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__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , 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__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))) , 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(), 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__0()) -> c_12(0^#()) , activate^#(n__length(X)) -> c_13(length^#(X)) , activate^#(n__s(X)) -> c_14(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , activate^#(n__nil()) -> c_17(nil^#()) , activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , isNatIList^#(X) -> c_29(X) , isNatIList^#(n__zeros()) -> c_30() , isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , nil^#() -> c_32() , isNatList^#(X) -> c_25(X) , isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNatList^#(n__nil()) -> c_27() , isNat^#(X) -> c_21(X) , isNat^#(n__0()) -> c_22() , isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , and^#(tt(), X) -> c_20(activate^#(X)) } 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__0()) -> c_12(0^#()) , activate^#(n__length(X)) -> c_13(length^#(X)) , activate^#(n__s(X)) -> c_14(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , activate^#(n__nil()) -> c_17(nil^#()) , activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , isNatIList^#(X) -> c_29(X) , isNatIList^#(n__zeros()) -> c_30() , isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , nil^#() -> c_32() , isNatList^#(X) -> c_25(X) , isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNatList^#(n__nil()) -> c_27() , isNat^#(X) -> c_21(X) , isNat^#(n__0()) -> c_22() , isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , and^#(tt(), X) -> c_20(activate^#(X)) } 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__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , 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__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))) , nil() -> n__nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,4,22,24,27,29} by applications of Pre({2,4,22,24,27,29}) = {3,6,7,9,10,11,12,16,17,18,19,20,21,25,28,30,31}. 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__0()) -> c_12(0^#()) , 13: activate^#(n__length(X)) -> c_13(length^#(X)) , 14: activate^#(n__s(X)) -> c_14(s^#(X)) , 15: activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , 16: activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , 17: activate^#(n__nil()) -> c_17(nil^#()) , 18: activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , 19: activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , 20: isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , 21: isNatIList^#(X) -> c_29(X) , 22: isNatIList^#(n__zeros()) -> c_30() , 23: isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 24: nil^#() -> c_32() , 25: isNatList^#(X) -> c_25(X) , 26: isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , 27: isNatList^#(n__nil()) -> c_27() , 28: isNat^#(X) -> c_21(X) , 29: isNat^#(n__0()) -> c_22() , 30: isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , 31: isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , 32: and^#(tt(), X) -> c_20(activate^#(X)) } 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__0()) -> c_12(0^#()) , activate^#(n__length(X)) -> c_13(length^#(X)) , activate^#(n__s(X)) -> c_14(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , activate^#(n__nil()) -> c_17(nil^#()) , activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , isNatIList^#(X) -> c_29(X) , isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(X) -> c_25(X) , isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNat^#(X) -> c_21(X) , isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , and^#(tt(), X) -> c_20(activate^#(X)) } 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__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , 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__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))) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , isNatIList^#(n__zeros()) -> c_30() , nil^#() -> c_32() , isNatList^#(n__nil()) -> c_27() , isNat^#(n__0()) -> c_22() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,10,15} by applications of Pre({7,10,15}) = {2,4,5,8,11,19,21,23,26}. 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__0()) -> c_12(0^#()) , 11: activate^#(n__length(X)) -> c_13(length^#(X)) , 12: activate^#(n__s(X)) -> c_14(s^#(X)) , 13: activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , 14: activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , 15: activate^#(n__nil()) -> c_17(nil^#()) , 16: activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , 17: activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , 18: isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , 19: isNatIList^#(X) -> c_29(X) , 20: isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , 21: isNatList^#(X) -> c_25(X) , 22: isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , 23: isNat^#(X) -> c_21(X) , 24: isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , 25: isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , 26: and^#(tt(), X) -> c_20(activate^#(X)) , 27: zeros^#() -> c_2() , 28: 0^#() -> c_4() , 29: isNatIList^#(n__zeros()) -> c_30() , 30: nil^#() -> c_32() , 31: isNatList^#(n__nil()) -> c_27() , 32: isNat^#(n__0()) -> c_22() } 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__length(X)) -> c_13(length^#(X)) , activate^#(n__s(X)) -> c_14(s^#(X)) , activate^#(n__cons(X1, X2)) -> c_15(cons^#(X1, X2)) , activate^#(n__isNatIList(X)) -> c_16(isNatIList^#(X)) , activate^#(n__isNatList(X)) -> c_18(isNatList^#(X)) , activate^#(n__isNat(X)) -> c_19(isNat^#(X)) , isNatIList^#(V) -> c_28(isNatList^#(activate(V))) , isNatIList^#(X) -> c_29(X) , isNatIList^#(n__cons(V1, V2)) -> c_31(and^#(isNat(activate(V1)), n__isNatIList(activate(V2)))) , isNatList^#(X) -> c_25(X) , isNatList^#(n__cons(V1, V2)) -> c_26(and^#(isNat(activate(V1)), n__isNatList(activate(V2)))) , isNat^#(X) -> c_21(X) , isNat^#(n__length(V1)) -> c_23(isNatList^#(activate(V1))) , isNat^#(n__s(V1)) -> c_24(isNat^#(activate(V1))) , and^#(tt(), X) -> c_20(activate^#(X)) } 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__0()) -> 0() , activate(n__length(X)) -> length(X) , activate(n__s(X)) -> s(X) , activate(n__cons(X1, X2)) -> cons(X1, X2) , activate(n__isNatIList(X)) -> isNatIList(X) , activate(n__nil()) -> nil() , activate(n__isNatList(X)) -> isNatList(X) , activate(n__isNat(X)) -> isNat(X) , 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__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))) , nil() -> n__nil() } Weak DPs: { zeros^#() -> c_2() , 0^#() -> c_4() , length^#(nil()) -> c_9(0^#()) , activate^#(n__0()) -> c_12(0^#()) , activate^#(n__nil()) -> c_17(nil^#()) , isNatIList^#(n__zeros()) -> c_30() , nil^#() -> c_32() , isNatList^#(n__nil()) -> c_27() , isNat^#(n__0()) -> c_22() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..