , 44: a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , 45: a__isNatIList^#(X) -> c_30() , 46: a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 47: a__isNatIList^#(zeros()) -> c_32() , 48: a__U52^#(X1, X2) -> c_35() , 49: a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , 50: a__U53^#(X) -> c_37() , 51: a__U53^#(tt()) -> c_38() , 52: a__U62^#(X1, X2) -> c_41() , 53: a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , 54: a__U63^#(X) -> c_43() , 55: a__U63^#(tt()) -> c_44() , 56: a__U71^#(X1, X2) -> c_45() , 57: a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , 58: a__length^#(X) -> c_47() , 59: a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , 60: a__length^#(nil()) -> c_49() , 61: mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , 62: mark^#(0()) -> c_51() , 63: mark^#(zeros()) -> c_52(a__zeros^#()) , 64: mark^#(tt()) -> c_53() , 65: mark^#(s(X)) -> c_54(mark^#(X)) , 66: mark^#(nil()) -> c_55() , 67: mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , 68: mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , 69: mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , 70: mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , 71: mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , 72: mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , 73: mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , 74: mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , 75: mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , 76: mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , 77: mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , 78: mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , 79: mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , 80: mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , 81: mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , 82: mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , 83: mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , 84: mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , 85: mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , 86: mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , 87: mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , 88: mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , 89: mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , 90: mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , 91: mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , 92: mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , 93: a__take^#(X1, X2) -> c_97() , 94: a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , 95: a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , 96: a__U81^#(X) -> c_82() , 97: a__U81^#(tt()) -> c_83() , 98: a__U91^#(X1, X2, X3, X4) -> c_84() , 99: a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(tt(), X) -> c_87(mark^#(X)) , a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , mark^#(zeros()) -> c_52(a__zeros^#()) , mark^#(s(X)) -> c_54(mark^#(X)) , mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } Weak DPs: { a__zeros^#() -> c_1() , a__zeros^#() -> c_2() , a__U11^#(X1, X2) -> c_3() , a__U12^#(X) -> c_5() , a__U12^#(tt()) -> c_6() , a__isNatList^#(X) -> c_7() , a__isNatList^#(nil()) -> c_9() , a__U51^#(X1, X2, X3) -> c_33() , a__and^#(X1, X2) -> c_86() , a__isNatKind^#(X) -> c_93() , a__isNatKind^#(0()) -> c_94() , a__U61^#(X1, X2, X3) -> c_39() , a__U21^#(X1, X2) -> c_11() , a__U22^#(X) -> c_13() , a__U22^#(tt()) -> c_14() , a__isNat^#(X) -> c_15() , a__isNat^#(0()) -> c_16() , a__isNatIListKind^#(X) -> c_88() , a__isNatIListKind^#(zeros()) -> c_90() , a__isNatIListKind^#(nil()) -> c_91() , a__U31^#(X1, X2) -> c_19() , a__U32^#(X) -> c_21() , a__U32^#(tt()) -> c_22() , a__U41^#(X1, X2, X3) -> c_23() , a__U42^#(X1, X2) -> c_25() , a__U43^#(X) -> c_27() , a__U43^#(tt()) -> c_28() , a__isNatIList^#(X) -> c_30() , a__isNatIList^#(zeros()) -> c_32() , a__U52^#(X1, X2) -> c_35() , a__U53^#(X) -> c_37() , a__U53^#(tt()) -> c_38() , a__U62^#(X1, X2) -> c_41() , a__U63^#(X) -> c_43() , a__U63^#(tt()) -> c_44() , a__U71^#(X1, X2) -> c_45() , a__length^#(X) -> c_47() , a__length^#(nil()) -> c_49() , mark^#(0()) -> c_51() , mark^#(tt()) -> c_53() , mark^#(nil()) -> c_55() , a__take^#(X1, X2) -> c_97() , a__U81^#(X) -> c_82() , a__U81^#(tt()) -> c_83() , a__U91^#(X1, X2, X3, X4) -> c_84() } Weak Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {24} by applications of Pre({24}) = {5,21,23,25,26,27,29,32,33,35,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,54}. Here rules are labeled as follows: DPs: { 1: a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , 2: a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 3: a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 4: a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , 5: a__and^#(tt(), X) -> c_87(mark^#(X)) , 6: a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , 7: a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , 8: a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , 9: a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , 10: a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , 11: a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , 12: a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 13: a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 14: a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , 15: a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , 16: a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , 17: a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , 18: a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , 19: a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , 20: a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , 21: a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , 22: a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , 23: mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , 24: mark^#(zeros()) -> c_52(a__zeros^#()) , 25: mark^#(s(X)) -> c_54(mark^#(X)) , 26: mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , 27: mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , 28: mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , 29: mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , 30: mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , 31: mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , 32: mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , 33: mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , 34: mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , 35: mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , 36: mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , 37: mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , 38: mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , 39: mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , 40: mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , 41: mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , 42: mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , 43: mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , 44: mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , 45: mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , 46: mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , 47: mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , 48: mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , 49: mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , 50: mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , 51: mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , 52: a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , 53: a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , 54: a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) , 55: a__zeros^#() -> c_1() , 56: a__zeros^#() -> c_2() , 57: a__U11^#(X1, X2) -> c_3() , 58: a__U12^#(X) -> c_5() , 59: a__U12^#(tt()) -> c_6() , 60: a__isNatList^#(X) -> c_7() , 61: a__isNatList^#(nil()) -> c_9() , 62: a__U51^#(X1, X2, X3) -> c_33() , 63: a__and^#(X1, X2) -> c_86() , 64: a__isNatKind^#(X) -> c_93() , 65: a__isNatKind^#(0()) -> c_94() , 66: a__U61^#(X1, X2, X3) -> c_39() , 67: a__U21^#(X1, X2) -> c_11() , 68: a__U22^#(X) -> c_13() , 69: a__U22^#(tt()) -> c_14() , 70: a__isNat^#(X) -> c_15() , 71: a__isNat^#(0()) -> c_16() , 72: a__isNatIListKind^#(X) -> c_88() , 73: a__isNatIListKind^#(zeros()) -> c_90() , 74: a__isNatIListKind^#(nil()) -> c_91() , 75: a__U31^#(X1, X2) -> c_19() , 76: a__U32^#(X) -> c_21() , 77: a__U32^#(tt()) -> c_22() , 78: a__U41^#(X1, X2, X3) -> c_23() , 79: a__U42^#(X1, X2) -> c_25() , 80: a__U43^#(X) -> c_27() , 81: a__U43^#(tt()) -> c_28() , 82: a__isNatIList^#(X) -> c_30() , 83: a__isNatIList^#(zeros()) -> c_32() , 84: a__U52^#(X1, X2) -> c_35() , 85: a__U53^#(X) -> c_37() , 86: a__U53^#(tt()) -> c_38() , 87: a__U62^#(X1, X2) -> c_41() , 88: a__U63^#(X) -> c_43() , 89: a__U63^#(tt()) -> c_44() , 90: a__U71^#(X1, X2) -> c_45() , 91: a__length^#(X) -> c_47() , 92: a__length^#(nil()) -> c_49() , 93: mark^#(0()) -> c_51() , 94: mark^#(tt()) -> c_53() , 95: mark^#(nil()) -> c_55() , 96: a__take^#(X1, X2) -> c_97() , 97: a__U81^#(X) -> c_82() , 98: a__U81^#(tt()) -> c_83() , 99: a__U91^#(X1, X2, X3, X4) -> c_84() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(tt(), X) -> c_87(mark^#(X)) , a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , mark^#(s(X)) -> c_54(mark^#(X)) , mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } Weak DPs: { a__zeros^#() -> c_1() , a__zeros^#() -> c_2() , a__U11^#(X1, X2) -> c_3() , a__U12^#(X) -> c_5() , a__U12^#(tt()) -> c_6() , a__isNatList^#(X) -> c_7() , a__isNatList^#(nil()) -> c_9() , a__U51^#(X1, X2, X3) -> c_33() , a__and^#(X1, X2) -> c_86() , a__isNatKind^#(X) -> c_93() , a__isNatKind^#(0()) -> c_94() , a__U61^#(X1, X2, X3) -> c_39() , a__U21^#(X1, X2) -> c_11() , a__U22^#(X) -> c_13() , a__U22^#(tt()) -> c_14() , a__isNat^#(X) -> c_15() , a__isNat^#(0()) -> c_16() , a__isNatIListKind^#(X) -> c_88() , a__isNatIListKind^#(zeros()) -> c_90() , a__isNatIListKind^#(nil()) -> c_91() , a__U31^#(X1, X2) -> c_19() , a__U32^#(X) -> c_21() , a__U32^#(tt()) -> c_22() , a__U41^#(X1, X2, X3) -> c_23() , a__U42^#(X1, X2) -> c_25() , a__U43^#(X) -> c_27() , a__U43^#(tt()) -> c_28() , a__isNatIList^#(X) -> c_30() , a__isNatIList^#(zeros()) -> c_32() , a__U52^#(X1, X2) -> c_35() , a__U53^#(X) -> c_37() , a__U53^#(tt()) -> c_38() , a__U62^#(X1, X2) -> c_41() , a__U63^#(X) -> c_43() , a__U63^#(tt()) -> c_44() , a__U71^#(X1, X2) -> c_45() , a__length^#(X) -> c_47() , a__length^#(nil()) -> c_49() , mark^#(0()) -> c_51() , mark^#(zeros()) -> c_52(a__zeros^#()) , mark^#(tt()) -> c_53() , mark^#(nil()) -> c_55() , a__take^#(X1, X2) -> c_97() , a__U81^#(X) -> c_82() , a__U81^#(tt()) -> c_83() , a__U91^#(X1, X2, X3, X4) -> c_84() } Weak Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { a__zeros^#() -> c_1() , a__zeros^#() -> c_2() , a__U11^#(X1, X2) -> c_3() , a__U12^#(X) -> c_5() , a__U12^#(tt()) -> c_6() , a__isNatList^#(X) -> c_7() , a__isNatList^#(nil()) -> c_9() , a__U51^#(X1, X2, X3) -> c_33() , a__and^#(X1, X2) -> c_86() , a__isNatKind^#(X) -> c_93() , a__isNatKind^#(0()) -> c_94() , a__U61^#(X1, X2, X3) -> c_39() , a__U21^#(X1, X2) -> c_11() , a__U22^#(X) -> c_13() , a__U22^#(tt()) -> c_14() , a__isNat^#(X) -> c_15() , a__isNat^#(0()) -> c_16() , a__isNatIListKind^#(X) -> c_88() , a__isNatIListKind^#(zeros()) -> c_90() , a__isNatIListKind^#(nil()) -> c_91() , a__U31^#(X1, X2) -> c_19() , a__U32^#(X) -> c_21() , a__U32^#(tt()) -> c_22() , a__U41^#(X1, X2, X3) -> c_23() , a__U42^#(X1, X2) -> c_25() , a__U43^#(X) -> c_27() , a__U43^#(tt()) -> c_28() , a__isNatIList^#(X) -> c_30() , a__isNatIList^#(zeros()) -> c_32() , a__U52^#(X1, X2) -> c_35() , a__U53^#(X) -> c_37() , a__U53^#(tt()) -> c_38() , a__U62^#(X1, X2) -> c_41() , a__U63^#(X) -> c_43() , a__U63^#(tt()) -> c_44() , a__U71^#(X1, X2) -> c_45() , a__length^#(X) -> c_47() , a__length^#(nil()) -> c_49() , mark^#(0()) -> c_51() , mark^#(zeros()) -> c_52(a__zeros^#()) , mark^#(tt()) -> c_53() , mark^#(nil()) -> c_55() , a__take^#(X1, X2) -> c_97() , a__U81^#(X) -> c_82() , a__U81^#(tt()) -> c_83() , a__U91^#(X1, X2, X3, X4) -> c_84() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__isNatList^#(cons(V1, V2)) -> c_8(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(take(V1, V2)) -> c_10(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(tt(), V1, V2) -> c_34(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(tt(), X) -> c_87(mark^#(X)) , a__isNatKind^#(s(V1)) -> c_95(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_96(a__isNatIListKind^#(V1)) , a__U61^#(tt(), V1, V2) -> c_40(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__isNat^#(s(V1)) -> c_17(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_18(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(cons(V1, V2)) -> c_89(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(take(V1, V2)) -> c_92(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U41^#(tt(), V1, V2) -> c_24(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__isNatIList^#(V) -> c_29(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(cons(V1, V2)) -> c_31(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U71^#(tt(), L) -> c_46(a__length^#(mark(L)), mark^#(L)) , a__length^#(cons(N, L)) -> c_48(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , mark^#(cons(X1, X2)) -> c_50(mark^#(X1)) , mark^#(s(X)) -> c_54(mark^#(X)) , mark^#(take(X1, X2)) -> c_56(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_57(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_58(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_59(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_60(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_61(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_62(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(isNatList(X)) -> c_64(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_65(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U31(X1, X2)) -> c_67(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_69(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_70(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(isNatIList(X)) -> c_72(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_73(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_74(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_76(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_77(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U71(X1, X2)) -> c_79(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_81(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_99(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U91^#(tt(), IL, M, N) -> c_85(mark^#(N)) } Weak Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { a__U11^#(tt(), V1) -> c_4(a__U12^#(a__isNatList(V1)), a__isNatList^#(V1)) , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1)), a__isNat^#(V1)) , a__U31^#(tt(), V) -> c_20(a__U32^#(a__isNatList(V)), a__isNatList^#(V)) , a__U42^#(tt(), V2) -> c_26(a__U43^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , a__U52^#(tt(), V2) -> c_36(a__U53^#(a__isNatList(V2)), a__isNatList^#(V2)) , a__U62^#(tt(), V2) -> c_42(a__U63^#(a__isNatIList(V2)), a__isNatIList^#(V2)) , mark^#(U12(X)) -> c_63(a__U12^#(mark(X)), mark^#(X)) , mark^#(U22(X)) -> c_66(a__U22^#(mark(X)), mark^#(X)) , mark^#(U32(X)) -> c_68(a__U32^#(mark(X)), mark^#(X)) , mark^#(U43(X)) -> c_71(a__U43^#(mark(X)), mark^#(X)) , mark^#(U53(X)) -> c_75(a__U53^#(mark(X)), mark^#(X)) , mark^#(U63(X)) -> c_78(a__U63^#(mark(X)), mark^#(X)) , mark^#(U81(X)) -> c_80(a__U81^#(mark(X)), mark^#(X)) , a__take^#(0(), IL) -> c_98(a__U81^#(a__and(a__isNatIList(IL), isNatIListKind(IL))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__U11^#(tt(), V1) -> c_1(a__isNatList^#(V1)) , a__isNatList^#(cons(V1, V2)) -> c_2(a__U51^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatList^#(take(V1, V2)) -> c_3(a__U61^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U51^#(tt(), V1, V2) -> c_4(a__U52^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__and^#(tt(), X) -> c_5(mark^#(X)) , a__isNatKind^#(s(V1)) -> c_6(a__isNatKind^#(V1)) , a__isNatKind^#(length(V1)) -> c_7(a__isNatIListKind^#(V1)) , a__U61^#(tt(), V1, V2) -> c_8(a__U62^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U21^#(tt(), V1) -> c_9(a__isNat^#(V1)) , a__isNat^#(s(V1)) -> c_10(a__U21^#(a__isNatKind(V1), V1), a__isNatKind^#(V1)) , a__isNat^#(length(V1)) -> c_11(a__U11^#(a__isNatIListKind(V1), V1), a__isNatIListKind^#(V1)) , a__isNatIListKind^#(cons(V1, V2)) -> c_12(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__isNatIListKind^#(take(V1, V2)) -> c_13(a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U31^#(tt(), V) -> c_14(a__isNatList^#(V)) , a__U41^#(tt(), V1, V2) -> c_15(a__U42^#(a__isNat(V1), V2), a__isNat^#(V1)) , a__U42^#(tt(), V2) -> c_16(a__isNatIList^#(V2)) , a__isNatIList^#(V) -> c_17(a__U31^#(a__isNatIListKind(V), V), a__isNatIListKind^#(V)) , a__isNatIList^#(cons(V1, V2)) -> c_18(a__U41^#(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2), a__and^#(a__isNatKind(V1), isNatIListKind(V2)), a__isNatKind^#(V1)) , a__U52^#(tt(), V2) -> c_19(a__isNatList^#(V2)) , a__U62^#(tt(), V2) -> c_20(a__isNatIList^#(V2)) , a__U71^#(tt(), L) -> c_21(a__length^#(mark(L)), mark^#(L)) , a__length^#(cons(N, L)) -> c_22(a__U71^#(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L), a__and^#(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), a__and^#(a__isNatList(L), isNatIListKind(L)), a__isNatList^#(L)) , mark^#(cons(X1, X2)) -> c_23(mark^#(X1)) , mark^#(s(X)) -> c_24(mark^#(X)) , mark^#(take(X1, X2)) -> c_25(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , mark^#(length(X)) -> c_26(a__length^#(mark(X)), mark^#(X)) , mark^#(isNatIListKind(X)) -> c_27(a__isNatIListKind^#(X)) , mark^#(and(X1, X2)) -> c_28(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(isNat(X)) -> c_29(a__isNat^#(X)) , mark^#(isNatKind(X)) -> c_30(a__isNatKind^#(X)) , mark^#(U11(X1, X2)) -> c_31(a__U11^#(mark(X1), X2), mark^#(X1)) , mark^#(U12(X)) -> c_32(mark^#(X)) , mark^#(isNatList(X)) -> c_33(a__isNatList^#(X)) , mark^#(U21(X1, X2)) -> c_34(a__U21^#(mark(X1), X2), mark^#(X1)) , mark^#(U22(X)) -> c_35(mark^#(X)) , mark^#(U31(X1, X2)) -> c_36(a__U31^#(mark(X1), X2), mark^#(X1)) , mark^#(U32(X)) -> c_37(mark^#(X)) , mark^#(U41(X1, X2, X3)) -> c_38(a__U41^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U42(X1, X2)) -> c_39(a__U42^#(mark(X1), X2), mark^#(X1)) , mark^#(U43(X)) -> c_40(mark^#(X)) , mark^#(isNatIList(X)) -> c_41(a__isNatIList^#(X)) , mark^#(U51(X1, X2, X3)) -> c_42(a__U51^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U52(X1, X2)) -> c_43(a__U52^#(mark(X1), X2), mark^#(X1)) , mark^#(U53(X)) -> c_44(mark^#(X)) , mark^#(U61(X1, X2, X3)) -> c_45(a__U61^#(mark(X1), X2, X3), mark^#(X1)) , mark^#(U62(X1, X2)) -> c_46(a__U62^#(mark(X1), X2), mark^#(X1)) , mark^#(U63(X)) -> c_47(mark^#(X)) , mark^#(U71(X1, X2)) -> c_48(a__U71^#(mark(X1), X2), mark^#(X1)) , mark^#(U81(X)) -> c_49(mark^#(X)) , mark^#(U91(X1, X2, X3, X4)) -> c_50(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1)) , a__take^#(0(), IL) -> c_51(a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__take^#(s(M), cons(N, IL)) -> c_52(a__U91^#(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N), a__and^#(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), a__and^#(a__isNatIList(IL), isNatIListKind(IL)), a__isNatIList^#(IL)) , a__U91^#(tt(), IL, M, N) -> c_53(mark^#(N)) } Weak Trs: { a__zeros() -> cons(0(), zeros()) , a__zeros() -> zeros() , a__U11(X1, X2) -> U11(X1, X2) , a__U11(tt(), V1) -> a__U12(a__isNatList(V1)) , a__U12(X) -> U12(X) , a__U12(tt()) -> tt() , a__isNatList(X) -> isNatList(X) , a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatList(nil()) -> tt() , a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__U21(X1, X2) -> U21(X1, X2) , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) , a__U22(X) -> U22(X) , a__U22(tt()) -> tt() , a__isNat(X) -> isNat(X) , a__isNat(0()) -> tt() , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) , a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) , a__U31(X1, X2) -> U31(X1, X2) , a__U31(tt(), V) -> a__U32(a__isNatList(V)) , a__U32(X) -> U32(X) , a__U32(tt()) -> tt() , a__U41(X1, X2, X3) -> U41(X1, X2, X3) , a__U41(tt(), V1, V2) -> a__U42(a__isNat(V1), V2) , a__U42(X1, X2) -> U42(X1, X2) , a__U42(tt(), V2) -> a__U43(a__isNatIList(V2)) , a__U43(X) -> U43(X) , a__U43(tt()) -> tt() , a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) , a__isNatIList(X) -> isNatIList(X) , a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) , a__isNatIList(zeros()) -> tt() , a__U51(X1, X2, X3) -> U51(X1, X2, X3) , a__U51(tt(), V1, V2) -> a__U52(a__isNat(V1), V2) , a__U52(X1, X2) -> U52(X1, X2) , a__U52(tt(), V2) -> a__U53(a__isNatList(V2)) , a__U53(X) -> U53(X) , a__U53(tt()) -> tt() , a__U61(X1, X2, X3) -> U61(X1, X2, X3) , a__U61(tt(), V1, V2) -> a__U62(a__isNat(V1), V2) , a__U62(X1, X2) -> U62(X1, X2) , a__U62(tt(), V2) -> a__U63(a__isNatIList(V2)) , a__U63(X) -> U63(X) , a__U63(tt()) -> tt() , a__U71(X1, X2) -> U71(X1, X2) , a__U71(tt(), L) -> s(a__length(mark(L))) , a__length(X) -> length(X) , a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) , a__length(nil()) -> 0() , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(0()) -> 0() , mark(zeros()) -> a__zeros() , mark(tt()) -> tt() , mark(s(X)) -> s(mark(X)) , mark(nil()) -> nil() , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) , mark(length(X)) -> a__length(mark(X)) , mark(isNatIListKind(X)) -> a__isNatIListKind(X) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(isNat(X)) -> a__isNat(X) , mark(isNatKind(X)) -> a__isNatKind(X) , mark(U11(X1, X2)) -> a__U11(mark(X1), X2) , mark(U12(X)) -> a__U12(mark(X)) , mark(isNatList(X)) -> a__isNatList(X) , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) , mark(U22(X)) -> a__U22(mark(X)) , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) , mark(U32(X)) -> a__U32(mark(X)) , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) , mark(U42(X1, X2)) -> a__U42(mark(X1), X2) , mark(U43(X)) -> a__U43(mark(X)) , mark(isNatIList(X)) -> a__isNatIList(X) , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) , mark(U52(X1, X2)) -> a__U52(mark(X1), X2) , mark(U53(X)) -> a__U53(mark(X)) , mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) , mark(U62(X1, X2)) -> a__U62(mark(X1), X2) , mark(U63(X)) -> a__U63(mark(X)) , mark(U71(X1, X2)) -> a__U71(mark(X1), X2) , mark(U81(X)) -> a__U81(mark(X)) , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) , a__U81(X) -> U81(X) , a__U81(tt()) -> nil() , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) , a__U91(tt(), IL, M, N) -> cons(mark(N), take(M, IL)) , a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , a__isNatIListKind(X) -> isNatIListKind(X) , a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatIListKind(zeros()) -> tt() , a__isNatIListKind(nil()) -> tt() , a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) , a__isNatKind(X) -> isNatKind(X) , a__isNatKind(0()) -> tt() , a__isNatKind(s(V1)) -> a__isNatKind(V1) , a__isNatKind(length(V1)) -> a__isNatIListKind(V1) , a__take(X1, X2) -> take(X1, X2) , a__take(0(), IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) , a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..