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(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)) -> U61(active(X1), X2) , active(U61(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , 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(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) -> mark(U61(X1, X2)) , U61(ok(X1), ok(X2)) -> ok(U61(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)) , 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)) -> U61(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , proper(nil()) -> ok(nil()) , 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 perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-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^#(U21(X1, X2)) -> c_9(U21^#(active(X1), X2)) , active^#(U21(tt(), V1)) -> c_10(U22^#(isNat(V1))) , active^#(U22(X)) -> c_11(U22^#(active(X))) , active^#(U22(tt())) -> c_12() , active^#(isNat(0())) -> c_13() , active^#(isNat(s(V1))) -> c_14(U21^#(isNatKind(V1), V1)) , active^#(isNat(length(V1))) -> c_15(U11^#(isNatIListKind(V1), V1)) , active^#(U31(X1, X2)) -> c_16(U31^#(active(X1), X2)) , active^#(U31(tt(), V)) -> c_17(U32^#(isNatList(V))) , active^#(U32(X)) -> c_18(U32^#(active(X))) , active^#(U32(tt())) -> c_19() , active^#(U41(X1, X2, X3)) -> c_20(U41^#(active(X1), X2, X3)) , active^#(U41(tt(), V1, V2)) -> c_21(U42^#(isNat(V1), V2)) , active^#(U42(X1, X2)) -> c_22(U42^#(active(X1), X2)) , active^#(U42(tt(), V2)) -> c_23(U43^#(isNatIList(V2))) , active^#(U43(X)) -> c_24(U43^#(active(X))) , active^#(U43(tt())) -> c_25() , active^#(isNatIList(V)) -> c_26(U31^#(isNatIListKind(V), V)) , active^#(isNatIList(zeros())) -> c_27() , active^#(isNatIList(cons(V1, V2))) -> c_28(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U51(X1, X2, X3)) -> c_29(U51^#(active(X1), X2, X3)) , active^#(U51(tt(), V1, V2)) -> c_30(U52^#(isNat(V1), V2)) , active^#(U52(X1, X2)) -> c_31(U52^#(active(X1), X2)) , active^#(U52(tt(), V2)) -> c_32(U53^#(isNatList(V2))) , active^#(U53(X)) -> c_33(U53^#(active(X))) , active^#(U53(tt())) -> c_34() , active^#(U61(X1, X2)) -> c_35(U61^#(active(X1), X2)) , active^#(U61(tt(), L)) -> c_36(s^#(length(L))) , active^#(s(X)) -> c_37(s^#(active(X))) , active^#(length(X)) -> c_38(length^#(active(X))) , active^#(length(cons(N, L))) -> c_39(U61^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active^#(length(nil())) -> c_40() , active^#(and(X1, X2)) -> c_41(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_42(X) , active^#(isNatIListKind(zeros())) -> c_43() , active^#(isNatIListKind(cons(V1, V2))) -> c_44(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatIListKind(nil())) -> c_45() , active^#(isNatKind(0())) -> c_46() , active^#(isNatKind(s(V1))) -> c_47(isNatKind^#(V1)) , active^#(isNatKind(length(V1))) -> c_48(isNatIListKind^#(V1)) , cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) , U12^#(mark(X)) -> c_53(U12^#(X)) , U12^#(ok(X)) -> c_54(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) , U22^#(mark(X)) -> c_58(U22^#(X)) , U22^#(ok(X)) -> c_59(U22^#(X)) , U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) , U32^#(mark(X)) -> c_63(U32^#(X)) , U32^#(ok(X)) -> c_64(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) , U43^#(mark(X)) -> c_69(U43^#(X)) , U43^#(ok(X)) -> c_70(U43^#(X)) , U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) , U53^#(mark(X)) -> c_76(U53^#(X)) , U53^#(ok(X)) -> c_77(U53^#(X)) , U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) , U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) , s^#(mark(X)) -> c_80(s^#(X)) , s^#(ok(X)) -> c_81(s^#(X)) , length^#(mark(X)) -> c_82(length^#(X)) , length^#(ok(X)) -> c_83(length^#(X)) , and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_55(isNatList^#(X)) , isNat^#(ok(X)) -> c_60(isNat^#(X)) , isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) , proper^#(zeros()) -> c_88() , proper^#(cons(X1, X2)) -> c_89(cons^#(proper(X1), proper(X2))) , proper^#(0()) -> c_90() , proper^#(U11(X1, X2)) -> c_91(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_92() , proper^#(U12(X)) -> c_93(U12^#(proper(X))) , proper^#(isNatList(X)) -> c_94(isNatList^#(proper(X))) , proper^#(U21(X1, X2)) -> c_95(U21^#(proper(X1), proper(X2))) , proper^#(U22(X)) -> c_96(U22^#(proper(X))) , proper^#(isNat(X)) -> c_97(isNat^#(proper(X))) , proper^#(U31(X1, X2)) -> c_98(U31^#(proper(X1), proper(X2))) , proper^#(U32(X)) -> c_99(U32^#(proper(X))) , proper^#(U41(X1, X2, X3)) -> c_100(U41^#(proper(X1), proper(X2), proper(X3))) , proper^#(U42(X1, X2)) -> c_101(U42^#(proper(X1), proper(X2))) , proper^#(U43(X)) -> c_102(U43^#(proper(X))) , proper^#(isNatIList(X)) -> c_103(isNatIList^#(proper(X))) , proper^#(U51(X1, X2, X3)) -> c_104(U51^#(proper(X1), proper(X2), proper(X3))) , proper^#(U52(X1, X2)) -> c_105(U52^#(proper(X1), proper(X2))) , proper^#(U53(X)) -> c_106(U53^#(proper(X))) , proper^#(U61(X1, X2)) -> c_107(U61^#(proper(X1), proper(X2))) , proper^#(s(X)) -> c_108(s^#(proper(X))) , proper^#(length(X)) -> c_109(length^#(proper(X))) , proper^#(and(X1, X2)) -> c_110(and^#(proper(X1), proper(X2))) , proper^#(isNatIListKind(X)) -> c_111(isNatIListKind^#(proper(X))) , proper^#(isNatKind(X)) -> c_112(isNatKind^#(proper(X))) , proper^#(nil()) -> c_113() , top^#(mark(X)) -> c_114(top^#(proper(X))) , top^#(ok(X)) -> c_115(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^#(U21(X1, X2)) -> c_9(U21^#(active(X1), X2)) , active^#(U21(tt(), V1)) -> c_10(U22^#(isNat(V1))) , active^#(U22(X)) -> c_11(U22^#(active(X))) , active^#(U22(tt())) -> c_12() , active^#(isNat(0())) -> c_13() , active^#(isNat(s(V1))) -> c_14(U21^#(isNatKind(V1), V1)) , active^#(isNat(length(V1))) -> c_15(U11^#(isNatIListKind(V1), V1)) , active^#(U31(X1, X2)) -> c_16(U31^#(active(X1), X2)) , active^#(U31(tt(), V)) -> c_17(U32^#(isNatList(V))) , active^#(U32(X)) -> c_18(U32^#(active(X))) , active^#(U32(tt())) -> c_19() , active^#(U41(X1, X2, X3)) -> c_20(U41^#(active(X1), X2, X3)) , active^#(U41(tt(), V1, V2)) -> c_21(U42^#(isNat(V1), V2)) , active^#(U42(X1, X2)) -> c_22(U42^#(active(X1), X2)) , active^#(U42(tt(), V2)) -> c_23(U43^#(isNatIList(V2))) , active^#(U43(X)) -> c_24(U43^#(active(X))) , active^#(U43(tt())) -> c_25() , active^#(isNatIList(V)) -> c_26(U31^#(isNatIListKind(V), V)) , active^#(isNatIList(zeros())) -> c_27() , active^#(isNatIList(cons(V1, V2))) -> c_28(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) , active^#(U51(X1, X2, X3)) -> c_29(U51^#(active(X1), X2, X3)) , active^#(U51(tt(), V1, V2)) -> c_30(U52^#(isNat(V1), V2)) , active^#(U52(X1, X2)) -> c_31(U52^#(active(X1), X2)) , active^#(U52(tt(), V2)) -> c_32(U53^#(isNatList(V2))) , active^#(U53(X)) -> c_33(U53^#(active(X))) , active^#(U53(tt())) -> c_34() , active^#(U61(X1, X2)) -> c_35(U61^#(active(X1), X2)) , active^#(U61(tt(), L)) -> c_36(s^#(length(L))) , active^#(s(X)) -> c_37(s^#(active(X))) , active^#(length(X)) -> c_38(length^#(active(X))) , active^#(length(cons(N, L))) -> c_39(U61^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active^#(length(nil())) -> c_40() , active^#(and(X1, X2)) -> c_41(and^#(active(X1), X2)) , active^#(and(tt(), X)) -> c_42(X) , active^#(isNatIListKind(zeros())) -> c_43() , active^#(isNatIListKind(cons(V1, V2))) -> c_44(and^#(isNatKind(V1), isNatIListKind(V2))) , active^#(isNatIListKind(nil())) -> c_45() , active^#(isNatKind(0())) -> c_46() , active^#(isNatKind(s(V1))) -> c_47(isNatKind^#(V1)) , active^#(isNatKind(length(V1))) -> c_48(isNatIListKind^#(V1)) , cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) , U12^#(mark(X)) -> c_53(U12^#(X)) , U12^#(ok(X)) -> c_54(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) , U22^#(mark(X)) -> c_58(U22^#(X)) , U22^#(ok(X)) -> c_59(U22^#(X)) , U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) , U32^#(mark(X)) -> c_63(U32^#(X)) , U32^#(ok(X)) -> c_64(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) , U43^#(mark(X)) -> c_69(U43^#(X)) , U43^#(ok(X)) -> c_70(U43^#(X)) , U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) , U53^#(mark(X)) -> c_76(U53^#(X)) , U53^#(ok(X)) -> c_77(U53^#(X)) , U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) , U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) , s^#(mark(X)) -> c_80(s^#(X)) , s^#(ok(X)) -> c_81(s^#(X)) , length^#(mark(X)) -> c_82(length^#(X)) , length^#(ok(X)) -> c_83(length^#(X)) , and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_55(isNatList^#(X)) , isNat^#(ok(X)) -> c_60(isNat^#(X)) , isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) , proper^#(zeros()) -> c_88() , proper^#(cons(X1, X2)) -> c_89(cons^#(proper(X1), proper(X2))) , proper^#(0()) -> c_90() , proper^#(U11(X1, X2)) -> c_91(U11^#(proper(X1), proper(X2))) , proper^#(tt()) -> c_92() , proper^#(U12(X)) -> c_93(U12^#(proper(X))) , proper^#(isNatList(X)) -> c_94(isNatList^#(proper(X))) , proper^#(U21(X1, X2)) -> c_95(U21^#(proper(X1), proper(X2))) , proper^#(U22(X)) -> c_96(U22^#(proper(X))) , proper^#(isNat(X)) -> c_97(isNat^#(proper(X))) , proper^#(U31(X1, X2)) -> c_98(U31^#(proper(X1), proper(X2))) , proper^#(U32(X)) -> c_99(U32^#(proper(X))) , proper^#(U41(X1, X2, X3)) -> c_100(U41^#(proper(X1), proper(X2), proper(X3))) , proper^#(U42(X1, X2)) -> c_101(U42^#(proper(X1), proper(X2))) , proper^#(U43(X)) -> c_102(U43^#(proper(X))) , proper^#(isNatIList(X)) -> c_103(isNatIList^#(proper(X))) , proper^#(U51(X1, X2, X3)) -> c_104(U51^#(proper(X1), proper(X2), proper(X3))) , proper^#(U52(X1, X2)) -> c_105(U52^#(proper(X1), proper(X2))) , proper^#(U53(X)) -> c_106(U53^#(proper(X))) , proper^#(U61(X1, X2)) -> c_107(U61^#(proper(X1), proper(X2))) , proper^#(s(X)) -> c_108(s^#(proper(X))) , proper^#(length(X)) -> c_109(length^#(proper(X))) , proper^#(and(X1, X2)) -> c_110(and^#(proper(X1), proper(X2))) , proper^#(isNatIListKind(X)) -> c_111(isNatIListKind^#(proper(X))) , proper^#(isNatKind(X)) -> c_112(isNatKind^#(proper(X))) , proper^#(nil()) -> c_113() , top^#(mark(X)) -> c_114(top^#(proper(X))) , top^#(ok(X)) -> c_115(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(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)) -> U61(active(X1), X2) , active(U61(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , 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(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) -> mark(U61(X1, X2)) , U61(ok(X1), ok(X2)) -> ok(U61(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)) , 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)) -> U61(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , proper(nil()) -> ok(nil()) , 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_50(cons^#(X1, X2)) :50 -->_1 cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) :49 3: active^#(U11(X1, X2)) -> c_3(U11^#(active(X1), X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 4: active^#(U11(tt(), V1)) -> c_4(U12^#(isNatList(V1))) -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 5: active^#(U12(X)) -> c_5(U12^#(active(X))) -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 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_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 8: active^#(isNatList(nil())) -> c_8() 9: active^#(U21(X1, X2)) -> c_9(U21^#(active(X1), X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 10: active^#(U21(tt(), V1)) -> c_10(U22^#(isNat(V1))) -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 11: active^#(U22(X)) -> c_11(U22^#(active(X))) -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 12: active^#(U22(tt())) -> c_12() 13: active^#(isNat(0())) -> c_13() 14: active^#(isNat(s(V1))) -> c_14(U21^#(isNatKind(V1), V1)) -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 15: active^#(isNat(length(V1))) -> c_15(U11^#(isNatIListKind(V1), V1)) -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 16: active^#(U31(X1, X2)) -> c_16(U31^#(active(X1), X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 17: active^#(U31(tt(), V)) -> c_17(U32^#(isNatList(V))) -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 18: active^#(U32(X)) -> c_18(U32^#(active(X))) -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 19: active^#(U32(tt())) -> c_19() 20: active^#(U41(X1, X2, X3)) -> c_20(U41^#(active(X1), X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 21: active^#(U41(tt(), V1, V2)) -> c_21(U42^#(isNat(V1), V2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 22: active^#(U42(X1, X2)) -> c_22(U42^#(active(X1), X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 23: active^#(U42(tt(), V2)) -> c_23(U43^#(isNatIList(V2))) -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 24: active^#(U43(X)) -> c_24(U43^#(active(X))) -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 25: active^#(U43(tt())) -> c_25() 26: active^#(isNatIList(V)) -> c_26(U31^#(isNatIListKind(V), V)) -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 27: active^#(isNatIList(zeros())) -> c_27() 28: active^#(isNatIList(cons(V1, V2))) -> c_28(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 29: active^#(U51(X1, X2, X3)) -> c_29(U51^#(active(X1), X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 30: active^#(U51(tt(), V1, V2)) -> c_30(U52^#(isNat(V1), V2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 31: active^#(U52(X1, X2)) -> c_31(U52^#(active(X1), X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 32: active^#(U52(tt(), V2)) -> c_32(U53^#(isNatList(V2))) -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 33: active^#(U53(X)) -> c_33(U53^#(active(X))) -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 34: active^#(U53(tt())) -> c_34() 35: active^#(U61(X1, X2)) -> c_35(U61^#(active(X1), X2)) -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 36: active^#(U61(tt(), L)) -> c_36(s^#(length(L))) -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 37: active^#(s(X)) -> c_37(s^#(active(X))) -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 38: active^#(length(X)) -> c_38(length^#(active(X))) -->_1 length^#(ok(X)) -> c_83(length^#(X)) :80 -->_1 length^#(mark(X)) -> c_82(length^#(X)) :79 39: active^#(length(cons(N, L))) -> c_39(U61^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 40: active^#(length(nil())) -> c_40() 41: active^#(and(X1, X2)) -> c_41(and^#(active(X1), X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 42: active^#(and(tt(), X)) -> c_42(X) -->_1 top^#(ok(X)) -> c_115(top^#(active(X))) :115 -->_1 top^#(mark(X)) -> c_114(top^#(proper(X))) :114 -->_1 proper^#(isNatKind(X)) -> c_112(isNatKind^#(proper(X))) :112 -->_1 proper^#(isNatIListKind(X)) -> c_111(isNatIListKind^#(proper(X))) :111 -->_1 proper^#(and(X1, X2)) -> c_110(and^#(proper(X1), proper(X2))) :110 -->_1 proper^#(length(X)) -> c_109(length^#(proper(X))) :109 -->_1 proper^#(s(X)) -> c_108(s^#(proper(X))) :108 -->_1 proper^#(U61(X1, X2)) -> c_107(U61^#(proper(X1), proper(X2))) :107 -->_1 proper^#(U53(X)) -> c_106(U53^#(proper(X))) :106 -->_1 proper^#(U52(X1, X2)) -> c_105(U52^#(proper(X1), proper(X2))) :105 -->_1 proper^#(U51(X1, X2, X3)) -> c_104(U51^#(proper(X1), proper(X2), proper(X3))) :104 -->_1 proper^#(isNatIList(X)) -> c_103(isNatIList^#(proper(X))) :103 -->_1 proper^#(U43(X)) -> c_102(U43^#(proper(X))) :102 -->_1 proper^#(U42(X1, X2)) -> c_101(U42^#(proper(X1), proper(X2))) :101 -->_1 proper^#(U41(X1, X2, X3)) -> c_100(U41^#(proper(X1), proper(X2), proper(X3))) :100 -->_1 proper^#(U32(X)) -> c_99(U32^#(proper(X))) :99 -->_1 proper^#(U31(X1, X2)) -> c_98(U31^#(proper(X1), proper(X2))) :98 -->_1 proper^#(isNat(X)) -> c_97(isNat^#(proper(X))) :97 -->_1 proper^#(U22(X)) -> c_96(U22^#(proper(X))) :96 -->_1 proper^#(U21(X1, X2)) -> c_95(U21^#(proper(X1), proper(X2))) :95 -->_1 proper^#(isNatList(X)) -> c_94(isNatList^#(proper(X))) :94 -->_1 proper^#(U12(X)) -> c_93(U12^#(proper(X))) :93 -->_1 proper^#(U11(X1, X2)) -> c_91(U11^#(proper(X1), proper(X2))) :91 -->_1 proper^#(cons(X1, X2)) -> c_89(cons^#(proper(X1), proper(X2))) :89 -->_1 isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) :87 -->_1 isNat^#(ok(X)) -> c_60(isNat^#(X)) :86 -->_1 isNatList^#(ok(X)) -> c_55(isNatList^#(X)) :85 -->_1 isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) :84 -->_1 isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) :83 -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 -->_1 length^#(ok(X)) -> c_83(length^#(X)) :80 -->_1 length^#(mark(X)) -> c_82(length^#(X)) :79 -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 -->_1 cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) :50 -->_1 cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) :49 -->_1 active^#(isNatKind(length(V1))) -> c_48(isNatIListKind^#(V1)) :48 -->_1 active^#(isNatKind(s(V1))) -> c_47(isNatKind^#(V1)) :47 -->_1 active^#(isNatIListKind(cons(V1, V2))) -> c_44(and^#(isNatKind(V1), isNatIListKind(V2))) :44 -->_1 proper^#(nil()) -> c_113() :113 -->_1 proper^#(tt()) -> c_92() :92 -->_1 proper^#(0()) -> c_90() :90 -->_1 proper^#(zeros()) -> c_88() :88 -->_1 active^#(isNatKind(0())) -> c_46() :46 -->_1 active^#(isNatIListKind(nil())) -> c_45() :45 -->_1 active^#(isNatIListKind(zeros())) -> c_43() :43 -->_1 active^#(and(tt(), X)) -> c_42(X) :42 -->_1 active^#(and(X1, X2)) -> c_41(and^#(active(X1), X2)) :41 -->_1 active^#(length(nil())) -> c_40() :40 -->_1 active^#(length(cons(N, L))) -> c_39(U61^#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) :39 -->_1 active^#(length(X)) -> c_38(length^#(active(X))) :38 -->_1 active^#(s(X)) -> c_37(s^#(active(X))) :37 -->_1 active^#(U61(tt(), L)) -> c_36(s^#(length(L))) :36 -->_1 active^#(U61(X1, X2)) -> c_35(U61^#(active(X1), X2)) :35 -->_1 active^#(U53(tt())) -> c_34() :34 -->_1 active^#(U53(X)) -> c_33(U53^#(active(X))) :33 -->_1 active^#(U52(tt(), V2)) -> c_32(U53^#(isNatList(V2))) :32 -->_1 active^#(U52(X1, X2)) -> c_31(U52^#(active(X1), X2)) :31 -->_1 active^#(U51(tt(), V1, V2)) -> c_30(U52^#(isNat(V1), V2)) :30 -->_1 active^#(U51(X1, X2, X3)) -> c_29(U51^#(active(X1), X2, X3)) :29 -->_1 active^#(isNatIList(cons(V1, V2))) -> c_28(U41^#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) :28 -->_1 active^#(isNatIList(zeros())) -> c_27() :27 -->_1 active^#(isNatIList(V)) -> c_26(U31^#(isNatIListKind(V), V)) :26 -->_1 active^#(U43(tt())) -> c_25() :25 -->_1 active^#(U43(X)) -> c_24(U43^#(active(X))) :24 -->_1 active^#(U42(tt(), V2)) -> c_23(U43^#(isNatIList(V2))) :23 -->_1 active^#(U42(X1, X2)) -> c_22(U42^#(active(X1), X2)) :22 -->_1 active^#(U41(tt(), V1, V2)) -> c_21(U42^#(isNat(V1), V2)) :21 -->_1 active^#(U41(X1, X2, X3)) -> c_20(U41^#(active(X1), X2, X3)) :20 -->_1 active^#(U32(tt())) -> c_19() :19 -->_1 active^#(U32(X)) -> c_18(U32^#(active(X))) :18 -->_1 active^#(U31(tt(), V)) -> c_17(U32^#(isNatList(V))) :17 -->_1 active^#(U31(X1, X2)) -> c_16(U31^#(active(X1), X2)) :16 -->_1 active^#(isNat(length(V1))) -> c_15(U11^#(isNatIListKind(V1), V1)) :15 -->_1 active^#(isNat(s(V1))) -> c_14(U21^#(isNatKind(V1), V1)) :14 -->_1 active^#(isNat(0())) -> c_13() :13 -->_1 active^#(U22(tt())) -> c_12() :12 -->_1 active^#(U22(X)) -> c_11(U22^#(active(X))) :11 -->_1 active^#(U21(tt(), V1)) -> c_10(U22^#(isNat(V1))) :10 -->_1 active^#(U21(X1, X2)) -> c_9(U21^#(active(X1), X2)) :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 43: active^#(isNatIListKind(zeros())) -> c_43() 44: active^#(isNatIListKind(cons(V1, V2))) -> c_44(and^#(isNatKind(V1), isNatIListKind(V2))) -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 45: active^#(isNatIListKind(nil())) -> c_45() 46: active^#(isNatKind(0())) -> c_46() 47: active^#(isNatKind(s(V1))) -> c_47(isNatKind^#(V1)) -->_1 isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) :83 48: active^#(isNatKind(length(V1))) -> c_48(isNatIListKind^#(V1)) -->_1 isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) :84 49: cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) -->_1 cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) :50 -->_1 cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) :49 50: cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) -->_1 cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) :50 -->_1 cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) :49 51: U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 52: U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 53: U12^#(mark(X)) -> c_53(U12^#(X)) -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 54: U12^#(ok(X)) -> c_54(U12^#(X)) -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 55: U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 56: U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 57: U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 58: U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 59: U22^#(mark(X)) -> c_58(U22^#(X)) -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 60: U22^#(ok(X)) -> c_59(U22^#(X)) -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 61: U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 62: U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 63: U32^#(mark(X)) -> c_63(U32^#(X)) -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 64: U32^#(ok(X)) -> c_64(U32^#(X)) -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 65: U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 66: U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 67: U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 68: U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 69: U43^#(mark(X)) -> c_69(U43^#(X)) -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 70: U43^#(ok(X)) -> c_70(U43^#(X)) -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 71: U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 72: U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 73: U53^#(mark(X)) -> c_76(U53^#(X)) -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 74: U53^#(ok(X)) -> c_77(U53^#(X)) -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 75: U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 76: U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 77: s^#(mark(X)) -> c_80(s^#(X)) -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 78: s^#(ok(X)) -> c_81(s^#(X)) -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 79: length^#(mark(X)) -> c_82(length^#(X)) -->_1 length^#(ok(X)) -> c_83(length^#(X)) :80 -->_1 length^#(mark(X)) -> c_82(length^#(X)) :79 80: length^#(ok(X)) -> c_83(length^#(X)) -->_1 length^#(ok(X)) -> c_83(length^#(X)) :80 -->_1 length^#(mark(X)) -> c_82(length^#(X)) :79 81: and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 82: and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 83: isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) -->_1 isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) :83 84: isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) -->_1 isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) :84 85: isNatList^#(ok(X)) -> c_55(isNatList^#(X)) -->_1 isNatList^#(ok(X)) -> c_55(isNatList^#(X)) :85 86: isNat^#(ok(X)) -> c_60(isNat^#(X)) -->_1 isNat^#(ok(X)) -> c_60(isNat^#(X)) :86 87: isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) -->_1 isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) :87 88: proper^#(zeros()) -> c_88() 89: proper^#(cons(X1, X2)) -> c_89(cons^#(proper(X1), proper(X2))) -->_1 cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) :50 -->_1 cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) :49 90: proper^#(0()) -> c_90() 91: proper^#(U11(X1, X2)) -> c_91(U11^#(proper(X1), proper(X2))) -->_1 U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) :52 -->_1 U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) :51 92: proper^#(tt()) -> c_92() 93: proper^#(U12(X)) -> c_93(U12^#(proper(X))) -->_1 U12^#(ok(X)) -> c_54(U12^#(X)) :54 -->_1 U12^#(mark(X)) -> c_53(U12^#(X)) :53 94: proper^#(isNatList(X)) -> c_94(isNatList^#(proper(X))) -->_1 isNatList^#(ok(X)) -> c_55(isNatList^#(X)) :85 95: proper^#(U21(X1, X2)) -> c_95(U21^#(proper(X1), proper(X2))) -->_1 U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) :58 -->_1 U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) :57 96: proper^#(U22(X)) -> c_96(U22^#(proper(X))) -->_1 U22^#(ok(X)) -> c_59(U22^#(X)) :60 -->_1 U22^#(mark(X)) -> c_58(U22^#(X)) :59 97: proper^#(isNat(X)) -> c_97(isNat^#(proper(X))) -->_1 isNat^#(ok(X)) -> c_60(isNat^#(X)) :86 98: proper^#(U31(X1, X2)) -> c_98(U31^#(proper(X1), proper(X2))) -->_1 U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) :62 -->_1 U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) :61 99: proper^#(U32(X)) -> c_99(U32^#(proper(X))) -->_1 U32^#(ok(X)) -> c_64(U32^#(X)) :64 -->_1 U32^#(mark(X)) -> c_63(U32^#(X)) :63 100: proper^#(U41(X1, X2, X3)) -> c_100(U41^#(proper(X1), proper(X2), proper(X3))) -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) :66 -->_1 U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) :65 101: proper^#(U42(X1, X2)) -> c_101(U42^#(proper(X1), proper(X2))) -->_1 U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) :68 -->_1 U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) :67 102: proper^#(U43(X)) -> c_102(U43^#(proper(X))) -->_1 U43^#(ok(X)) -> c_70(U43^#(X)) :70 -->_1 U43^#(mark(X)) -> c_69(U43^#(X)) :69 103: proper^#(isNatIList(X)) -> c_103(isNatIList^#(proper(X))) -->_1 isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) :87 104: proper^#(U51(X1, X2, X3)) -> c_104(U51^#(proper(X1), proper(X2), proper(X3))) -->_1 U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) :56 -->_1 U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) :55 105: proper^#(U52(X1, X2)) -> c_105(U52^#(proper(X1), proper(X2))) -->_1 U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) :72 -->_1 U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) :71 106: proper^#(U53(X)) -> c_106(U53^#(proper(X))) -->_1 U53^#(ok(X)) -> c_77(U53^#(X)) :74 -->_1 U53^#(mark(X)) -> c_76(U53^#(X)) :73 107: proper^#(U61(X1, X2)) -> c_107(U61^#(proper(X1), proper(X2))) -->_1 U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) :76 -->_1 U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) :75 108: proper^#(s(X)) -> c_108(s^#(proper(X))) -->_1 s^#(ok(X)) -> c_81(s^#(X)) :78 -->_1 s^#(mark(X)) -> c_80(s^#(X)) :77 109: proper^#(length(X)) -> c_109(length^#(proper(X))) -->_1 length^#(ok(X)) -> c_83(length^#(X)) :80 -->_1 length^#(mark(X)) -> c_82(length^#(X)) :79 110: proper^#(and(X1, X2)) -> c_110(and^#(proper(X1), proper(X2))) -->_1 and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) :82 -->_1 and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) :81 111: proper^#(isNatIListKind(X)) -> c_111(isNatIListKind^#(proper(X))) -->_1 isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) :84 112: proper^#(isNatKind(X)) -> c_112(isNatKind^#(proper(X))) -->_1 isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) :83 113: proper^#(nil()) -> c_113() 114: top^#(mark(X)) -> c_114(top^#(proper(X))) -->_1 top^#(ok(X)) -> c_115(top^#(active(X))) :115 -->_1 top^#(mark(X)) -> c_114(top^#(proper(X))) :114 115: top^#(ok(X)) -> c_115(top^#(active(X))) -->_1 top^#(ok(X)) -> c_115(top^#(active(X))) :115 -->_1 top^#(mark(X)) -> c_114(top^#(proper(X))) :114 Only the nodes {1,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,90,92,113,114,115} are reachable from nodes {1,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,90,92,113,114,115} 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_49(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) , U12^#(mark(X)) -> c_53(U12^#(X)) , U12^#(ok(X)) -> c_54(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) , U22^#(mark(X)) -> c_58(U22^#(X)) , U22^#(ok(X)) -> c_59(U22^#(X)) , U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) , U32^#(mark(X)) -> c_63(U32^#(X)) , U32^#(ok(X)) -> c_64(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) , U43^#(mark(X)) -> c_69(U43^#(X)) , U43^#(ok(X)) -> c_70(U43^#(X)) , U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) , U53^#(mark(X)) -> c_76(U53^#(X)) , U53^#(ok(X)) -> c_77(U53^#(X)) , U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) , U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) , s^#(mark(X)) -> c_80(s^#(X)) , s^#(ok(X)) -> c_81(s^#(X)) , length^#(mark(X)) -> c_82(length^#(X)) , length^#(ok(X)) -> c_83(length^#(X)) , and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_55(isNatList^#(X)) , isNat^#(ok(X)) -> c_60(isNat^#(X)) , isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) , proper^#(zeros()) -> c_88() , proper^#(0()) -> c_90() , proper^#(tt()) -> c_92() , proper^#(nil()) -> c_113() , top^#(mark(X)) -> c_114(top^#(proper(X))) , top^#(ok(X)) -> c_115(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(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)) -> U61(active(X1), X2) , active(U61(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , 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(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) -> mark(U61(X1, X2)) , U61(ok(X1), ok(X2)) -> ok(U61(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)) , 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)) -> U61(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , proper(nil()) -> ok(nil()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,41,42,43,44} by applications of Pre({1,41,42,43,44}) = {}. Here rules are labeled as follows: DPs: { 1: active^#(zeros()) -> c_1(cons^#(0(), zeros())) , 2: cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) , 3: cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) , 4: U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) , 5: U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) , 6: U12^#(mark(X)) -> c_53(U12^#(X)) , 7: U12^#(ok(X)) -> c_54(U12^#(X)) , 8: U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) , 9: U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) , 10: U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) , 11: U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) , 12: U22^#(mark(X)) -> c_58(U22^#(X)) , 13: U22^#(ok(X)) -> c_59(U22^#(X)) , 14: U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) , 15: U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) , 16: U32^#(mark(X)) -> c_63(U32^#(X)) , 17: U32^#(ok(X)) -> c_64(U32^#(X)) , 18: U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) , 19: U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) , 20: U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) , 21: U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) , 22: U43^#(mark(X)) -> c_69(U43^#(X)) , 23: U43^#(ok(X)) -> c_70(U43^#(X)) , 24: U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) , 25: U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) , 26: U53^#(mark(X)) -> c_76(U53^#(X)) , 27: U53^#(ok(X)) -> c_77(U53^#(X)) , 28: U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) , 29: U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) , 30: s^#(mark(X)) -> c_80(s^#(X)) , 31: s^#(ok(X)) -> c_81(s^#(X)) , 32: length^#(mark(X)) -> c_82(length^#(X)) , 33: length^#(ok(X)) -> c_83(length^#(X)) , 34: and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) , 35: and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) , 36: isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) , 37: isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) , 38: isNatList^#(ok(X)) -> c_55(isNatList^#(X)) , 39: isNat^#(ok(X)) -> c_60(isNat^#(X)) , 40: isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) , 41: proper^#(zeros()) -> c_88() , 42: proper^#(0()) -> c_90() , 43: proper^#(tt()) -> c_92() , 44: proper^#(nil()) -> c_113() , 45: top^#(mark(X)) -> c_114(top^#(proper(X))) , 46: top^#(ok(X)) -> c_115(top^#(active(X))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cons^#(mark(X1), X2) -> c_49(cons^#(X1, X2)) , cons^#(ok(X1), ok(X2)) -> c_50(cons^#(X1, X2)) , U11^#(mark(X1), X2) -> c_51(U11^#(X1, X2)) , U11^#(ok(X1), ok(X2)) -> c_52(U11^#(X1, X2)) , U12^#(mark(X)) -> c_53(U12^#(X)) , U12^#(ok(X)) -> c_54(U12^#(X)) , U51^#(mark(X1), X2, X3) -> c_72(U51^#(X1, X2, X3)) , U51^#(ok(X1), ok(X2), ok(X3)) -> c_73(U51^#(X1, X2, X3)) , U21^#(mark(X1), X2) -> c_56(U21^#(X1, X2)) , U21^#(ok(X1), ok(X2)) -> c_57(U21^#(X1, X2)) , U22^#(mark(X)) -> c_58(U22^#(X)) , U22^#(ok(X)) -> c_59(U22^#(X)) , U31^#(mark(X1), X2) -> c_61(U31^#(X1, X2)) , U31^#(ok(X1), ok(X2)) -> c_62(U31^#(X1, X2)) , U32^#(mark(X)) -> c_63(U32^#(X)) , U32^#(ok(X)) -> c_64(U32^#(X)) , U41^#(mark(X1), X2, X3) -> c_65(U41^#(X1, X2, X3)) , U41^#(ok(X1), ok(X2), ok(X3)) -> c_66(U41^#(X1, X2, X3)) , U42^#(mark(X1), X2) -> c_67(U42^#(X1, X2)) , U42^#(ok(X1), ok(X2)) -> c_68(U42^#(X1, X2)) , U43^#(mark(X)) -> c_69(U43^#(X)) , U43^#(ok(X)) -> c_70(U43^#(X)) , U52^#(mark(X1), X2) -> c_74(U52^#(X1, X2)) , U52^#(ok(X1), ok(X2)) -> c_75(U52^#(X1, X2)) , U53^#(mark(X)) -> c_76(U53^#(X)) , U53^#(ok(X)) -> c_77(U53^#(X)) , U61^#(mark(X1), X2) -> c_78(U61^#(X1, X2)) , U61^#(ok(X1), ok(X2)) -> c_79(U61^#(X1, X2)) , s^#(mark(X)) -> c_80(s^#(X)) , s^#(ok(X)) -> c_81(s^#(X)) , length^#(mark(X)) -> c_82(length^#(X)) , length^#(ok(X)) -> c_83(length^#(X)) , and^#(mark(X1), X2) -> c_84(and^#(X1, X2)) , and^#(ok(X1), ok(X2)) -> c_85(and^#(X1, X2)) , isNatKind^#(ok(X)) -> c_87(isNatKind^#(X)) , isNatIListKind^#(ok(X)) -> c_86(isNatIListKind^#(X)) , isNatList^#(ok(X)) -> c_55(isNatList^#(X)) , isNat^#(ok(X)) -> c_60(isNat^#(X)) , isNatIList^#(ok(X)) -> c_71(isNatIList^#(X)) , top^#(mark(X)) -> c_114(top^#(proper(X))) , top^#(ok(X)) -> c_115(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(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)) -> U61(active(X1), X2) , active(U61(tt(), L)) -> mark(s(length(L))) , active(s(X)) -> s(active(X)) , active(length(X)) -> length(active(X)) , active(length(cons(N, L))) -> mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) , active(length(nil())) -> mark(0()) , 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(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) -> mark(U61(X1, X2)) , U61(ok(X1), ok(X2)) -> ok(U61(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)) , 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)) -> U61(proper(X1), proper(X2)) , proper(s(X)) -> s(proper(X)) , proper(length(X)) -> length(proper(X)) , proper(and(X1, X2)) -> and(proper(X1), proper(X2)) , proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) , proper(isNatKind(X)) -> isNatKind(proper(X)) , proper(nil()) -> ok(nil()) , top(mark(X)) -> top(proper(X)) , top(ok(X)) -> top(active(X)) } Weak DPs: { active^#(zeros()) -> c_1(cons^#(0(), zeros())) , proper^#(zeros()) -> c_88() , proper^#(0()) -> c_90() , proper^#(tt()) -> c_92() , proper^#(nil()) -> c_113() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..