MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } 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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , zeros^#() -> c_2() , cons^#(X1, X2) -> c_3(X1, X2) , 0^#() -> c_4() , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , U12^#(tt()) -> c_6() , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , isNatList^#(n__nil()) -> c_8() , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__nil()) -> c_16(nil^#()) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , length^#(nil()) -> c_39(0^#()) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__0()) -> c_47() , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U32^#(tt()) -> c_25() , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , U43^#(tt()) -> c_28() , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__zeros()) -> c_30() , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U53^#(tt()) -> c_34() , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , zeros^#() -> c_2() , cons^#(X1, X2) -> c_3(X1, X2) , 0^#() -> c_4() , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , U12^#(tt()) -> c_6() , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , isNatList^#(n__nil()) -> c_8() , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__nil()) -> c_16(nil^#()) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , length^#(nil()) -> c_39(0^#()) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__0()) -> c_47() , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U32^#(tt()) -> c_25() , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , U43^#(tt()) -> c_28() , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__zeros()) -> c_30() , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U53^#(tt()) -> c_34() , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,4,6,8,25,27,28,32,36,37,41,44,46,49} by applications of Pre({2,4,6,8,25,27,28,32,36,37,41,44,46,49}) = {3,5,10,11,12,16,17,19,20,22,23,24,29,31,33,34,35,40,43,48}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(cons^#(0(), n__zeros())) , 2: zeros^#() -> c_2() , 3: cons^#(X1, X2) -> c_3(X1, X2) , 4: 0^#() -> c_4() , 5: U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , 6: U12^#(tt()) -> c_6() , 7: isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 8: isNatList^#(n__nil()) -> c_8() , 9: U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , 10: activate^#(X) -> c_9(X) , 11: activate^#(n__zeros()) -> c_10(zeros^#()) , 12: activate^#(n__0()) -> c_11(0^#()) , 13: activate^#(n__length(X)) -> c_12(length^#(X)) , 14: activate^#(n__s(X)) -> c_13(s^#(X)) , 15: activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , 16: activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , 17: activate^#(n__nil()) -> c_16(nil^#()) , 18: activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , 19: activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , 20: length^#(X) -> c_37(X) , 21: length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , 22: length^#(nil()) -> c_39(0^#()) , 23: s^#(X) -> c_36(X) , 24: isNatIListKind^#(X) -> c_42(X) , 25: isNatIListKind^#(n__zeros()) -> c_43() , 26: isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , 27: isNatIListKind^#(n__nil()) -> c_45() , 28: nil^#() -> c_50() , 29: and^#(X1, X2) -> c_40(X1, X2) , 30: and^#(tt(), X) -> c_41(activate^#(X)) , 31: isNatKind^#(X) -> c_46(X) , 32: isNatKind^#(n__0()) -> c_47() , 33: isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , 34: isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , 35: U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , 36: U22^#(tt()) -> c_20() , 37: isNat^#(n__0()) -> c_21() , 38: isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , 39: isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , 40: U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , 41: U32^#(tt()) -> c_25() , 42: U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , 43: U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , 44: U43^#(tt()) -> c_28() , 45: isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , 46: isNatIList^#(n__zeros()) -> c_30() , 47: isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 48: U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , 49: U53^#(tt()) -> c_34() , 50: U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , cons^#(X1, X2) -> c_3(X1, X2) , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__nil()) -> c_16(nil^#()) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , length^#(nil()) -> c_39(0^#()) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , U12^#(tt()) -> c_6() , isNatList^#(n__nil()) -> c_8() , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , isNatKind^#(n__0()) -> c_47() , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , U32^#(tt()) -> c_25() , U43^#(tt()) -> c_28() , isNatIList^#(n__zeros()) -> c_30() , U53^#(tt()) -> c_34() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,8,13,18,27,30,32,35} by applications of Pre({3,8,13,18,27,30,32,35}) = {2,5,6,9,16,19,20,22,23,24,28,29,31,33}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(cons^#(0(), n__zeros())) , 2: cons^#(X1, X2) -> c_3(X1, X2) , 3: U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , 4: isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 5: U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , 6: activate^#(X) -> c_9(X) , 7: activate^#(n__zeros()) -> c_10(zeros^#()) , 8: activate^#(n__0()) -> c_11(0^#()) , 9: activate^#(n__length(X)) -> c_12(length^#(X)) , 10: activate^#(n__s(X)) -> c_13(s^#(X)) , 11: activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , 12: activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , 13: activate^#(n__nil()) -> c_16(nil^#()) , 14: activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , 15: activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , 16: length^#(X) -> c_37(X) , 17: length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , 18: length^#(nil()) -> c_39(0^#()) , 19: s^#(X) -> c_36(X) , 20: isNatIListKind^#(X) -> c_42(X) , 21: isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , 22: and^#(X1, X2) -> c_40(X1, X2) , 23: and^#(tt(), X) -> c_41(activate^#(X)) , 24: isNatKind^#(X) -> c_46(X) , 25: isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , 26: isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , 27: U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , 28: isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , 29: isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , 30: U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , 31: U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , 32: U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , 33: isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , 34: isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 35: U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , 36: U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) , 37: zeros^#() -> c_2() , 38: 0^#() -> c_4() , 39: U12^#(tt()) -> c_6() , 40: isNatList^#(n__nil()) -> c_8() , 41: isNatIListKind^#(n__zeros()) -> c_43() , 42: isNatIListKind^#(n__nil()) -> c_45() , 43: nil^#() -> c_50() , 44: isNatKind^#(n__0()) -> c_47() , 45: U22^#(tt()) -> c_20() , 46: isNat^#(n__0()) -> c_21() , 47: U32^#(tt()) -> c_25() , 48: U43^#(tt()) -> c_28() , 49: isNatIList^#(n__zeros()) -> c_30() , 50: U53^#(tt()) -> c_34() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , cons^#(X1, X2) -> c_3(X1, X2) , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , U12^#(tt()) -> c_6() , isNatList^#(n__nil()) -> c_8() , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__nil()) -> c_16(nil^#()) , length^#(nil()) -> c_39(0^#()) , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , isNatKind^#(n__0()) -> c_47() , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U32^#(tt()) -> c_25() , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , U43^#(tt()) -> c_28() , isNatIList^#(n__zeros()) -> c_30() , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U53^#(tt()) -> c_34() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,23,24,25,26} by applications of Pre({4,23,24,25,26}) = {2,3,5,13,15,16,18,20,27}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(cons^#(0(), n__zeros())) , 2: cons^#(X1, X2) -> c_3(X1, X2) , 3: isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 4: U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , 5: activate^#(X) -> c_9(X) , 6: activate^#(n__zeros()) -> c_10(zeros^#()) , 7: activate^#(n__length(X)) -> c_12(length^#(X)) , 8: activate^#(n__s(X)) -> c_13(s^#(X)) , 9: activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , 10: activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , 11: activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , 12: activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , 13: length^#(X) -> c_37(X) , 14: length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , 15: s^#(X) -> c_36(X) , 16: isNatIListKind^#(X) -> c_42(X) , 17: isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , 18: and^#(X1, X2) -> c_40(X1, X2) , 19: and^#(tt(), X) -> c_41(activate^#(X)) , 20: isNatKind^#(X) -> c_46(X) , 21: isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , 22: isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , 23: isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , 24: isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , 25: U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , 26: isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , 27: isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 28: U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) , 29: zeros^#() -> c_2() , 30: 0^#() -> c_4() , 31: U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , 32: U12^#(tt()) -> c_6() , 33: isNatList^#(n__nil()) -> c_8() , 34: activate^#(n__0()) -> c_11(0^#()) , 35: activate^#(n__nil()) -> c_16(nil^#()) , 36: length^#(nil()) -> c_39(0^#()) , 37: isNatIListKind^#(n__zeros()) -> c_43() , 38: isNatIListKind^#(n__nil()) -> c_45() , 39: nil^#() -> c_50() , 40: isNatKind^#(n__0()) -> c_47() , 41: U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , 42: U22^#(tt()) -> c_20() , 43: isNat^#(n__0()) -> c_21() , 44: U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , 45: U32^#(tt()) -> c_25() , 46: U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , 47: U43^#(tt()) -> c_28() , 48: isNatIList^#(n__zeros()) -> c_30() , 49: U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , 50: U53^#(tt()) -> c_34() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , cons^#(X1, X2) -> c_3(X1, X2) , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , U12^#(tt()) -> c_6() , isNatList^#(n__nil()) -> c_8() , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__nil()) -> c_16(nil^#()) , length^#(nil()) -> c_39(0^#()) , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , isNatKind^#(n__0()) -> c_47() , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U32^#(tt()) -> c_25() , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , U43^#(tt()) -> c_28() , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__zeros()) -> c_30() , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U53^#(tt()) -> c_34() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,22} by applications of Pre({3,22}) = {2,4,12,14,15,17,19}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(cons^#(0(), n__zeros())) , 2: cons^#(X1, X2) -> c_3(X1, X2) , 3: isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 4: activate^#(X) -> c_9(X) , 5: activate^#(n__zeros()) -> c_10(zeros^#()) , 6: activate^#(n__length(X)) -> c_12(length^#(X)) , 7: activate^#(n__s(X)) -> c_13(s^#(X)) , 8: activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , 9: activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , 10: activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , 11: activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , 12: length^#(X) -> c_37(X) , 13: length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , 14: s^#(X) -> c_36(X) , 15: isNatIListKind^#(X) -> c_42(X) , 16: isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , 17: and^#(X1, X2) -> c_40(X1, X2) , 18: and^#(tt(), X) -> c_41(activate^#(X)) , 19: isNatKind^#(X) -> c_46(X) , 20: isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , 21: isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , 22: isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , 23: U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) , 24: zeros^#() -> c_2() , 25: 0^#() -> c_4() , 26: U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , 27: U12^#(tt()) -> c_6() , 28: isNatList^#(n__nil()) -> c_8() , 29: U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , 30: activate^#(n__0()) -> c_11(0^#()) , 31: activate^#(n__nil()) -> c_16(nil^#()) , 32: length^#(nil()) -> c_39(0^#()) , 33: isNatIListKind^#(n__zeros()) -> c_43() , 34: isNatIListKind^#(n__nil()) -> c_45() , 35: nil^#() -> c_50() , 36: isNatKind^#(n__0()) -> c_47() , 37: U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , 38: U22^#(tt()) -> c_20() , 39: isNat^#(n__0()) -> c_21() , 40: isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , 41: isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , 42: U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , 43: U32^#(tt()) -> c_25() , 44: U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , 45: U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , 46: U43^#(tt()) -> c_28() , 47: isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , 48: isNatIList^#(n__zeros()) -> c_30() , 49: U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , 50: U53^#(tt()) -> c_34() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(cons^#(0(), n__zeros())) , cons^#(X1, X2) -> c_3(X1, X2) , activate^#(X) -> c_9(X) , activate^#(n__zeros()) -> c_10(zeros^#()) , activate^#(n__length(X)) -> c_12(length^#(X)) , activate^#(n__s(X)) -> c_13(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_14(cons^#(X1, X2)) , activate^#(n__isNatIListKind(X)) -> c_15(isNatIListKind^#(X)) , activate^#(n__and(X1, X2)) -> c_17(and^#(X1, X2)) , activate^#(n__isNatKind(X)) -> c_18(isNatKind^#(X)) , length^#(X) -> c_37(X) , length^#(cons(N, L)) -> c_38(U61^#(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))) , s^#(X) -> c_36(X) , isNatIListKind^#(X) -> c_42(X) , isNatIListKind^#(n__cons(V1, V2)) -> c_44(and^#(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))) , and^#(X1, X2) -> c_40(X1, X2) , and^#(tt(), X) -> c_41(activate^#(X)) , isNatKind^#(X) -> c_46(X) , isNatKind^#(n__length(V1)) -> c_48(isNatIListKind^#(activate(V1))) , isNatKind^#(n__s(V1)) -> c_49(isNatKind^#(activate(V1))) , U61^#(tt(), L) -> c_35(s^#(length(activate(L)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), V1) -> U12(isNatList(activate(V1))) , U12(tt()) -> tt() , isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , isNatList(n__nil()) -> tt() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIListKind(X)) -> isNatIListKind(X) , activate(n__nil()) -> nil() , activate(n__and(X1, X2)) -> and(X1, X2) , activate(n__isNatKind(X)) -> isNatKind(X) , U21(tt(), V1) -> U22(isNat(activate(V1))) , U22(tt()) -> tt() , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) , U31(tt(), V) -> U32(isNatList(activate(V))) , U32(tt()) -> tt() , U41(tt(), V1, V2) -> U42(isNat(activate(V1)), activate(V2)) , U42(tt(), V2) -> U43(isNatIList(activate(V2))) , U43(tt()) -> tt() , isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) , U51(tt(), V1, V2) -> U52(isNat(activate(V1)), activate(V2)) , U52(tt(), V2) -> U53(isNatList(activate(V2))) , U53(tt()) -> tt() , U61(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) , length(nil()) -> 0() , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNatIListKind(X) -> n__isNatIListKind(X) , isNatIListKind(n__zeros()) -> tt() , isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) , isNatIListKind(n__nil()) -> tt() , isNatKind(X) -> n__isNatKind(X) , isNatKind(n__0()) -> tt() , isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) , isNatKind(n__s(V1)) -> isNatKind(activate(V1)) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , U11^#(tt(), V1) -> c_5(U12^#(isNatList(activate(V1)))) , U12^#(tt()) -> c_6() , isNatList^#(n__cons(V1, V2)) -> c_7(U51^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , isNatList^#(n__nil()) -> c_8() , U51^#(tt(), V1, V2) -> c_32(U52^#(isNat(activate(V1)), activate(V2))) , activate^#(n__0()) -> c_11(0^#()) , activate^#(n__nil()) -> c_16(nil^#()) , length^#(nil()) -> c_39(0^#()) , isNatIListKind^#(n__zeros()) -> c_43() , isNatIListKind^#(n__nil()) -> c_45() , nil^#() -> c_50() , isNatKind^#(n__0()) -> c_47() , U21^#(tt(), V1) -> c_19(U22^#(isNat(activate(V1)))) , U22^#(tt()) -> c_20() , isNat^#(n__0()) -> c_21() , isNat^#(n__length(V1)) -> c_22(U11^#(isNatIListKind(activate(V1)), activate(V1))) , isNat^#(n__s(V1)) -> c_23(U21^#(isNatKind(activate(V1)), activate(V1))) , U31^#(tt(), V) -> c_24(U32^#(isNatList(activate(V)))) , U32^#(tt()) -> c_25() , U41^#(tt(), V1, V2) -> c_26(U42^#(isNat(activate(V1)), activate(V2))) , U42^#(tt(), V2) -> c_27(U43^#(isNatIList(activate(V2)))) , U43^#(tt()) -> c_28() , isNatIList^#(V) -> c_29(U31^#(isNatIListKind(activate(V)), activate(V))) , isNatIList^#(n__zeros()) -> c_30() , isNatIList^#(n__cons(V1, V2)) -> c_31(U41^#(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))) , U52^#(tt(), V2) -> c_33(U53^#(isNatList(activate(V2)))) , U53^#(tt()) -> c_34() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..