MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { active(zeros()) -> mark(cons(0(), zeros())) , active(cons(X1, X2)) -> cons(active(X1), X2) , active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), V1)) -> mark(U12(isNatList(V1))) , active(U12(X)) -> U12(active(X)) , active(U12(tt())) -> mark(tt()) , active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(isNatList(nil())) -> mark(tt()) , active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U21(X1, X2)) -> U21(active(X1), X2) , active(U21(tt(), V1)) -> mark(U22(isNat(V1))) , active(U22(X)) -> U22(active(X)) , active(U22(tt())) -> mark(tt()) , active(isNat(0())) -> mark(tt()) , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) , active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) , active(U31(X1, X2)) -> U31(active(X1), X2) , active(U31(tt(), V)) -> mark(U32(isNatList(V))) , active(U32(X)) -> U32(active(X)) , active(U32(tt())) -> mark(tt()) , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) , active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)) , active(U42(X1, X2)) -> U42(active(X1), X2) , active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))) , active(U43(X)) -> U43(active(X)) , active(U43(tt())) -> mark(tt()) , active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) , active(isNatIList(zeros())) -> mark(tt()) , active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) , active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)) , active(U52(X1, X2)) -> U52(active(X1), X2) , active(U52(tt(), V2)) -> mark(U53(isNatList(V2))) , active(U53(X)) -> U53(active(X)) , active(U53(tt())) -> mark(tt()) , active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3) , active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)) , active(U62(X1, X2)) -> U62(active(X1), X2) , active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))) , active(U63(X)) -> U63(active(X)) , active(U63(tt())) -> mark(tt()) , active(U71(X1, X2)) -> U71(active(X1), X2) , active(U71(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , active(U81(X)) -> U81(active(X)) , active(U81(tt())) -> mark(nil()) , active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) , active(U91(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))) , active(take(X1, X2)) -> take(X1, active(X2)) , active(take(X1, X2)) -> take(active(X1), X2) , active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))) , active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNatIListKind(zeros())) -> mark(tt()) , active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatIListKind(nil())) -> mark(tt()) , active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatKind(0())) -> mark(tt()) , active(isNatKind(s(V1))) -> mark(isNatKind(V1)) , active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) , cons(mark(X1), X2) -> mark(cons(X1, X2)) , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U12(mark(X)) -> mark(U12(X)) , U12(ok(X)) -> ok(U12(X)) , isNatList(ok(X)) -> ok(isNatList(X)) , U21(mark(X1), X2) -> mark(U21(X1, X2)) , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) , U22(mark(X)) -> mark(U22(X)) , U22(ok(X)) -> ok(U22(X)) , isNat(ok(X)) -> ok(isNat(X)) , U31(mark(X1), X2) -> mark(U31(X1, X2)) , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) , U32(mark(X)) -> mark(U32(X)) , U32(ok(X)) -> ok(U32(X)) , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) , U42(mark(X1), X2) -> mark(U42(X1, X2)) , U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) , U43(mark(X)) -> mark(U43(X)) , U43(ok(X)) -> ok(U43(X)) , isNatIList(ok(X)) -> ok(isNatIList(X)) , U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) , U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) , U52(mark(X1), X2) -> mark(U52(X1, X2)) , U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) , U53(mark(X)) -> mark(U53(X)) , U53(ok(X)) -> ok(U53(X)) , U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)) , U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)) , U62(mark(X1), X2) -> mark(U62(X1, X2)) , U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) , U63(mark(X)) -> mark(U63(X)) , U63(ok(X)) -> ok(U63(X)) , U71(mark(X1), X2) -> mark(U71(X1, X2)) , U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) , s(mark(X)) -> mark(s(X)) , s(ok(X)) -> ok(s(X)) , length(mark(X)) -> mark(length(X)) , length(ok(X)) -> ok(length(X)) , U81(mark(X)) -> mark(U81(X)) , U81(ok(X)) -> ok(U81(X)) , U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) , U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) , take(X1, mark(X2)) -> mark(take(X1, X2)) , take(mark(X1), X2) -> mark(take(X1, X2)) , take(ok(X1), ok(X2)) -> ok(take(X1, X2)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) , isNatKind(ok(X)) -> ok(isNatKind(X)) , proper(zeros()) -> ok(zeros()) , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) , proper(0()) -> ok(0()) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U12(X)) -> U12(proper(X)) , proper(isNatList(X)) -> isNatList(proper(X)) , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) , proper(U22(X)) -> U22(proper(X)) , proper(isNat(X)) -> isNat(proper(X)) , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) , proper(U32(X)) -> U32(proper(X)) , proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) , proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) , proper(U43(X)) -> U43(proper(X)) , proper(isNatIList(X)) -> isNatIList(proper(X)) , proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) , proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) , proper(U53(X)) -> U53(proper(X)) , proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)) , proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) , proper(U63(X)) -> U63(proper(X)) , proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(U81(X)) -> U81(proper(X)) , proper(nil()) -> ok(nil()) , proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) , proper(take(X1, X2)) -> take(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { active^#(zeros()) -> c_1(cons^#(0(), zeros())) , active^#(cons(X1, X2)) -> c_2(cons^#(active(X1), X2)) , active^#(U11(X1, X2)) -> c_3(U11^#(active(X1), X2)) , active^#(U11(tt(), V1)) -> c_4(U12^#(isNatList(V1))) , active^#(U12(X)) -> c_5(U12^#(active(X))) , active^#(U12(tt())) -> c_6() , active^#(isNatList(cons(V1, V2))) -> c_7(U51^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(isNatList(nil())) -> c_8() , active^#(isNatList(take(V1, V2))) -> c_9(U61^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) , active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) , active^#(U22(X)) -> c_12(U22^#(active(X))) , active^#(U22(tt())) -> c_13() , active^#(isNat(0())) -> c_14() , active^#(isNat(s(V1))) -> c_15(U21^#(isNatKind(V1), V1)) , active^#(isNat(length(V1))) -> c_16(U11^#(isNatIListKind(V1), V1)) , active^#(U31(X1, X2)) -> c_17(U31^#(active(X1), X2)) , active^#(U31(tt(), V)) -> c_18(U32^#(isNatList(V))) , active^#(U32(X)) -> c_19(U32^#(active(X))) , active^#(U32(tt())) -> c_20() , active^#(U41(X1, X2, X3)) -> c_21(U41^#(active(X1), X2, X3)) , active^#(U41(tt(), V1, V2)) -> c_22(U42^#(isNat(V1), V2)) , active^#(U42(X1, X2)) -> c_23(U42^#(active(X1), X2)) , active^#(U42(tt(), V2)) -> c_24(U43^#(isNatIList(V2))) , active^#(U43(X)) -> c_25(U43^#(active(X))) , active^#(U43(tt())) -> c_26() , active^#(isNatIList(V)) -> c_27(U31^#(isNatIListKind(V), V)) , active^#(isNatIList(zeros())) -> c_28() , active^#(isNatIList(cons(V1, V2))) -> c_29(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U51(X1, X2, X3)) -> c_30(U51^#(active(X1), X2, X3)) , active^#(U51(tt(), V1, V2)) -> c_31(U52^#(isNat(V1), V2)) , active^#(U52(X1, X2)) -> c_32(U52^#(active(X1), X2)) , active^#(U52(tt(), V2)) -> c_33(U53^#(isNatList(V2))) , active^#(U53(X)) -> c_34(U53^#(active(X))) , active^#(U53(tt())) -> c_35() , active^#(U61(X1, X2, X3)) -> c_36(U61^#(active(X1), X2, X3)) , active^#(U61(tt(), V1, V2)) -> c_37(U62^#(isNat(V1), V2)) , active^#(U62(X1, X2)) -> c_38(U62^#(active(X1), X2)) , active^#(U62(tt(), V2)) -> c_39(U63^#(isNatIList(V2))) , active^#(U63(X)) -> c_40(U63^#(active(X))) , active^#(U63(tt())) -> c_41() , active^#(U71(X1, X2)) -> c_42(U71^#(active(X1), X2)) , active^#(U71(tt(), L)) -> c_43(s^#(length(L))) , active^#(s(X)) -> c_44(s^#(active(X))) , active^#(length(X)) -> c_45(length^#(active(X))) , active^#(length(cons(N, L))) -> c_46(U71^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active^#(length(nil())) -> c_47() , active^#(U81(X)) -> c_48(U81^#(active(X))) , active^#(U81(tt())) -> c_49() , active^#(U91(X1, X2, X3, X4)) -> c_50(U91^#(active(X1), X2, X3, X4)) , active^#(U91(tt(), IL, M, N)) -> c_51(cons^#(N, take(M, IL))) , active^#(take(X1, X2)) -> c_52(take^#(X1, active(X2))) , active^#(take(X1, X2)) -> c_53(take^#(active(X1), X2)) , active^#(take(0(), IL)) -> c_54(U81^#(and(isNatIList(IL), isNatIListKind(IL)))) , active^#(take(s(M), cons(N, IL))) -> c_55(U91^#(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active^#(and(X1, X2)) -> c_56(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_57(X) , active^#(isNatIListKind(zeros())) -> c_58() , active^#(isNatIListKind(cons(V1, V2))) -> c_59(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatIListKind(nil())) -> c_60() , active^#(isNatIListKind(take(V1, V2))) -> c_61(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatKind(0())) -> c_62() , active^#(isNatKind(s(V1))) -> c_63(isNatKind^#(V1)) , active^#(isNatKind(length(V1))) -> c_64(isNatIListKind^#(V1)) , cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) , U12^#(mark(X)) -> c_69(U12^#(X)) , U12^#(ok(X)) -> c_70(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) , U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) , U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) , U22^#(mark(X)) -> c_74(U22^#(X)) , U22^#(ok(X)) -> c_75(U22^#(X)) , U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) , U32^#(mark(X)) -> c_79(U32^#(X)) , U32^#(ok(X)) -> c_80(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) , U43^#(mark(X)) -> c_85(U43^#(X)) , U43^#(ok(X)) -> c_86(U43^#(X)) , U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) , U53^#(mark(X)) -> c_92(U53^#(X)) , U53^#(ok(X)) -> c_93(U53^#(X)) , U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) , U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) , U63^#(mark(X)) -> c_98(U63^#(X)) , U63^#(ok(X)) -> c_99(U63^#(X)) , U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) , U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) , s^#(mark(X)) -> c_102(s^#(X)) , s^#(ok(X)) -> c_103(s^#(X)) , length^#(mark(X)) -> c_104(length^#(X)) , length^#(ok(X)) -> c_105(length^#(X)) , U81^#(mark(X)) -> c_106(U81^#(X)) , U81^#(ok(X)) -> c_107(U81^#(X)) , U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) , U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) , take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) , take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) , take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) , and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_71(isNatList^#(X)) , isNat^#(ok(X)) -> c_76(isNat^#(X)) , isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) , proper^#(zeros()) -> c_117() , proper^#(cons(X1, X2)) -> c_118(cons^#(proper(X1), proper(X2))) , proper^#(0()) -> c_119() , proper^#(U11(X1, X2)) -> c_120(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_121() , proper^#(U12(X)) -> c_122(U12^#(proper(X))) , proper^#(isNatList(X)) -> c_123(isNatList^#(proper(X))) , proper^#(U21(X1, X2)) -> c_124(U21^#(proper(X1), proper(X2))) , proper^#(U22(X)) -> c_125(U22^#(proper(X))) , proper^#(isNat(X)) -> c_126(isNat^#(proper(X))) , proper^#(U31(X1, X2)) -> c_127(U31^#(proper(X1), proper(X2))) , proper^#(U32(X)) -> c_128(U32^#(proper(X))) , proper^#(U41(X1, X2, X3)) -> c_129(U41^#(proper(X1), proper(X2), proper(X3))) , proper^#(U42(X1, X2)) -> c_130(U42^#(proper(X1), proper(X2))) , proper^#(U43(X)) -> c_131(U43^#(proper(X))) , proper^#(isNatIList(X)) -> c_132(isNatIList^#(proper(X))) , proper^#(U51(X1, X2, X3)) -> c_133(U51^#(proper(X1), proper(X2), proper(X3))) , proper^#(U52(X1, X2)) -> c_134(U52^#(proper(X1), proper(X2))) , proper^#(U53(X)) -> c_135(U53^#(proper(X))) , proper^#(U61(X1, X2, X3)) -> c_136(U61^#(proper(X1), proper(X2), proper(X3))) , proper^#(U62(X1, X2)) -> c_137(U62^#(proper(X1), proper(X2))) , proper^#(U63(X)) -> c_138(U63^#(proper(X))) , proper^#(U71(X1, X2)) -> c_139(U71^#(proper(X1), proper(X2))) , proper^#(s(X)) -> c_140(s^#(proper(X))) , proper^#(length(X)) -> c_141(length^#(proper(X))) , proper^#(U81(X)) -> c_142(U81^#(proper(X))) , proper^#(nil()) -> c_143() , proper^#(U91(X1, X2, X3, X4)) -> c_144(U91^#(proper(X1), proper(X2), proper(X3), proper(X4))) , proper^#(take(X1, X2)) -> c_145(take^#(proper(X1), proper(X2))) , proper^#(and(X1, X2)) -> c_146(and^#(proper(X1), proper(X2))) , proper^#(isNatIListKind(X)) -> c_147(isNatIListKind^#(proper(X))) , proper^#(isNatKind(X)) -> c_148(isNatKind^#(proper(X))) , top^#(mark(X)) -> c_149(top^#(proper(X))) , top^#(ok(X)) -> c_150(top^#(active(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { active^#(zeros()) -> c_1(cons^#(0(), zeros())) , active^#(cons(X1, X2)) -> c_2(cons^#(active(X1), X2)) , active^#(U11(X1, X2)) -> c_3(U11^#(active(X1), X2)) , active^#(U11(tt(), V1)) -> c_4(U12^#(isNatList(V1))) , active^#(U12(X)) -> c_5(U12^#(active(X))) , active^#(U12(tt())) -> c_6() , active^#(isNatList(cons(V1, V2))) -> c_7(U51^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(isNatList(nil())) -> c_8() , active^#(isNatList(take(V1, V2))) -> c_9(U61^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) , active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) , active^#(U22(X)) -> c_12(U22^#(active(X))) , active^#(U22(tt())) -> c_13() , active^#(isNat(0())) -> c_14() , active^#(isNat(s(V1))) -> c_15(U21^#(isNatKind(V1), V1)) , active^#(isNat(length(V1))) -> c_16(U11^#(isNatIListKind(V1), V1)) , active^#(U31(X1, X2)) -> c_17(U31^#(active(X1), X2)) , active^#(U31(tt(), V)) -> c_18(U32^#(isNatList(V))) , active^#(U32(X)) -> c_19(U32^#(active(X))) , active^#(U32(tt())) -> c_20() , active^#(U41(X1, X2, X3)) -> c_21(U41^#(active(X1), X2, X3)) , active^#(U41(tt(), V1, V2)) -> c_22(U42^#(isNat(V1), V2)) , active^#(U42(X1, X2)) -> c_23(U42^#(active(X1), X2)) , active^#(U42(tt(), V2)) -> c_24(U43^#(isNatIList(V2))) , active^#(U43(X)) -> c_25(U43^#(active(X))) , active^#(U43(tt())) -> c_26() , active^#(isNatIList(V)) -> c_27(U31^#(isNatIListKind(V), V)) , active^#(isNatIList(zeros())) -> c_28() , active^#(isNatIList(cons(V1, V2))) -> c_29(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U51(X1, X2, X3)) -> c_30(U51^#(active(X1), X2, X3)) , active^#(U51(tt(), V1, V2)) -> c_31(U52^#(isNat(V1), V2)) , active^#(U52(X1, X2)) -> c_32(U52^#(active(X1), X2)) , active^#(U52(tt(), V2)) -> c_33(U53^#(isNatList(V2))) , active^#(U53(X)) -> c_34(U53^#(active(X))) , active^#(U53(tt())) -> c_35() , active^#(U61(X1, X2, X3)) -> c_36(U61^#(active(X1), X2, X3)) , active^#(U61(tt(), V1, V2)) -> c_37(U62^#(isNat(V1), V2)) , active^#(U62(X1, X2)) -> c_38(U62^#(active(X1), X2)) , active^#(U62(tt(), V2)) -> c_39(U63^#(isNatIList(V2))) , active^#(U63(X)) -> c_40(U63^#(active(X))) , active^#(U63(tt())) -> c_41() , active^#(U71(X1, X2)) -> c_42(U71^#(active(X1), X2)) , active^#(U71(tt(), L)) -> c_43(s^#(length(L))) , active^#(s(X)) -> c_44(s^#(active(X))) , active^#(length(X)) -> c_45(length^#(active(X))) , active^#(length(cons(N, L))) -> c_46(U71^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active^#(length(nil())) -> c_47() , active^#(U81(X)) -> c_48(U81^#(active(X))) , active^#(U81(tt())) -> c_49() , active^#(U91(X1, X2, X3, X4)) -> c_50(U91^#(active(X1), X2, X3, X4)) , active^#(U91(tt(), IL, M, N)) -> c_51(cons^#(N, take(M, IL))) , active^#(take(X1, X2)) -> c_52(take^#(X1, active(X2))) , active^#(take(X1, X2)) -> c_53(take^#(active(X1), X2)) , active^#(take(0(), IL)) -> c_54(U81^#(and(isNatIList(IL), isNatIListKind(IL)))) , active^#(take(s(M), cons(N, IL))) -> c_55(U91^#(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active^#(and(X1, X2)) -> c_56(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_57(X) , active^#(isNatIListKind(zeros())) -> c_58() , active^#(isNatIListKind(cons(V1, V2))) -> c_59(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatIListKind(nil())) -> c_60() , active^#(isNatIListKind(take(V1, V2))) -> c_61(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatKind(0())) -> c_62() , active^#(isNatKind(s(V1))) -> c_63(isNatKind^#(V1)) , active^#(isNatKind(length(V1))) -> c_64(isNatIListKind^#(V1)) , cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) , U12^#(mark(X)) -> c_69(U12^#(X)) , U12^#(ok(X)) -> c_70(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) , U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) , U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) , U22^#(mark(X)) -> c_74(U22^#(X)) , U22^#(ok(X)) -> c_75(U22^#(X)) , U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) , U32^#(mark(X)) -> c_79(U32^#(X)) , U32^#(ok(X)) -> c_80(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) , U43^#(mark(X)) -> c_85(U43^#(X)) , U43^#(ok(X)) -> c_86(U43^#(X)) , U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) , U53^#(mark(X)) -> c_92(U53^#(X)) , U53^#(ok(X)) -> c_93(U53^#(X)) , U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) , U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) , U63^#(mark(X)) -> c_98(U63^#(X)) , U63^#(ok(X)) -> c_99(U63^#(X)) , U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) , U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) , s^#(mark(X)) -> c_102(s^#(X)) , s^#(ok(X)) -> c_103(s^#(X)) , length^#(mark(X)) -> c_104(length^#(X)) , length^#(ok(X)) -> c_105(length^#(X)) , U81^#(mark(X)) -> c_106(U81^#(X)) , U81^#(ok(X)) -> c_107(U81^#(X)) , U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) , U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) , take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) , take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) , take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) , and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_71(isNatList^#(X)) , isNat^#(ok(X)) -> c_76(isNat^#(X)) , isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) , proper^#(zeros()) -> c_117() , proper^#(cons(X1, X2)) -> c_118(cons^#(proper(X1), proper(X2))) , proper^#(0()) -> c_119() , proper^#(U11(X1, X2)) -> c_120(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_121() , proper^#(U12(X)) -> c_122(U12^#(proper(X))) , proper^#(isNatList(X)) -> c_123(isNatList^#(proper(X))) , proper^#(U21(X1, X2)) -> c_124(U21^#(proper(X1), proper(X2))) , proper^#(U22(X)) -> c_125(U22^#(proper(X))) , proper^#(isNat(X)) -> c_126(isNat^#(proper(X))) , proper^#(U31(X1, X2)) -> c_127(U31^#(proper(X1), proper(X2))) , proper^#(U32(X)) -> c_128(U32^#(proper(X))) , proper^#(U41(X1, X2, X3)) -> c_129(U41^#(proper(X1), proper(X2), proper(X3))) , proper^#(U42(X1, X2)) -> c_130(U42^#(proper(X1), proper(X2))) , proper^#(U43(X)) -> c_131(U43^#(proper(X))) , proper^#(isNatIList(X)) -> c_132(isNatIList^#(proper(X))) , proper^#(U51(X1, X2, X3)) -> c_133(U51^#(proper(X1), proper(X2), proper(X3))) , proper^#(U52(X1, X2)) -> c_134(U52^#(proper(X1), proper(X2))) , proper^#(U53(X)) -> c_135(U53^#(proper(X))) , proper^#(U61(X1, X2, X3)) -> c_136(U61^#(proper(X1), proper(X2), proper(X3))) , proper^#(U62(X1, X2)) -> c_137(U62^#(proper(X1), proper(X2))) , proper^#(U63(X)) -> c_138(U63^#(proper(X))) , proper^#(U71(X1, X2)) -> c_139(U71^#(proper(X1), proper(X2))) , proper^#(s(X)) -> c_140(s^#(proper(X))) , proper^#(length(X)) -> c_141(length^#(proper(X))) , proper^#(U81(X)) -> c_142(U81^#(proper(X))) , proper^#(nil()) -> c_143() , proper^#(U91(X1, X2, X3, X4)) -> c_144(U91^#(proper(X1), proper(X2), proper(X3), proper(X4))) , proper^#(take(X1, X2)) -> c_145(take^#(proper(X1), proper(X2))) , proper^#(and(X1, X2)) -> c_146(and^#(proper(X1), proper(X2))) , proper^#(isNatIListKind(X)) -> c_147(isNatIListKind^#(proper(X))) , proper^#(isNatKind(X)) -> c_148(isNatKind^#(proper(X))) , top^#(mark(X)) -> c_149(top^#(proper(X))) , top^#(ok(X)) -> c_150(top^#(active(X))) } Strict Trs: { active(zeros()) -> mark(cons(0(), zeros())) , active(cons(X1, X2)) -> cons(active(X1), X2) , active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), V1)) -> mark(U12(isNatList(V1))) , active(U12(X)) -> U12(active(X)) , active(U12(tt())) -> mark(tt()) , active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(isNatList(nil())) -> mark(tt()) , active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U21(X1, X2)) -> U21(active(X1), X2) , active(U21(tt(), V1)) -> mark(U22(isNat(V1))) , active(U22(X)) -> U22(active(X)) , active(U22(tt())) -> mark(tt()) , active(isNat(0())) -> mark(tt()) , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) , active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) , active(U31(X1, X2)) -> U31(active(X1), X2) , active(U31(tt(), V)) -> mark(U32(isNatList(V))) , active(U32(X)) -> U32(active(X)) , active(U32(tt())) -> mark(tt()) , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) , active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)) , active(U42(X1, X2)) -> U42(active(X1), X2) , active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))) , active(U43(X)) -> U43(active(X)) , active(U43(tt())) -> mark(tt()) , active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) , active(isNatIList(zeros())) -> mark(tt()) , active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) , active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)) , active(U52(X1, X2)) -> U52(active(X1), X2) , active(U52(tt(), V2)) -> mark(U53(isNatList(V2))) , active(U53(X)) -> U53(active(X)) , active(U53(tt())) -> mark(tt()) , active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3) , active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)) , active(U62(X1, X2)) -> U62(active(X1), X2) , active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))) , active(U63(X)) -> U63(active(X)) , active(U63(tt())) -> mark(tt()) , active(U71(X1, X2)) -> U71(active(X1), X2) , active(U71(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , active(U81(X)) -> U81(active(X)) , active(U81(tt())) -> mark(nil()) , active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) , active(U91(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))) , active(take(X1, X2)) -> take(X1, active(X2)) , active(take(X1, X2)) -> take(active(X1), X2) , active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))) , active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNatIListKind(zeros())) -> mark(tt()) , active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatIListKind(nil())) -> mark(tt()) , active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatKind(0())) -> mark(tt()) , active(isNatKind(s(V1))) -> mark(isNatKind(V1)) , active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) , cons(mark(X1), X2) -> mark(cons(X1, X2)) , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U12(mark(X)) -> mark(U12(X)) , U12(ok(X)) -> ok(U12(X)) , isNatList(ok(X)) -> ok(isNatList(X)) , U21(mark(X1), X2) -> mark(U21(X1, X2)) , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) , U22(mark(X)) -> mark(U22(X)) , U22(ok(X)) -> ok(U22(X)) , isNat(ok(X)) -> ok(isNat(X)) , U31(mark(X1), X2) -> mark(U31(X1, X2)) , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) , U32(mark(X)) -> mark(U32(X)) , U32(ok(X)) -> ok(U32(X)) , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) , U42(mark(X1), X2) -> mark(U42(X1, X2)) , U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) , U43(mark(X)) -> mark(U43(X)) , U43(ok(X)) -> ok(U43(X)) , isNatIList(ok(X)) -> ok(isNatIList(X)) , U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) , U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) , U52(mark(X1), X2) -> mark(U52(X1, X2)) , U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) , U53(mark(X)) -> mark(U53(X)) , U53(ok(X)) -> ok(U53(X)) , U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)) , U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)) , U62(mark(X1), X2) -> mark(U62(X1, X2)) , U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) , U63(mark(X)) -> mark(U63(X)) , U63(ok(X)) -> ok(U63(X)) , U71(mark(X1), X2) -> mark(U71(X1, X2)) , U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) , s(mark(X)) -> mark(s(X)) , s(ok(X)) -> ok(s(X)) , length(mark(X)) -> mark(length(X)) , length(ok(X)) -> ok(length(X)) , U81(mark(X)) -> mark(U81(X)) , U81(ok(X)) -> ok(U81(X)) , U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) , U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) , take(X1, mark(X2)) -> mark(take(X1, X2)) , take(mark(X1), X2) -> mark(take(X1, X2)) , take(ok(X1), ok(X2)) -> ok(take(X1, X2)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) , isNatKind(ok(X)) -> ok(isNatKind(X)) , proper(zeros()) -> ok(zeros()) , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) , proper(0()) -> ok(0()) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U12(X)) -> U12(proper(X)) , proper(isNatList(X)) -> isNatList(proper(X)) , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) , proper(U22(X)) -> U22(proper(X)) , proper(isNat(X)) -> isNat(proper(X)) , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) , proper(U32(X)) -> U32(proper(X)) , proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) , proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) , proper(U43(X)) -> U43(proper(X)) , proper(isNatIList(X)) -> isNatIList(proper(X)) , proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) , proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) , proper(U53(X)) -> U53(proper(X)) , proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)) , proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) , proper(U63(X)) -> U63(proper(X)) , proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(U81(X)) -> U81(proper(X)) , proper(nil()) -> ok(nil()) , proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) , proper(take(X1, X2)) -> take(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE Consider the dependency graph: 1: active^#(zeros()) -> c_1(cons^#(0(), zeros())) 2: active^#(cons(X1, X2)) -> c_2(cons^#(active(X1), X2)) -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 3: active^#(U11(X1, X2)) -> c_3(U11^#(active(X1), X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 4: active^#(U11(tt(), V1)) -> c_4(U12^#(isNatList(V1))) -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 5: active^#(U12(X)) -> c_5(U12^#(active(X))) -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 6: active^#(U12(tt())) -> c_6() 7: active^#(isNatList(cons(V1, V2))) -> c_7(U51^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 8: active^#(isNatList(nil())) -> c_8() 9: active^#(isNatList(take(V1, V2))) -> c_9(U61^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 10: active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 11: active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 12: active^#(U22(X)) -> c_12(U22^#(active(X))) -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 13: active^#(U22(tt())) -> c_13() 14: active^#(isNat(0())) -> c_14() 15: active^#(isNat(s(V1))) -> c_15(U21^#(isNatKind(V1), V1)) -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 16: active^#(isNat(length(V1))) -> c_16(U11^#(isNatIListKind(V1), V1)) -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 17: active^#(U31(X1, X2)) -> c_17(U31^#(active(X1), X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 18: active^#(U31(tt(), V)) -> c_18(U32^#(isNatList(V))) -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 19: active^#(U32(X)) -> c_19(U32^#(active(X))) -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 20: active^#(U32(tt())) -> c_20() 21: active^#(U41(X1, X2, X3)) -> c_21(U41^#(active(X1), X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 22: active^#(U41(tt(), V1, V2)) -> c_22(U42^#(isNat(V1), V2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 23: active^#(U42(X1, X2)) -> c_23(U42^#(active(X1), X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 24: active^#(U42(tt(), V2)) -> c_24(U43^#(isNatIList(V2))) -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 25: active^#(U43(X)) -> c_25(U43^#(active(X))) -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 26: active^#(U43(tt())) -> c_26() 27: active^#(isNatIList(V)) -> c_27(U31^#(isNatIListKind(V), V)) -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 28: active^#(isNatIList(zeros())) -> c_28() 29: active^#(isNatIList(cons(V1, V2))) -> c_29(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 30: active^#(U51(X1, X2, X3)) -> c_30(U51^#(active(X1), X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 31: active^#(U51(tt(), V1, V2)) -> c_31(U52^#(isNat(V1), V2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 32: active^#(U52(X1, X2)) -> c_32(U52^#(active(X1), X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 33: active^#(U52(tt(), V2)) -> c_33(U53^#(isNatList(V2))) -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 34: active^#(U53(X)) -> c_34(U53^#(active(X))) -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 35: active^#(U53(tt())) -> c_35() 36: active^#(U61(X1, X2, X3)) -> c_36(U61^#(active(X1), X2, X3)) -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 37: active^#(U61(tt(), V1, V2)) -> c_37(U62^#(isNat(V1), V2)) -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 38: active^#(U62(X1, X2)) -> c_38(U62^#(active(X1), X2)) -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 39: active^#(U62(tt(), V2)) -> c_39(U63^#(isNatIList(V2))) -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 40: active^#(U63(X)) -> c_40(U63^#(active(X))) -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 41: active^#(U63(tt())) -> c_41() 42: active^#(U71(X1, X2)) -> c_42(U71^#(active(X1), X2)) -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 43: active^#(U71(tt(), L)) -> c_43(s^#(length(L))) -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 44: active^#(s(X)) -> c_44(s^#(active(X))) -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 45: active^#(length(X)) -> c_45(length^#(active(X))) -->_1 length^#(ok(X)) -> c_105(length^#(X)) :102 -->_1 length^#(mark(X)) -> c_104(length^#(X)) :101 46: active^#(length(cons(N, L))) -> c_46(U71^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 47: active^#(length(nil())) -> c_47() 48: active^#(U81(X)) -> c_48(U81^#(active(X))) -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 49: active^#(U81(tt())) -> c_49() 50: active^#(U91(X1, X2, X3, X4)) -> c_50(U91^#(active(X1), X2, X3, X4)) -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 51: active^#(U91(tt(), IL, M, N)) -> c_51(cons^#(N, take(M, IL))) -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 52: active^#(take(X1, X2)) -> c_52(take^#(X1, active(X2))) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 53: active^#(take(X1, X2)) -> c_53(take^#(active(X1), X2)) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 54: active^#(take(0(), IL)) -> c_54(U81^#(and(isNatIList(IL), isNatIListKind(IL)))) -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 55: active^#(take(s(M), cons(N, IL))) -> c_55(U91^#(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 56: active^#(and(X1, X2)) -> c_56(and^#(active(X1), X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 57: active^#(and(tt(), X)) -> c_57(X) -->_1 top^#(ok(X)) -> c_150(top^#(active(X))) :150 -->_1 top^#(mark(X)) -> c_149(top^#(proper(X))) :149 -->_1 proper^#(isNatKind(X)) -> c_148(isNatKind^#(proper(X))) :148 -->_1 proper^#(isNatIListKind(X)) -> c_147(isNatIListKind^#(proper(X))) :147 -->_1 proper^#(and(X1, X2)) -> c_146(and^#(proper(X1), proper(X2))) :146 -->_1 proper^#(take(X1, X2)) -> c_145(take^#(proper(X1), proper(X2))) :145 -->_1 proper^#(U91(X1, X2, X3, X4)) -> c_144(U91^#(proper(X1), proper(X2), proper(X3), proper(X4))) :144 -->_1 proper^#(U81(X)) -> c_142(U81^#(proper(X))) :142 -->_1 proper^#(length(X)) -> c_141(length^#(proper(X))) :141 -->_1 proper^#(s(X)) -> c_140(s^#(proper(X))) :140 -->_1 proper^#(U71(X1, X2)) -> c_139(U71^#(proper(X1), proper(X2))) :139 -->_1 proper^#(U63(X)) -> c_138(U63^#(proper(X))) :138 -->_1 proper^#(U62(X1, X2)) -> c_137(U62^#(proper(X1), proper(X2))) :137 -->_1 proper^#(U61(X1, X2, X3)) -> c_136(U61^#(proper(X1), proper(X2), proper(X3))) :136 -->_1 proper^#(U53(X)) -> c_135(U53^#(proper(X))) :135 -->_1 proper^#(U52(X1, X2)) -> c_134(U52^#(proper(X1), proper(X2))) :134 -->_1 proper^#(U51(X1, X2, X3)) -> c_133(U51^#(proper(X1), proper(X2), proper(X3))) :133 -->_1 proper^#(isNatIList(X)) -> c_132(isNatIList^#(proper(X))) :132 -->_1 proper^#(U43(X)) -> c_131(U43^#(proper(X))) :131 -->_1 proper^#(U42(X1, X2)) -> c_130(U42^#(proper(X1), proper(X2))) :130 -->_1 proper^#(U41(X1, X2, X3)) -> c_129(U41^#(proper(X1), proper(X2), proper(X3))) :129 -->_1 proper^#(U32(X)) -> c_128(U32^#(proper(X))) :128 -->_1 proper^#(U31(X1, X2)) -> c_127(U31^#(proper(X1), proper(X2))) :127 -->_1 proper^#(isNat(X)) -> c_126(isNat^#(proper(X))) :126 -->_1 proper^#(U22(X)) -> c_125(U22^#(proper(X))) :125 -->_1 proper^#(U21(X1, X2)) -> c_124(U21^#(proper(X1), proper(X2))) :124 -->_1 proper^#(isNatList(X)) -> c_123(isNatList^#(proper(X))) :123 -->_1 proper^#(U12(X)) -> c_122(U12^#(proper(X))) :122 -->_1 proper^#(U11(X1, X2)) -> c_120(U11^#(proper(X1), proper(X2))) :120 -->_1 proper^#(cons(X1, X2)) -> c_118(cons^#(proper(X1), proper(X2))) :118 -->_1 isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) :116 -->_1 isNat^#(ok(X)) -> c_76(isNat^#(X)) :115 -->_1 isNatList^#(ok(X)) -> c_71(isNatList^#(X)) :114 -->_1 isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) :113 -->_1 isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) :112 -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 -->_1 length^#(ok(X)) -> c_105(length^#(X)) :102 -->_1 length^#(mark(X)) -> c_104(length^#(X)) :101 -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 -->_1 active^#(isNatKind(length(V1))) -> c_64(isNatIListKind^#(V1)) :64 -->_1 active^#(isNatKind(s(V1))) -> c_63(isNatKind^#(V1)) :63 -->_1 active^#(isNatIListKind(take(V1, V2))) -> c_61(and^#(isNatKind(V1), isNatIListKind(V2))) :61 -->_1 active^#(isNatIListKind(cons(V1, V2))) -> c_59(and^#(isNatKind(V1), isNatIListKind(V2))) :59 -->_1 proper^#(nil()) -> c_143() :143 -->_1 proper^#(tt()) -> c_121() :121 -->_1 proper^#(0()) -> c_119() :119 -->_1 proper^#(zeros()) -> c_117() :117 -->_1 active^#(isNatKind(0())) -> c_62() :62 -->_1 active^#(isNatIListKind(nil())) -> c_60() :60 -->_1 active^#(isNatIListKind(zeros())) -> c_58() :58 -->_1 active^#(and(tt(), X)) -> c_57(X) :57 -->_1 active^#(and(X1, X2)) -> c_56(and^#(active(X1), X2)) :56 -->_1 active^#(take(s(M), cons(N, IL))) -> c_55(U91^#(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) :55 -->_1 active^#(take(0(), IL)) -> c_54(U81^#(and(isNatIList(IL), isNatIListKind(IL)))) :54 -->_1 active^#(take(X1, X2)) -> c_53(take^#(active(X1), X2)) :53 -->_1 active^#(take(X1, X2)) -> c_52(take^#(X1, active(X2))) :52 -->_1 active^#(U91(tt(), IL, M, N)) -> c_51(cons^#(N, take(M, IL))) :51 -->_1 active^#(U91(X1, X2, X3, X4)) -> c_50(U91^#(active(X1), X2, X3, X4)) :50 -->_1 active^#(U81(tt())) -> c_49() :49 -->_1 active^#(U81(X)) -> c_48(U81^#(active(X))) :48 -->_1 active^#(length(nil())) -> c_47() :47 -->_1 active^#(length(cons(N, L))) -> c_46(U71^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) :46 -->_1 active^#(length(X)) -> c_45(length^#(active(X))) :45 -->_1 active^#(s(X)) -> c_44(s^#(active(X))) :44 -->_1 active^#(U71(tt(), L)) -> c_43(s^#(length(L))) :43 -->_1 active^#(U71(X1, X2)) -> c_42(U71^#(active(X1), X2)) :42 -->_1 active^#(U63(tt())) -> c_41() :41 -->_1 active^#(U63(X)) -> c_40(U63^#(active(X))) :40 -->_1 active^#(U62(tt(), V2)) -> c_39(U63^#(isNatIList(V2))) :39 -->_1 active^#(U62(X1, X2)) -> c_38(U62^#(active(X1), X2)) :38 -->_1 active^#(U61(tt(), V1, V2)) -> c_37(U62^#(isNat(V1), V2)) :37 -->_1 active^#(U61(X1, X2, X3)) -> c_36(U61^#(active(X1), X2, X3)) :36 -->_1 active^#(U53(tt())) -> c_35() :35 -->_1 active^#(U53(X)) -> c_34(U53^#(active(X))) :34 -->_1 active^#(U52(tt(), V2)) -> c_33(U53^#(isNatList(V2))) :33 -->_1 active^#(U52(X1, X2)) -> c_32(U52^#(active(X1), X2)) :32 -->_1 active^#(U51(tt(), V1, V2)) -> c_31(U52^#(isNat(V1), V2)) :31 -->_1 active^#(U51(X1, X2, X3)) -> c_30(U51^#(active(X1), X2, X3)) :30 -->_1 active^#(isNatIList(cons(V1, V2))) -> c_29(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) :29 -->_1 active^#(isNatIList(zeros())) -> c_28() :28 -->_1 active^#(isNatIList(V)) -> c_27(U31^#(isNatIListKind(V), V)) :27 -->_1 active^#(U43(tt())) -> c_26() :26 -->_1 active^#(U43(X)) -> c_25(U43^#(active(X))) :25 -->_1 active^#(U42(tt(), V2)) -> c_24(U43^#(isNatIList(V2))) :24 -->_1 active^#(U42(X1, X2)) -> c_23(U42^#(active(X1), X2)) :23 -->_1 active^#(U41(tt(), V1, V2)) -> c_22(U42^#(isNat(V1), V2)) :22 -->_1 active^#(U41(X1, X2, X3)) -> c_21(U41^#(active(X1), X2, X3)) :21 -->_1 active^#(U32(tt())) -> c_20() :20 -->_1 active^#(U32(X)) -> c_19(U32^#(active(X))) :19 -->_1 active^#(U31(tt(), V)) -> c_18(U32^#(isNatList(V))) :18 -->_1 active^#(U31(X1, X2)) -> c_17(U31^#(active(X1), X2)) :17 -->_1 active^#(isNat(length(V1))) -> c_16(U11^#(isNatIListKind(V1), V1)) :16 -->_1 active^#(isNat(s(V1))) -> c_15(U21^#(isNatKind(V1), V1)) :15 -->_1 active^#(isNat(0())) -> c_14() :14 -->_1 active^#(U22(tt())) -> c_13() :13 -->_1 active^#(U22(X)) -> c_12(U22^#(active(X))) :12 -->_1 active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) :11 -->_1 active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) :10 -->_1 active^#(isNatList(take(V1, V2))) -> c_9(U61^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) :9 -->_1 active^#(isNatList(nil())) -> c_8() :8 -->_1 active^#(isNatList(cons(V1, V2))) -> c_7(U51^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) :7 -->_1 active^#(U12(tt())) -> c_6() :6 -->_1 active^#(U12(X)) -> c_5(U12^#(active(X))) :5 -->_1 active^#(U11(tt(), V1)) -> c_4(U12^#(isNatList(V1))) :4 -->_1 active^#(U11(X1, X2)) -> c_3(U11^#(active(X1), X2)) :3 -->_1 active^#(cons(X1, X2)) -> c_2(cons^#(active(X1), X2)) :2 -->_1 active^#(zeros()) -> c_1(cons^#(0(), zeros())) :1 58: active^#(isNatIListKind(zeros())) -> c_58() 59: active^#(isNatIListKind(cons(V1, V2))) -> c_59(and^#(isNatKind(V1), isNatIListKind(V2))) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 60: active^#(isNatIListKind(nil())) -> c_60() 61: active^#(isNatIListKind(take(V1, V2))) -> c_61(and^#(isNatKind(V1), isNatIListKind(V2))) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 62: active^#(isNatKind(0())) -> c_62() 63: active^#(isNatKind(s(V1))) -> c_63(isNatKind^#(V1)) -->_1 isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) :112 64: active^#(isNatKind(length(V1))) -> c_64(isNatIListKind^#(V1)) -->_1 isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) :113 65: cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 66: cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 67: U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 68: U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 69: U12^#(mark(X)) -> c_69(U12^#(X)) -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 70: U12^#(ok(X)) -> c_70(U12^#(X)) -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 71: U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 72: U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 73: U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 74: U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 75: U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 76: U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 77: U22^#(mark(X)) -> c_74(U22^#(X)) -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 78: U22^#(ok(X)) -> c_75(U22^#(X)) -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 79: U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 80: U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 81: U32^#(mark(X)) -> c_79(U32^#(X)) -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 82: U32^#(ok(X)) -> c_80(U32^#(X)) -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 83: U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 84: U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 85: U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 86: U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 87: U43^#(mark(X)) -> c_85(U43^#(X)) -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 88: U43^#(ok(X)) -> c_86(U43^#(X)) -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 89: U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 90: U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 91: U53^#(mark(X)) -> c_92(U53^#(X)) -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 92: U53^#(ok(X)) -> c_93(U53^#(X)) -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 93: U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 94: U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 95: U63^#(mark(X)) -> c_98(U63^#(X)) -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 96: U63^#(ok(X)) -> c_99(U63^#(X)) -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 97: U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 98: U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 99: s^#(mark(X)) -> c_102(s^#(X)) -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 100: s^#(ok(X)) -> c_103(s^#(X)) -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 101: length^#(mark(X)) -> c_104(length^#(X)) -->_1 length^#(ok(X)) -> c_105(length^#(X)) :102 -->_1 length^#(mark(X)) -> c_104(length^#(X)) :101 102: length^#(ok(X)) -> c_105(length^#(X)) -->_1 length^#(ok(X)) -> c_105(length^#(X)) :102 -->_1 length^#(mark(X)) -> c_104(length^#(X)) :101 103: U81^#(mark(X)) -> c_106(U81^#(X)) -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 104: U81^#(ok(X)) -> c_107(U81^#(X)) -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 105: U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 106: U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 107: take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 108: take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 109: take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 110: and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 111: and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 112: isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) -->_1 isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) :112 113: isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) -->_1 isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) :113 114: isNatList^#(ok(X)) -> c_71(isNatList^#(X)) -->_1 isNatList^#(ok(X)) -> c_71(isNatList^#(X)) :114 115: isNat^#(ok(X)) -> c_76(isNat^#(X)) -->_1 isNat^#(ok(X)) -> c_76(isNat^#(X)) :115 116: isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) -->_1 isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) :116 117: proper^#(zeros()) -> c_117() 118: proper^#(cons(X1, X2)) -> c_118(cons^#(proper(X1), proper(X2))) -->_1 cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) :66 -->_1 cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) :65 119: proper^#(0()) -> c_119() 120: proper^#(U11(X1, X2)) -> c_120(U11^#(proper(X1), proper(X2))) -->_1 U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) :68 -->_1 U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) :67 121: proper^#(tt()) -> c_121() 122: proper^#(U12(X)) -> c_122(U12^#(proper(X))) -->_1 U12^#(ok(X)) -> c_70(U12^#(X)) :70 -->_1 U12^#(mark(X)) -> c_69(U12^#(X)) :69 123: proper^#(isNatList(X)) -> c_123(isNatList^#(proper(X))) -->_1 isNatList^#(ok(X)) -> c_71(isNatList^#(X)) :114 124: proper^#(U21(X1, X2)) -> c_124(U21^#(proper(X1), proper(X2))) -->_1 U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) :76 -->_1 U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) :75 125: proper^#(U22(X)) -> c_125(U22^#(proper(X))) -->_1 U22^#(ok(X)) -> c_75(U22^#(X)) :78 -->_1 U22^#(mark(X)) -> c_74(U22^#(X)) :77 126: proper^#(isNat(X)) -> c_126(isNat^#(proper(X))) -->_1 isNat^#(ok(X)) -> c_76(isNat^#(X)) :115 127: proper^#(U31(X1, X2)) -> c_127(U31^#(proper(X1), proper(X2))) -->_1 U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) :80 -->_1 U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) :79 128: proper^#(U32(X)) -> c_128(U32^#(proper(X))) -->_1 U32^#(ok(X)) -> c_80(U32^#(X)) :82 -->_1 U32^#(mark(X)) -> c_79(U32^#(X)) :81 129: proper^#(U41(X1, X2, X3)) -> c_129(U41^#(proper(X1), proper(X2), proper(X3))) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) :84 -->_1 U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) :83 130: proper^#(U42(X1, X2)) -> c_130(U42^#(proper(X1), proper(X2))) -->_1 U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) :86 -->_1 U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) :85 131: proper^#(U43(X)) -> c_131(U43^#(proper(X))) -->_1 U43^#(ok(X)) -> c_86(U43^#(X)) :88 -->_1 U43^#(mark(X)) -> c_85(U43^#(X)) :87 132: proper^#(isNatIList(X)) -> c_132(isNatIList^#(proper(X))) -->_1 isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) :116 133: proper^#(U51(X1, X2, X3)) -> c_133(U51^#(proper(X1), proper(X2), proper(X3))) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) :72 -->_1 U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) :71 134: proper^#(U52(X1, X2)) -> c_134(U52^#(proper(X1), proper(X2))) -->_1 U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) :90 -->_1 U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) :89 135: proper^#(U53(X)) -> c_135(U53^#(proper(X))) -->_1 U53^#(ok(X)) -> c_93(U53^#(X)) :92 -->_1 U53^#(mark(X)) -> c_92(U53^#(X)) :91 136: proper^#(U61(X1, X2, X3)) -> c_136(U61^#(proper(X1), proper(X2), proper(X3))) -->_1 U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) :74 -->_1 U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) :73 137: proper^#(U62(X1, X2)) -> c_137(U62^#(proper(X1), proper(X2))) -->_1 U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) :94 -->_1 U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) :93 138: proper^#(U63(X)) -> c_138(U63^#(proper(X))) -->_1 U63^#(ok(X)) -> c_99(U63^#(X)) :96 -->_1 U63^#(mark(X)) -> c_98(U63^#(X)) :95 139: proper^#(U71(X1, X2)) -> c_139(U71^#(proper(X1), proper(X2))) -->_1 U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) :98 -->_1 U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) :97 140: proper^#(s(X)) -> c_140(s^#(proper(X))) -->_1 s^#(ok(X)) -> c_103(s^#(X)) :100 -->_1 s^#(mark(X)) -> c_102(s^#(X)) :99 141: proper^#(length(X)) -> c_141(length^#(proper(X))) -->_1 length^#(ok(X)) -> c_105(length^#(X)) :102 -->_1 length^#(mark(X)) -> c_104(length^#(X)) :101 142: proper^#(U81(X)) -> c_142(U81^#(proper(X))) -->_1 U81^#(ok(X)) -> c_107(U81^#(X)) :104 -->_1 U81^#(mark(X)) -> c_106(U81^#(X)) :103 143: proper^#(nil()) -> c_143() 144: proper^#(U91(X1, X2, X3, X4)) -> c_144(U91^#(proper(X1), proper(X2), proper(X3), proper(X4))) -->_1 U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) :106 -->_1 U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) :105 145: proper^#(take(X1, X2)) -> c_145(take^#(proper(X1), proper(X2))) -->_1 take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) :109 -->_1 take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) :108 -->_1 take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) :107 146: proper^#(and(X1, X2)) -> c_146(and^#(proper(X1), proper(X2))) -->_1 and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) :111 -->_1 and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) :110 147: proper^#(isNatIListKind(X)) -> c_147(isNatIListKind^#(proper(X))) -->_1 isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) :113 148: proper^#(isNatKind(X)) -> c_148(isNatKind^#(proper(X))) -->_1 isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) :112 149: top^#(mark(X)) -> c_149(top^#(proper(X))) -->_1 top^#(ok(X)) -> c_150(top^#(active(X))) :150 -->_1 top^#(mark(X)) -> c_149(top^#(proper(X))) :149 150: top^#(ok(X)) -> c_150(top^#(active(X))) -->_1 top^#(ok(X)) -> c_150(top^#(active(X))) :150 -->_1 top^#(mark(X)) -> c_149(top^#(proper(X))) :149 Only the nodes {1,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,109,108,110,111,112,113,114,115,116,117,119,121,143,149,150} are reachable from nodes {1,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,119,121,143,149,150} that start derivation from marked basic terms. The nodes not reachable are removed from the problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { active^#(zeros()) -> c_1(cons^#(0(), zeros())) , cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) , U12^#(mark(X)) -> c_69(U12^#(X)) , U12^#(ok(X)) -> c_70(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) , U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) , U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) , U22^#(mark(X)) -> c_74(U22^#(X)) , U22^#(ok(X)) -> c_75(U22^#(X)) , U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) , U32^#(mark(X)) -> c_79(U32^#(X)) , U32^#(ok(X)) -> c_80(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) , U43^#(mark(X)) -> c_85(U43^#(X)) , U43^#(ok(X)) -> c_86(U43^#(X)) , U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) , U53^#(mark(X)) -> c_92(U53^#(X)) , U53^#(ok(X)) -> c_93(U53^#(X)) , U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) , U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) , U63^#(mark(X)) -> c_98(U63^#(X)) , U63^#(ok(X)) -> c_99(U63^#(X)) , U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) , U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) , s^#(mark(X)) -> c_102(s^#(X)) , s^#(ok(X)) -> c_103(s^#(X)) , length^#(mark(X)) -> c_104(length^#(X)) , length^#(ok(X)) -> c_105(length^#(X)) , U81^#(mark(X)) -> c_106(U81^#(X)) , U81^#(ok(X)) -> c_107(U81^#(X)) , U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) , U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) , take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) , take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) , take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) , and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_71(isNatList^#(X)) , isNat^#(ok(X)) -> c_76(isNat^#(X)) , isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) , proper^#(zeros()) -> c_117() , proper^#(0()) -> c_119() , proper^#(tt()) -> c_121() , proper^#(nil()) -> c_143() , top^#(mark(X)) -> c_149(top^#(proper(X))) , top^#(ok(X)) -> c_150(top^#(active(X))) } Strict Trs: { active(zeros()) -> mark(cons(0(), zeros())) , active(cons(X1, X2)) -> cons(active(X1), X2) , active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), V1)) -> mark(U12(isNatList(V1))) , active(U12(X)) -> U12(active(X)) , active(U12(tt())) -> mark(tt()) , active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(isNatList(nil())) -> mark(tt()) , active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U21(X1, X2)) -> U21(active(X1), X2) , active(U21(tt(), V1)) -> mark(U22(isNat(V1))) , active(U22(X)) -> U22(active(X)) , active(U22(tt())) -> mark(tt()) , active(isNat(0())) -> mark(tt()) , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) , active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) , active(U31(X1, X2)) -> U31(active(X1), X2) , active(U31(tt(), V)) -> mark(U32(isNatList(V))) , active(U32(X)) -> U32(active(X)) , active(U32(tt())) -> mark(tt()) , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) , active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)) , active(U42(X1, X2)) -> U42(active(X1), X2) , active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))) , active(U43(X)) -> U43(active(X)) , active(U43(tt())) -> mark(tt()) , active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) , active(isNatIList(zeros())) -> mark(tt()) , active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) , active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)) , active(U52(X1, X2)) -> U52(active(X1), X2) , active(U52(tt(), V2)) -> mark(U53(isNatList(V2))) , active(U53(X)) -> U53(active(X)) , active(U53(tt())) -> mark(tt()) , active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3) , active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)) , active(U62(X1, X2)) -> U62(active(X1), X2) , active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))) , active(U63(X)) -> U63(active(X)) , active(U63(tt())) -> mark(tt()) , active(U71(X1, X2)) -> U71(active(X1), X2) , active(U71(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , active(U81(X)) -> U81(active(X)) , active(U81(tt())) -> mark(nil()) , active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) , active(U91(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))) , active(take(X1, X2)) -> take(X1, active(X2)) , active(take(X1, X2)) -> take(active(X1), X2) , active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))) , active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNatIListKind(zeros())) -> mark(tt()) , active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatIListKind(nil())) -> mark(tt()) , active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatKind(0())) -> mark(tt()) , active(isNatKind(s(V1))) -> mark(isNatKind(V1)) , active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) , cons(mark(X1), X2) -> mark(cons(X1, X2)) , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U12(mark(X)) -> mark(U12(X)) , U12(ok(X)) -> ok(U12(X)) , isNatList(ok(X)) -> ok(isNatList(X)) , U21(mark(X1), X2) -> mark(U21(X1, X2)) , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) , U22(mark(X)) -> mark(U22(X)) , U22(ok(X)) -> ok(U22(X)) , isNat(ok(X)) -> ok(isNat(X)) , U31(mark(X1), X2) -> mark(U31(X1, X2)) , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) , U32(mark(X)) -> mark(U32(X)) , U32(ok(X)) -> ok(U32(X)) , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) , U42(mark(X1), X2) -> mark(U42(X1, X2)) , U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) , U43(mark(X)) -> mark(U43(X)) , U43(ok(X)) -> ok(U43(X)) , isNatIList(ok(X)) -> ok(isNatIList(X)) , U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) , U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) , U52(mark(X1), X2) -> mark(U52(X1, X2)) , U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) , U53(mark(X)) -> mark(U53(X)) , U53(ok(X)) -> ok(U53(X)) , U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)) , U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)) , U62(mark(X1), X2) -> mark(U62(X1, X2)) , U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) , U63(mark(X)) -> mark(U63(X)) , U63(ok(X)) -> ok(U63(X)) , U71(mark(X1), X2) -> mark(U71(X1, X2)) , U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) , s(mark(X)) -> mark(s(X)) , s(ok(X)) -> ok(s(X)) , length(mark(X)) -> mark(length(X)) , length(ok(X)) -> ok(length(X)) , U81(mark(X)) -> mark(U81(X)) , U81(ok(X)) -> ok(U81(X)) , U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) , U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) , take(X1, mark(X2)) -> mark(take(X1, X2)) , take(mark(X1), X2) -> mark(take(X1, X2)) , take(ok(X1), ok(X2)) -> ok(take(X1, X2)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) , isNatKind(ok(X)) -> ok(isNatKind(X)) , proper(zeros()) -> ok(zeros()) , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) , proper(0()) -> ok(0()) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U12(X)) -> U12(proper(X)) , proper(isNatList(X)) -> isNatList(proper(X)) , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) , proper(U22(X)) -> U22(proper(X)) , proper(isNat(X)) -> isNat(proper(X)) , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) , proper(U32(X)) -> U32(proper(X)) , proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) , proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) , proper(U43(X)) -> U43(proper(X)) , proper(isNatIList(X)) -> isNatIList(proper(X)) , proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) , proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) , proper(U53(X)) -> U53(proper(X)) , proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)) , proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) , proper(U63(X)) -> U63(proper(X)) , proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(U81(X)) -> U81(proper(X)) , proper(nil()) -> ok(nil()) , proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) , proper(take(X1, X2)) -> take(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,54,55,56,57} by applications of Pre({1,54,55,56,57}) = {}. Here rules are labeled as follows: DPs: { 1: active^#(zeros()) -> c_1(cons^#(0(), zeros())) , 2: cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) , 3: cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) , 4: U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) , 5: U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) , 6: U12^#(mark(X)) -> c_69(U12^#(X)) , 7: U12^#(ok(X)) -> c_70(U12^#(X)) , 8: U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) , 9: U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) , 10: U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) , 11: U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) , 12: U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) , 13: U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) , 14: U22^#(mark(X)) -> c_74(U22^#(X)) , 15: U22^#(ok(X)) -> c_75(U22^#(X)) , 16: U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) , 17: U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) , 18: U32^#(mark(X)) -> c_79(U32^#(X)) , 19: U32^#(ok(X)) -> c_80(U32^#(X)) , 20: U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) , 21: U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) , 22: U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) , 23: U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) , 24: U43^#(mark(X)) -> c_85(U43^#(X)) , 25: U43^#(ok(X)) -> c_86(U43^#(X)) , 26: U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) , 27: U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) , 28: U53^#(mark(X)) -> c_92(U53^#(X)) , 29: U53^#(ok(X)) -> c_93(U53^#(X)) , 30: U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) , 31: U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) , 32: U63^#(mark(X)) -> c_98(U63^#(X)) , 33: U63^#(ok(X)) -> c_99(U63^#(X)) , 34: U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) , 35: U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) , 36: s^#(mark(X)) -> c_102(s^#(X)) , 37: s^#(ok(X)) -> c_103(s^#(X)) , 38: length^#(mark(X)) -> c_104(length^#(X)) , 39: length^#(ok(X)) -> c_105(length^#(X)) , 40: U81^#(mark(X)) -> c_106(U81^#(X)) , 41: U81^#(ok(X)) -> c_107(U81^#(X)) , 42: U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) , 43: U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) , 44: take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) , 45: take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) , 46: take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) , 47: and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) , 48: and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) , 49: isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) , 50: isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) , 51: isNatList^#(ok(X)) -> c_71(isNatList^#(X)) , 52: isNat^#(ok(X)) -> c_76(isNat^#(X)) , 53: isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) , 54: proper^#(zeros()) -> c_117() , 55: proper^#(0()) -> c_119() , 56: proper^#(tt()) -> c_121() , 57: proper^#(nil()) -> c_143() , 58: top^#(mark(X)) -> c_149(top^#(proper(X))) , 59: top^#(ok(X)) -> c_150(top^#(active(X))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cons^#(mark(X1), X2) -> c_65(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_66(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_67(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_68(U11^#(X1, X2)) , U12^#(mark(X)) -> c_69(U12^#(X)) , U12^#(ok(X)) -> c_70(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_88(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_89(U51^#(X1, X2, X3)) , U61^#(mark(X1), X2, X3) -> c_94(U61^#(X1, X2, X3)) , U61^#(ok(X1), ok(X2), ok(X3)) -> c_95(U61^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_72(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_73(U21^#(X1, X2)) , U22^#(mark(X)) -> c_74(U22^#(X)) , U22^#(ok(X)) -> c_75(U22^#(X)) , U31^#(mark(X1), X2) -> c_77(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_78(U31^#(X1, X2)) , U32^#(mark(X)) -> c_79(U32^#(X)) , U32^#(ok(X)) -> c_80(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_81(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_82(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_83(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_84(U42^#(X1, X2)) , U43^#(mark(X)) -> c_85(U43^#(X)) , U43^#(ok(X)) -> c_86(U43^#(X)) , U52^#(mark(X1), X2) -> c_90(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_91(U52^#(X1, X2)) , U53^#(mark(X)) -> c_92(U53^#(X)) , U53^#(ok(X)) -> c_93(U53^#(X)) , U62^#(mark(X1), X2) -> c_96(U62^#(X1, X2)) , U62^#(ok(X1), ok(X2)) -> c_97(U62^#(X1, X2)) , U63^#(mark(X)) -> c_98(U63^#(X)) , U63^#(ok(X)) -> c_99(U63^#(X)) , U71^#(mark(X1), X2) -> c_100(U71^#(X1, X2)) , U71^#(ok(X1), ok(X2)) -> c_101(U71^#(X1, X2)) , s^#(mark(X)) -> c_102(s^#(X)) , s^#(ok(X)) -> c_103(s^#(X)) , length^#(mark(X)) -> c_104(length^#(X)) , length^#(ok(X)) -> c_105(length^#(X)) , U81^#(mark(X)) -> c_106(U81^#(X)) , U81^#(ok(X)) -> c_107(U81^#(X)) , U91^#(mark(X1), X2, X3, X4) -> c_108(U91^#(X1, X2, X3, X4)) , U91^#(ok(X1), ok(X2), ok(X3), ok(X4)) -> c_109(U91^#(X1, X2, X3, X4)) , take^#(X1, mark(X2)) -> c_110(take^#(X1, X2)) , take^#(mark(X1), X2) -> c_111(take^#(X1, X2)) , take^#(ok(X1), ok(X2)) -> c_112(take^#(X1, X2)) , and^#(mark(X1), X2) -> c_113(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_114(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_116(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_115(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_71(isNatList^#(X)) , isNat^#(ok(X)) -> c_76(isNat^#(X)) , isNatIList^#(ok(X)) -> c_87(isNatIList^#(X)) , top^#(mark(X)) -> c_149(top^#(proper(X))) , top^#(ok(X)) -> c_150(top^#(active(X))) } Strict Trs: { active(zeros()) -> mark(cons(0(), zeros())) , active(cons(X1, X2)) -> cons(active(X1), X2) , active(U11(X1, X2)) -> U11(active(X1), X2) , active(U11(tt(), V1)) -> mark(U12(isNatList(V1))) , active(U12(X)) -> U12(active(X)) , active(U12(tt())) -> mark(tt()) , active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(isNatList(nil())) -> mark(tt()) , active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U21(X1, X2)) -> U21(active(X1), X2) , active(U21(tt(), V1)) -> mark(U22(isNat(V1))) , active(U22(X)) -> U22(active(X)) , active(U22(tt())) -> mark(tt()) , active(isNat(0())) -> mark(tt()) , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) , active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) , active(U31(X1, X2)) -> U31(active(X1), X2) , active(U31(tt(), V)) -> mark(U32(isNatList(V))) , active(U32(X)) -> U32(active(X)) , active(U32(tt())) -> mark(tt()) , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) , active(U41(tt(), V1, V2)) -> mark(U42(isNat(V1), V2)) , active(U42(X1, X2)) -> U42(active(X1), X2) , active(U42(tt(), V2)) -> mark(U43(isNatIList(V2))) , active(U43(X)) -> U43(active(X)) , active(U43(tt())) -> mark(tt()) , active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) , active(isNatIList(zeros())) -> mark(tt()) , active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) , active(U51(tt(), V1, V2)) -> mark(U52(isNat(V1), V2)) , active(U52(X1, X2)) -> U52(active(X1), X2) , active(U52(tt(), V2)) -> mark(U53(isNatList(V2))) , active(U53(X)) -> U53(active(X)) , active(U53(tt())) -> mark(tt()) , active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3) , active(U61(tt(), V1, V2)) -> mark(U62(isNat(V1), V2)) , active(U62(X1, X2)) -> U62(active(X1), X2) , active(U62(tt(), V2)) -> mark(U63(isNatIList(V2))) , active(U63(X)) -> U63(active(X)) , active(U63(tt())) -> mark(tt()) , active(U71(X1, X2)) -> U71(active(X1), X2) , active(U71(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , active(U81(X)) -> U81(active(X)) , active(U81(tt())) -> mark(nil()) , active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4) , active(U91(tt(), IL, M, N)) -> mark(cons(N, take(M, IL))) , active(take(X1, X2)) -> take(X1, active(X2)) , active(take(X1, X2)) -> take(active(X1), X2) , active(take(0(), IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL)))) , active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) , active(and(X1, X2)) -> and(active(X1), X2) , active(and(tt(), X)) -> mark(X) , active(isNatIListKind(zeros())) -> mark(tt()) , active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatIListKind(nil())) -> mark(tt()) , active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2))) , active(isNatKind(0())) -> mark(tt()) , active(isNatKind(s(V1))) -> mark(isNatKind(V1)) , active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) , cons(mark(X1), X2) -> mark(cons(X1, X2)) , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) , U11(mark(X1), X2) -> mark(U11(X1, X2)) , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) , U12(mark(X)) -> mark(U12(X)) , U12(ok(X)) -> ok(U12(X)) , isNatList(ok(X)) -> ok(isNatList(X)) , U21(mark(X1), X2) -> mark(U21(X1, X2)) , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) , U22(mark(X)) -> mark(U22(X)) , U22(ok(X)) -> ok(U22(X)) , isNat(ok(X)) -> ok(isNat(X)) , U31(mark(X1), X2) -> mark(U31(X1, X2)) , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) , U32(mark(X)) -> mark(U32(X)) , U32(ok(X)) -> ok(U32(X)) , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) , U42(mark(X1), X2) -> mark(U42(X1, X2)) , U42(ok(X1), ok(X2)) -> ok(U42(X1, X2)) , U43(mark(X)) -> mark(U43(X)) , U43(ok(X)) -> ok(U43(X)) , isNatIList(ok(X)) -> ok(isNatIList(X)) , U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) , U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) , U52(mark(X1), X2) -> mark(U52(X1, X2)) , U52(ok(X1), ok(X2)) -> ok(U52(X1, X2)) , U53(mark(X)) -> mark(U53(X)) , U53(ok(X)) -> ok(U53(X)) , U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3)) , U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3)) , U62(mark(X1), X2) -> mark(U62(X1, X2)) , U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) , U63(mark(X)) -> mark(U63(X)) , U63(ok(X)) -> ok(U63(X)) , U71(mark(X1), X2) -> mark(U71(X1, X2)) , U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) , s(mark(X)) -> mark(s(X)) , s(ok(X)) -> ok(s(X)) , length(mark(X)) -> mark(length(X)) , length(ok(X)) -> ok(length(X)) , U81(mark(X)) -> mark(U81(X)) , U81(ok(X)) -> ok(U81(X)) , U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4)) , U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4)) , take(X1, mark(X2)) -> mark(take(X1, X2)) , take(mark(X1), X2) -> mark(take(X1, X2)) , take(ok(X1), ok(X2)) -> ok(take(X1, X2)) , and(mark(X1), X2) -> mark(and(X1, X2)) , and(ok(X1), ok(X2)) -> ok(and(X1, X2)) , isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) , isNatKind(ok(X)) -> ok(isNatKind(X)) , proper(zeros()) -> ok(zeros()) , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) , proper(0()) -> ok(0()) , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) , proper(tt()) -> ok(tt()) , proper(U12(X)) -> U12(proper(X)) , proper(isNatList(X)) -> isNatList(proper(X)) , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) , proper(U22(X)) -> U22(proper(X)) , proper(isNat(X)) -> isNat(proper(X)) , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) , proper(U32(X)) -> U32(proper(X)) , proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) , proper(U42(X1, X2)) -> U42(proper(X1), proper(X2)) , proper(U43(X)) -> U43(proper(X)) , proper(isNatIList(X)) -> isNatIList(proper(X)) , proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) , proper(U52(X1, X2)) -> U52(proper(X1), proper(X2)) , proper(U53(X)) -> U53(proper(X)) , proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3)) , proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) , proper(U63(X)) -> U63(proper(X)) , proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(U81(X)) -> U81(proper(X)) , proper(nil()) -> ok(nil()) , proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4)) , proper(take(X1, X2)) -> take(proper(X1), proper(X2)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Weak DPs: { active^#(zeros()) -> c_1(cons^#(0(), zeros())) , proper^#(zeros()) -> c_117() , proper^#(0()) -> c_119() , proper^#(tt()) -> c_121() , proper^#(nil()) -> c_143() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..