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(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)
  , 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 minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

3) '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^#(activate(X)))
     , activate^#(n__s(X)) -> c_14(s^#(activate(X)))
     , activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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^#(activate(X)))
     , activate^#(n__s(X)) -> c_14(s^#(activate(X)))
     , activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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(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)
     , 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^#(activate(X)))
       , 14: activate^#(n__s(X)) -> c_14(s^#(activate(X)))
       , 15: activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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^#(activate(X)))
     , activate^#(n__s(X)) -> c_14(s^#(activate(X)))
     , activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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(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)
     , 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^#(activate(X)))
       , 12: activate^#(n__s(X)) -> c_14(s^#(activate(X)))
       , 13: activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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^#(activate(X)))
     , activate^#(n__s(X)) -> c_14(s^#(activate(X)))
     , activate^#(n__cons(X1, X2)) -> c_15(cons^#(activate(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(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)
     , 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..