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(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) , length(nil()) -> 0() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) , activate(n__0()) -> 0() , activate(n__length(X)) -> length(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , activate(n__and(X1, X2)) -> and(activate(X1), X2) , U21(tt()) -> nil() , nil() -> n__nil() , U31(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> isNatList(activate(V1)) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNatList(X) -> n__isNatList(X) , isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) , isNatList(n__nil()) -> tt() , isNatIList(V) -> isNatList(activate(V)) , isNatIList(X) -> n__isNatIList(X) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , take(X1, X2) -> n__take(X1, X2) , take(0(), IL) -> U21(isNatIList(IL)) , take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) } 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(), L) -> c_5(s^#(length(activate(L)))) , s^#(X) -> c_6(X) , length^#(X) -> c_7(X) , length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , length^#(nil()) -> c_9(0^#()) , activate^#(X) -> c_10(X) , activate^#(n__zeros()) -> c_11(zeros^#()) , activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , activate^#(n__0()) -> c_13(0^#()) , activate^#(n__length(X)) -> c_14(length^#(activate(X))) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , activate^#(n__nil()) -> c_18(nil^#()) , activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , take^#(X1, X2) -> c_39(X1, X2) , take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , isNatIList^#(X) -> c_36(X) , isNatIList^#(n__zeros()) -> c_37() , isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , nil^#() -> c_23() , isNatList^#(X) -> c_31(X) , isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNatList^#(n__nil()) -> c_34() , isNat^#(X) -> c_27(X) , isNat^#(n__0()) -> c_28() , isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , and^#(X1, X2) -> c_25(X1, X2) , and^#(tt(), X) -> c_26(activate^#(X)) , U21^#(tt()) -> c_22(nil^#()) , U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } 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(), L) -> c_5(s^#(length(activate(L)))) , s^#(X) -> c_6(X) , length^#(X) -> c_7(X) , length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , length^#(nil()) -> c_9(0^#()) , activate^#(X) -> c_10(X) , activate^#(n__zeros()) -> c_11(zeros^#()) , activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , activate^#(n__0()) -> c_13(0^#()) , activate^#(n__length(X)) -> c_14(length^#(activate(X))) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , activate^#(n__nil()) -> c_18(nil^#()) , activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , take^#(X1, X2) -> c_39(X1, X2) , take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , isNatIList^#(X) -> c_36(X) , isNatIList^#(n__zeros()) -> c_37() , isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , nil^#() -> c_23() , isNatList^#(X) -> c_31(X) , isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNatList^#(n__nil()) -> c_34() , isNat^#(X) -> c_27(X) , isNat^#(n__0()) -> c_28() , isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , and^#(X1, X2) -> c_25(X1, X2) , and^#(tt(), X) -> c_26(activate^#(X)) , U21^#(tt()) -> c_22(nil^#()) , U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) , length(nil()) -> 0() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) , activate(n__0()) -> 0() , activate(n__length(X)) -> length(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , activate(n__and(X1, X2)) -> and(activate(X1), X2) , U21(tt()) -> nil() , nil() -> n__nil() , U31(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> isNatList(activate(V1)) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNatList(X) -> n__isNatList(X) , isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) , isNatList(n__nil()) -> tt() , isNatIList(V) -> isNatList(activate(V)) , isNatIList(X) -> n__isNatIList(X) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , take(X1, X2) -> n__take(X1, X2) , take(0(), IL) -> U21(isNatIList(IL)) , take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,4,27,29,33,35} by applications of Pre({2,4,27,29,33,35}) = {3,6,7,9,10,11,13,17,18,19,20,22,25,26,30,34,36,37,38,40}. 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(), L) -> c_5(s^#(length(activate(L)))) , 6: s^#(X) -> c_6(X) , 7: length^#(X) -> c_7(X) , 8: length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , 9: length^#(nil()) -> c_9(0^#()) , 10: activate^#(X) -> c_10(X) , 11: activate^#(n__zeros()) -> c_11(zeros^#()) , 12: activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , 13: activate^#(n__0()) -> c_13(0^#()) , 14: activate^#(n__length(X)) -> c_14(length^#(activate(X))) , 15: activate^#(n__s(X)) -> c_15(s^#(activate(X))) , 16: activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , 17: activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , 18: activate^#(n__nil()) -> c_18(nil^#()) , 19: activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , 20: activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , 21: activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , 22: take^#(X1, X2) -> c_39(X1, X2) , 23: take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , 24: take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , 25: isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , 26: isNatIList^#(X) -> c_36(X) , 27: isNatIList^#(n__zeros()) -> c_37() , 28: isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 29: nil^#() -> c_23() , 30: isNatList^#(X) -> c_31(X) , 31: isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 32: isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , 33: isNatList^#(n__nil()) -> c_34() , 34: isNat^#(X) -> c_27(X) , 35: isNat^#(n__0()) -> c_28() , 36: isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , 37: isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , 38: and^#(X1, X2) -> c_25(X1, X2) , 39: and^#(tt(), X) -> c_26(activate^#(X)) , 40: U21^#(tt()) -> c_22(nil^#()) , 41: U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } 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(), L) -> c_5(s^#(length(activate(L)))) , s^#(X) -> c_6(X) , length^#(X) -> c_7(X) , length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , length^#(nil()) -> c_9(0^#()) , activate^#(X) -> c_10(X) , activate^#(n__zeros()) -> c_11(zeros^#()) , activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , activate^#(n__0()) -> c_13(0^#()) , activate^#(n__length(X)) -> c_14(length^#(activate(X))) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , activate^#(n__nil()) -> c_18(nil^#()) , activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , take^#(X1, X2) -> c_39(X1, X2) , take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , isNatIList^#(X) -> c_36(X) , isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(X) -> c_31(X) , isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNat^#(X) -> c_27(X) , isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , and^#(X1, X2) -> c_25(X1, X2) , and^#(tt(), X) -> c_26(activate^#(X)) , U21^#(tt()) -> c_22(nil^#()) , U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) , length(nil()) -> 0() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) , activate(n__0()) -> 0() , activate(n__length(X)) -> length(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , activate(n__and(X1, X2)) -> and(activate(X1), X2) , U21(tt()) -> nil() , nil() -> n__nil() , U31(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> isNatList(activate(V1)) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNatList(X) -> n__isNatList(X) , isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) , isNatList(n__nil()) -> tt() , isNatIList(V) -> isNatList(activate(V)) , isNatIList(X) -> n__isNatIList(X) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , take(X1, X2) -> n__take(X1, X2) , take(0(), IL) -> U21(isNatIList(IL)) , take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , isNatIList^#(n__zeros()) -> c_37() , nil^#() -> c_23() , isNatList^#(n__nil()) -> c_34() , isNat^#(n__0()) -> c_28() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,11,16,34} by applications of Pre({7,11,16,34}) = {2,4,5,8,12,20,21,24,26,29,32,33}. 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(), L) -> c_5(s^#(length(activate(L)))) , 4: s^#(X) -> c_6(X) , 5: length^#(X) -> c_7(X) , 6: length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , 7: length^#(nil()) -> c_9(0^#()) , 8: activate^#(X) -> c_10(X) , 9: activate^#(n__zeros()) -> c_11(zeros^#()) , 10: activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , 11: activate^#(n__0()) -> c_13(0^#()) , 12: activate^#(n__length(X)) -> c_14(length^#(activate(X))) , 13: activate^#(n__s(X)) -> c_15(s^#(activate(X))) , 14: activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , 15: activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , 16: activate^#(n__nil()) -> c_18(nil^#()) , 17: activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , 18: activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , 19: activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , 20: take^#(X1, X2) -> c_39(X1, X2) , 21: take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , 22: take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , 23: isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , 24: isNatIList^#(X) -> c_36(X) , 25: isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 26: isNatList^#(X) -> c_31(X) , 27: isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 28: isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , 29: isNat^#(X) -> c_27(X) , 30: isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , 31: isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , 32: and^#(X1, X2) -> c_25(X1, X2) , 33: and^#(tt(), X) -> c_26(activate^#(X)) , 34: U21^#(tt()) -> c_22(nil^#()) , 35: U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) , 36: zeros^#() -> c_2() , 37: 0^#() -> c_4() , 38: isNatIList^#(n__zeros()) -> c_37() , 39: nil^#() -> c_23() , 40: isNatList^#(n__nil()) -> c_34() , 41: isNat^#(n__0()) -> c_28() } 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(), L) -> c_5(s^#(length(activate(L)))) , s^#(X) -> c_6(X) , length^#(X) -> c_7(X) , length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , activate^#(X) -> c_10(X) , activate^#(n__zeros()) -> c_11(zeros^#()) , activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , activate^#(n__length(X)) -> c_14(length^#(activate(X))) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , take^#(X1, X2) -> c_39(X1, X2) , take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , isNatIList^#(X) -> c_36(X) , isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(X) -> c_31(X) , isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNat^#(X) -> c_27(X) , isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , and^#(X1, X2) -> c_25(X1, X2) , and^#(tt(), X) -> c_26(activate^#(X)) , U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) , length(nil()) -> 0() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) , activate(n__0()) -> 0() , activate(n__length(X)) -> length(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , activate(n__and(X1, X2)) -> and(activate(X1), X2) , U21(tt()) -> nil() , nil() -> n__nil() , U31(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> isNatList(activate(V1)) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNatList(X) -> n__isNatList(X) , isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) , isNatList(n__nil()) -> tt() , isNatIList(V) -> isNatList(activate(V)) , isNatIList(X) -> n__isNatIList(X) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , take(X1, X2) -> n__take(X1, X2) , take(0(), IL) -> U21(isNatIList(IL)) , take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , length^#(nil()) -> c_9(0^#()) , activate^#(n__0()) -> c_13(0^#()) , activate^#(n__nil()) -> c_18(nil^#()) , isNatIList^#(n__zeros()) -> c_37() , nil^#() -> c_23() , isNatList^#(n__nil()) -> c_34() , isNat^#(n__0()) -> c_28() , U21^#(tt()) -> c_22(nil^#()) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {18} by applications of Pre({18}) = {2,4,5,7,9,17,21,23,26,29}. 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(), L) -> c_5(s^#(length(activate(L)))) , 4: s^#(X) -> c_6(X) , 5: length^#(X) -> c_7(X) , 6: length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , 7: activate^#(X) -> c_10(X) , 8: activate^#(n__zeros()) -> c_11(zeros^#()) , 9: activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , 10: activate^#(n__length(X)) -> c_14(length^#(activate(X))) , 11: activate^#(n__s(X)) -> c_15(s^#(activate(X))) , 12: activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , 13: activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , 14: activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , 15: activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , 16: activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , 17: take^#(X1, X2) -> c_39(X1, X2) , 18: take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , 19: take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , 20: isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , 21: isNatIList^#(X) -> c_36(X) , 22: isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 23: isNatList^#(X) -> c_31(X) , 24: isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 25: isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , 26: isNat^#(X) -> c_27(X) , 27: isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , 28: isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , 29: and^#(X1, X2) -> c_25(X1, X2) , 30: and^#(tt(), X) -> c_26(activate^#(X)) , 31: U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) , 32: zeros^#() -> c_2() , 33: 0^#() -> c_4() , 34: length^#(nil()) -> c_9(0^#()) , 35: activate^#(n__0()) -> c_13(0^#()) , 36: activate^#(n__nil()) -> c_18(nil^#()) , 37: isNatIList^#(n__zeros()) -> c_37() , 38: nil^#() -> c_23() , 39: isNatList^#(n__nil()) -> c_34() , 40: isNat^#(n__0()) -> c_28() , 41: U21^#(tt()) -> c_22(nil^#()) } 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(), L) -> c_5(s^#(length(activate(L)))) , s^#(X) -> c_6(X) , length^#(X) -> c_7(X) , length^#(cons(N, L)) -> c_8(U11^#(and(isNatList(activate(L)), n__isNat(N)), activate(L))) , activate^#(X) -> c_10(X) , activate^#(n__zeros()) -> c_11(zeros^#()) , activate^#(n__take(X1, X2)) -> c_12(take^#(activate(X1), activate(X2))) , activate^#(n__length(X)) -> c_14(length^#(activate(X))) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__cons(X1, X2)) -> c_16(cons^#(activate(X1), X2)) , activate^#(n__isNatIList(X)) -> c_17(isNatIList^#(X)) , activate^#(n__isNatList(X)) -> c_19(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_20(isNat^#(X)) , activate^#(n__and(X1, X2)) -> c_21(and^#(activate(X1), X2)) , take^#(X1, X2) -> c_39(X1, X2) , take^#(s(M), cons(N, IL)) -> c_41(U31^#(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N)) , isNatIList^#(V) -> c_35(isNatList^#(activate(V))) , isNatIList^#(X) -> c_36(X) , isNatIList^#(n__cons(V1, V2)) -> c_38(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(X) -> c_31(X) , isNatList^#(n__take(V1, V2)) -> c_32(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(n__cons(V1, V2)) -> c_33(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNat^#(X) -> c_27(X) , isNat^#(n__length(V1)) -> c_29(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_30(isNat^#(activate(V1))) , and^#(X1, X2) -> c_25(X1, X2) , and^#(tt(), X) -> c_26(activate^#(X)) , U31^#(tt(), IL, M, N) -> c_24(cons^#(activate(N), n__take(activate(M), activate(IL)))) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , cons(X1, X2) -> n__cons(X1, X2) , 0() -> n__0() , U11(tt(), L) -> s(length(activate(L))) , s(X) -> n__s(X) , length(X) -> n__length(X) , length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) , length(nil()) -> 0() , activate(X) -> X , activate(n__zeros()) -> zeros() , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) , activate(n__0()) -> 0() , activate(n__length(X)) -> length(activate(X)) , activate(n__s(X)) -> s(activate(X)) , activate(n__cons(X1, X2)) -> cons(activate(X1), X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , activate(n__and(X1, X2)) -> and(activate(X1), X2) , U21(tt()) -> nil() , nil() -> n__nil() , U31(tt(), IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) , and(X1, X2) -> n__and(X1, X2) , and(tt(), X) -> activate(X) , isNat(X) -> n__isNat(X) , isNat(n__0()) -> tt() , isNat(n__length(V1)) -> isNatList(activate(V1)) , isNat(n__s(V1)) -> isNat(activate(V1)) , isNatList(X) -> n__isNatList(X) , isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) , isNatList(n__nil()) -> tt() , isNatIList(V) -> isNatList(activate(V)) , isNatIList(X) -> n__isNatIList(X) , isNatIList(n__zeros()) -> tt() , isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) , take(X1, X2) -> n__take(X1, X2) , take(0(), IL) -> U21(isNatIList(IL)) , take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , length^#(nil()) -> c_9(0^#()) , activate^#(n__0()) -> c_13(0^#()) , activate^#(n__nil()) -> c_18(nil^#()) , take^#(0(), IL) -> c_40(U21^#(isNatIList(IL))) , isNatIList^#(n__zeros()) -> c_37() , nil^#() -> c_23() , isNatList^#(n__nil()) -> c_34() , isNat^#(n__0()) -> c_28() , U21^#(tt()) -> c_22(nil^#()) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..