MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { active(and(X1, X2)) -> and(X1, active(X2))
  , active(and(X1, X2)) -> and(active(X1), X2)
  , active(and(tt(), T)) -> mark(T)
  , active(isNatIList(IL)) -> mark(isNatList(IL))
  , active(isNatIList(zeros())) -> mark(tt())
  , active(isNatIList(cons(N, IL))) ->
    mark(and(isNat(N), isNatIList(IL)))
  , active(isNatList(cons(N, L))) ->
    mark(and(isNat(N), isNatList(L)))
  , active(isNatList(nil())) -> mark(tt())
  , active(isNatList(take(N, IL))) ->
    mark(and(isNat(N), isNatIList(IL)))
  , active(isNat(0())) -> mark(tt())
  , active(isNat(s(N))) -> mark(isNat(N))
  , active(isNat(length(L))) -> mark(isNatList(L))
  , active(s(X)) -> s(active(X))
  , active(length(X)) -> length(active(X))
  , active(length(cons(N, L))) ->
    mark(uLength(and(isNat(N), isNatList(L)), L))
  , active(zeros()) -> mark(cons(0(), zeros()))
  , active(cons(X1, X2)) -> cons(active(X1), X2)
  , active(take(X1, X2)) -> take(X1, active(X2))
  , active(take(X1, X2)) -> take(active(X1), X2)
  , active(take(0(), IL)) -> mark(uTake1(isNatIList(IL)))
  , active(take(s(M), cons(N, IL))) ->
    mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))),
                M,
                N,
                IL))
  , active(uTake1(X)) -> uTake1(active(X))
  , active(uTake1(tt())) -> mark(nil())
  , active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4)
  , active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL)))
  , active(uLength(X1, X2)) -> uLength(active(X1), X2)
  , active(uLength(tt(), L)) -> mark(s(length(L)))
  , and(X1, mark(X2)) -> mark(and(X1, X2))
  , and(mark(X1), X2) -> mark(and(X1, X2))
  , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
  , isNatIList(ok(X)) -> ok(isNatIList(X))
  , isNatList(ok(X)) -> ok(isNatList(X))
  , isNat(ok(X)) -> ok(isNat(X))
  , s(mark(X)) -> mark(s(X))
  , s(ok(X)) -> ok(s(X))
  , length(mark(X)) -> mark(length(X))
  , length(ok(X)) -> ok(length(X))
  , cons(mark(X1), X2) -> mark(cons(X1, X2))
  , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
  , take(X1, mark(X2)) -> mark(take(X1, X2))
  , take(mark(X1), X2) -> mark(take(X1, X2))
  , take(ok(X1), ok(X2)) -> ok(take(X1, X2))
  , uTake1(mark(X)) -> mark(uTake1(X))
  , uTake1(ok(X)) -> ok(uTake1(X))
  , uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4))
  , uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) ->
    ok(uTake2(X1, X2, X3, X4))
  , uLength(mark(X1), X2) -> mark(uLength(X1, X2))
  , uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2))
  , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
  , proper(tt()) -> ok(tt())
  , proper(isNatIList(X)) -> isNatIList(proper(X))
  , proper(isNatList(X)) -> isNatList(proper(X))
  , proper(isNat(X)) -> isNat(proper(X))
  , proper(0()) -> ok(0())
  , proper(s(X)) -> s(proper(X))
  , proper(length(X)) -> length(proper(X))
  , proper(zeros()) -> ok(zeros())
  , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
  , proper(nil()) -> ok(nil())
  , proper(take(X1, X2)) -> take(proper(X1), proper(X2))
  , proper(uTake1(X)) -> uTake1(proper(X))
  , proper(uTake2(X1, X2, X3, X4)) ->
    uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
  , proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2))
  , top(mark(X)) -> top(proper(X))
  , top(ok(X)) -> top(active(X)) }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
   following reason:
   
   Computation stopped due to timeout after 60.0 seconds.

2) 'Best' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
      failed due to the following reason:
      
      Computation stopped due to timeout after 30.0 seconds.
   
   2) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) '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
      
      2) '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
      
   
   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:
     { active^#(and(X1, X2)) -> c_1(and^#(X1, active(X2)))
     , active^#(and(X1, X2)) -> c_2(and^#(active(X1), X2))
     , active^#(and(tt(), T)) -> c_3(T)
     , active^#(isNatIList(IL)) -> c_4(isNatList^#(IL))
     , active^#(isNatIList(zeros())) -> c_5()
     , active^#(isNatIList(cons(N, IL))) ->
       c_6(and^#(isNat(N), isNatIList(IL)))
     , active^#(isNatList(cons(N, L))) ->
       c_7(and^#(isNat(N), isNatList(L)))
     , active^#(isNatList(nil())) -> c_8()
     , active^#(isNatList(take(N, IL))) ->
       c_9(and^#(isNat(N), isNatIList(IL)))
     , active^#(isNat(0())) -> c_10()
     , active^#(isNat(s(N))) -> c_11(isNat^#(N))
     , active^#(isNat(length(L))) -> c_12(isNatList^#(L))
     , active^#(s(X)) -> c_13(s^#(active(X)))
     , active^#(length(X)) -> c_14(length^#(active(X)))
     , active^#(length(cons(N, L))) ->
       c_15(uLength^#(and(isNat(N), isNatList(L)), L))
     , active^#(zeros()) -> c_16(cons^#(0(), zeros()))
     , active^#(cons(X1, X2)) -> c_17(cons^#(active(X1), X2))
     , active^#(take(X1, X2)) -> c_18(take^#(X1, active(X2)))
     , active^#(take(X1, X2)) -> c_19(take^#(active(X1), X2))
     , active^#(take(0(), IL)) -> c_20(uTake1^#(isNatIList(IL)))
     , active^#(take(s(M), cons(N, IL))) ->
       c_21(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
                     M,
                     N,
                     IL))
     , active^#(uTake1(X)) -> c_22(uTake1^#(active(X)))
     , active^#(uTake1(tt())) -> c_23()
     , active^#(uTake2(X1, X2, X3, X4)) ->
       c_24(uTake2^#(active(X1), X2, X3, X4))
     , active^#(uTake2(tt(), M, N, IL)) -> c_25(cons^#(N, take(M, IL)))
     , active^#(uLength(X1, X2)) -> c_26(uLength^#(active(X1), X2))
     , active^#(uLength(tt(), L)) -> c_27(s^#(length(L)))
     , and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
     , and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
     , isNatList^#(ok(X)) -> c_32(isNatList^#(X))
     , isNat^#(ok(X)) -> c_33(isNat^#(X))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , length^#(mark(X)) -> c_36(length^#(X))
     , length^#(ok(X)) -> c_37(length^#(X))
     , uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
     , uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
     , cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
     , take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
     , take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
     , take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
     , uTake1^#(mark(X)) -> c_43(uTake1^#(X))
     , uTake1^#(ok(X)) -> c_44(uTake1^#(X))
     , uTake2^#(mark(X1), X2, X3, X4) -> c_45(uTake2^#(X1, X2, X3, X4))
     , uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       c_46(uTake2^#(X1, X2, X3, X4))
     , isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
     , proper^#(and(X1, X2)) -> c_49(and^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_50()
     , proper^#(isNatIList(X)) -> c_51(isNatIList^#(proper(X)))
     , proper^#(isNatList(X)) -> c_52(isNatList^#(proper(X)))
     , proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
     , proper^#(0()) -> c_54()
     , proper^#(s(X)) -> c_55(s^#(proper(X)))
     , proper^#(length(X)) -> c_56(length^#(proper(X)))
     , proper^#(zeros()) -> c_57()
     , proper^#(cons(X1, X2)) -> c_58(cons^#(proper(X1), proper(X2)))
     , proper^#(nil()) -> c_59()
     , proper^#(take(X1, X2)) -> c_60(take^#(proper(X1), proper(X2)))
     , proper^#(uTake1(X)) -> c_61(uTake1^#(proper(X)))
     , proper^#(uTake2(X1, X2, X3, X4)) ->
       c_62(uTake2^#(proper(X1), proper(X2), proper(X3), proper(X4)))
     , proper^#(uLength(X1, X2)) ->
       c_63(uLength^#(proper(X1), proper(X2)))
     , top^#(mark(X)) -> c_64(top^#(proper(X)))
     , top^#(ok(X)) -> c_65(top^#(active(X))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { active^#(and(X1, X2)) -> c_1(and^#(X1, active(X2)))
     , active^#(and(X1, X2)) -> c_2(and^#(active(X1), X2))
     , active^#(and(tt(), T)) -> c_3(T)
     , active^#(isNatIList(IL)) -> c_4(isNatList^#(IL))
     , active^#(isNatIList(zeros())) -> c_5()
     , active^#(isNatIList(cons(N, IL))) ->
       c_6(and^#(isNat(N), isNatIList(IL)))
     , active^#(isNatList(cons(N, L))) ->
       c_7(and^#(isNat(N), isNatList(L)))
     , active^#(isNatList(nil())) -> c_8()
     , active^#(isNatList(take(N, IL))) ->
       c_9(and^#(isNat(N), isNatIList(IL)))
     , active^#(isNat(0())) -> c_10()
     , active^#(isNat(s(N))) -> c_11(isNat^#(N))
     , active^#(isNat(length(L))) -> c_12(isNatList^#(L))
     , active^#(s(X)) -> c_13(s^#(active(X)))
     , active^#(length(X)) -> c_14(length^#(active(X)))
     , active^#(length(cons(N, L))) ->
       c_15(uLength^#(and(isNat(N), isNatList(L)), L))
     , active^#(zeros()) -> c_16(cons^#(0(), zeros()))
     , active^#(cons(X1, X2)) -> c_17(cons^#(active(X1), X2))
     , active^#(take(X1, X2)) -> c_18(take^#(X1, active(X2)))
     , active^#(take(X1, X2)) -> c_19(take^#(active(X1), X2))
     , active^#(take(0(), IL)) -> c_20(uTake1^#(isNatIList(IL)))
     , active^#(take(s(M), cons(N, IL))) ->
       c_21(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
                     M,
                     N,
                     IL))
     , active^#(uTake1(X)) -> c_22(uTake1^#(active(X)))
     , active^#(uTake1(tt())) -> c_23()
     , active^#(uTake2(X1, X2, X3, X4)) ->
       c_24(uTake2^#(active(X1), X2, X3, X4))
     , active^#(uTake2(tt(), M, N, IL)) -> c_25(cons^#(N, take(M, IL)))
     , active^#(uLength(X1, X2)) -> c_26(uLength^#(active(X1), X2))
     , active^#(uLength(tt(), L)) -> c_27(s^#(length(L)))
     , and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
     , and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
     , isNatList^#(ok(X)) -> c_32(isNatList^#(X))
     , isNat^#(ok(X)) -> c_33(isNat^#(X))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , length^#(mark(X)) -> c_36(length^#(X))
     , length^#(ok(X)) -> c_37(length^#(X))
     , uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
     , uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
     , cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
     , take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
     , take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
     , take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
     , uTake1^#(mark(X)) -> c_43(uTake1^#(X))
     , uTake1^#(ok(X)) -> c_44(uTake1^#(X))
     , uTake2^#(mark(X1), X2, X3, X4) -> c_45(uTake2^#(X1, X2, X3, X4))
     , uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       c_46(uTake2^#(X1, X2, X3, X4))
     , isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
     , proper^#(and(X1, X2)) -> c_49(and^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_50()
     , proper^#(isNatIList(X)) -> c_51(isNatIList^#(proper(X)))
     , proper^#(isNatList(X)) -> c_52(isNatList^#(proper(X)))
     , proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
     , proper^#(0()) -> c_54()
     , proper^#(s(X)) -> c_55(s^#(proper(X)))
     , proper^#(length(X)) -> c_56(length^#(proper(X)))
     , proper^#(zeros()) -> c_57()
     , proper^#(cons(X1, X2)) -> c_58(cons^#(proper(X1), proper(X2)))
     , proper^#(nil()) -> c_59()
     , proper^#(take(X1, X2)) -> c_60(take^#(proper(X1), proper(X2)))
     , proper^#(uTake1(X)) -> c_61(uTake1^#(proper(X)))
     , proper^#(uTake2(X1, X2, X3, X4)) ->
       c_62(uTake2^#(proper(X1), proper(X2), proper(X3), proper(X4)))
     , proper^#(uLength(X1, X2)) ->
       c_63(uLength^#(proper(X1), proper(X2)))
     , top^#(mark(X)) -> c_64(top^#(proper(X)))
     , top^#(ok(X)) -> c_65(top^#(active(X))) }
   Strict Trs:
     { active(and(X1, X2)) -> and(X1, active(X2))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), T)) -> mark(T)
     , active(isNatIList(IL)) -> mark(isNatList(IL))
     , active(isNatIList(zeros())) -> mark(tt())
     , active(isNatIList(cons(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNatList(cons(N, L))) ->
       mark(and(isNat(N), isNatList(L)))
     , active(isNatList(nil())) -> mark(tt())
     , active(isNatList(take(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(s(N))) -> mark(isNat(N))
     , active(isNat(length(L))) -> mark(isNatList(L))
     , active(s(X)) -> s(active(X))
     , active(length(X)) -> length(active(X))
     , active(length(cons(N, L))) ->
       mark(uLength(and(isNat(N), isNatList(L)), L))
     , active(zeros()) -> mark(cons(0(), zeros()))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(take(X1, X2)) -> take(X1, active(X2))
     , active(take(X1, X2)) -> take(active(X1), X2)
     , active(take(0(), IL)) -> mark(uTake1(isNatIList(IL)))
     , active(take(s(M), cons(N, IL))) ->
       mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))),
                   M,
                   N,
                   IL))
     , active(uTake1(X)) -> uTake1(active(X))
     , active(uTake1(tt())) -> mark(nil())
     , active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4)
     , active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL)))
     , active(uLength(X1, X2)) -> uLength(active(X1), X2)
     , active(uLength(tt(), L)) -> mark(s(length(L)))
     , and(X1, mark(X2)) -> mark(and(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatIList(ok(X)) -> ok(isNatIList(X))
     , isNatList(ok(X)) -> ok(isNatList(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , length(mark(X)) -> mark(length(X))
     , length(ok(X)) -> ok(length(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , take(X1, mark(X2)) -> mark(take(X1, X2))
     , take(mark(X1), X2) -> mark(take(X1, X2))
     , take(ok(X1), ok(X2)) -> ok(take(X1, X2))
     , uTake1(mark(X)) -> mark(uTake1(X))
     , uTake1(ok(X)) -> ok(uTake1(X))
     , uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4))
     , uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       ok(uTake2(X1, X2, X3, X4))
     , uLength(mark(X1), X2) -> mark(uLength(X1, X2))
     , uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(isNatIList(X)) -> isNatIList(proper(X))
     , proper(isNatList(X)) -> isNatList(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(0()) -> ok(0())
     , proper(s(X)) -> s(proper(X))
     , proper(length(X)) -> length(proper(X))
     , proper(zeros()) -> ok(zeros())
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , proper(take(X1, X2)) -> take(proper(X1), proper(X2))
     , proper(uTake1(X)) -> uTake1(proper(X))
     , proper(uTake2(X1, X2, X3, X4)) ->
       uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
     , proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Consider the dependency graph:
   
     1: active^#(and(X1, X2)) -> c_1(and^#(X1, active(X2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     2: active^#(and(X1, X2)) -> c_2(and^#(active(X1), X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     3: active^#(and(tt(), T)) -> c_3(T)
        -->_1 top^#(ok(X)) -> c_65(top^#(active(X))) :65
        -->_1 top^#(mark(X)) -> c_64(top^#(proper(X))) :64
        -->_1 proper^#(uLength(X1, X2)) ->
              c_63(uLength^#(proper(X1), proper(X2))) :63
        -->_1 proper^#(uTake2(X1, X2, X3, X4)) ->
              c_62(uTake2^#(proper(X1), proper(X2), proper(X3), proper(X4))) :62
        -->_1 proper^#(uTake1(X)) -> c_61(uTake1^#(proper(X))) :61
        -->_1 proper^#(take(X1, X2)) ->
              c_60(take^#(proper(X1), proper(X2))) :60
        -->_1 proper^#(cons(X1, X2)) ->
              c_58(cons^#(proper(X1), proper(X2))) :58
        -->_1 proper^#(length(X)) -> c_56(length^#(proper(X))) :56
        -->_1 proper^#(s(X)) -> c_55(s^#(proper(X))) :55
        -->_1 proper^#(isNat(X)) -> c_53(isNat^#(proper(X))) :53
        -->_1 proper^#(isNatList(X)) -> c_52(isNatList^#(proper(X))) :52
        -->_1 proper^#(isNatIList(X)) -> c_51(isNatIList^#(proper(X))) :51
        -->_1 proper^#(and(X1, X2)) ->
              c_49(and^#(proper(X1), proper(X2))) :49
        -->_1 isNatIList^#(ok(X)) -> c_31(isNatIList^#(X)) :48
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
        -->_1 length^#(ok(X)) -> c_37(length^#(X)) :36
        -->_1 length^#(mark(X)) -> c_36(length^#(X)) :35
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
        -->_1 isNat^#(ok(X)) -> c_33(isNat^#(X)) :32
        -->_1 isNatList^#(ok(X)) -> c_32(isNatList^#(X)) :31
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
        -->_1 active^#(uLength(tt(), L)) -> c_27(s^#(length(L))) :27
        -->_1 active^#(uLength(X1, X2)) ->
              c_26(uLength^#(active(X1), X2)) :26
        -->_1 active^#(uTake2(tt(), M, N, IL)) ->
              c_25(cons^#(N, take(M, IL))) :25
        -->_1 active^#(uTake2(X1, X2, X3, X4)) ->
              c_24(uTake2^#(active(X1), X2, X3, X4)) :24
        -->_1 active^#(uTake1(X)) -> c_22(uTake1^#(active(X))) :22
        -->_1 active^#(take(s(M), cons(N, IL))) ->
              c_21(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
                            M,
                            N,
                            IL)) :21
        -->_1 active^#(take(0(), IL)) -> c_20(uTake1^#(isNatIList(IL))) :20
        -->_1 active^#(take(X1, X2)) -> c_19(take^#(active(X1), X2)) :19
        -->_1 active^#(take(X1, X2)) -> c_18(take^#(X1, active(X2))) :18
        -->_1 active^#(cons(X1, X2)) -> c_17(cons^#(active(X1), X2)) :17
        -->_1 active^#(length(cons(N, L))) ->
              c_15(uLength^#(and(isNat(N), isNatList(L)), L)) :15
        -->_1 active^#(length(X)) -> c_14(length^#(active(X))) :14
        -->_1 active^#(s(X)) -> c_13(s^#(active(X))) :13
        -->_1 active^#(isNat(length(L))) -> c_12(isNatList^#(L)) :12
        -->_1 active^#(isNat(s(N))) -> c_11(isNat^#(N)) :11
        -->_1 active^#(isNatList(take(N, IL))) ->
              c_9(and^#(isNat(N), isNatIList(IL))) :9
        -->_1 active^#(isNatList(cons(N, L))) ->
              c_7(and^#(isNat(N), isNatList(L))) :7
        -->_1 active^#(isNatIList(cons(N, IL))) ->
              c_6(and^#(isNat(N), isNatIList(IL))) :6
        -->_1 active^#(isNatIList(IL)) -> c_4(isNatList^#(IL)) :4
        -->_1 proper^#(nil()) -> c_59() :59
        -->_1 proper^#(zeros()) -> c_57() :57
        -->_1 proper^#(0()) -> c_54() :54
        -->_1 proper^#(tt()) -> c_50() :50
        -->_1 active^#(uTake1(tt())) -> c_23() :23
        -->_1 active^#(zeros()) -> c_16(cons^#(0(), zeros())) :16
        -->_1 active^#(isNat(0())) -> c_10() :10
        -->_1 active^#(isNatList(nil())) -> c_8() :8
        -->_1 active^#(isNatIList(zeros())) -> c_5() :5
        -->_1 active^#(and(tt(), T)) -> c_3(T) :3
        -->_1 active^#(and(X1, X2)) -> c_2(and^#(active(X1), X2)) :2
        -->_1 active^#(and(X1, X2)) -> c_1(and^#(X1, active(X2))) :1
     
     4: active^#(isNatIList(IL)) -> c_4(isNatList^#(IL))
        -->_1 isNatList^#(ok(X)) -> c_32(isNatList^#(X)) :31
     
     5: active^#(isNatIList(zeros())) -> c_5()
     
     6: active^#(isNatIList(cons(N, IL))) ->
        c_6(and^#(isNat(N), isNatIList(IL)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     7: active^#(isNatList(cons(N, L))) ->
        c_7(and^#(isNat(N), isNatList(L)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     8: active^#(isNatList(nil())) -> c_8()
     
     9: active^#(isNatList(take(N, IL))) ->
        c_9(and^#(isNat(N), isNatIList(IL)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     10: active^#(isNat(0())) -> c_10()
     
     11: active^#(isNat(s(N))) -> c_11(isNat^#(N))
        -->_1 isNat^#(ok(X)) -> c_33(isNat^#(X)) :32
     
     12: active^#(isNat(length(L))) -> c_12(isNatList^#(L))
        -->_1 isNatList^#(ok(X)) -> c_32(isNatList^#(X)) :31
     
     13: active^#(s(X)) -> c_13(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     14: active^#(length(X)) -> c_14(length^#(active(X)))
        -->_1 length^#(ok(X)) -> c_37(length^#(X)) :36
        -->_1 length^#(mark(X)) -> c_36(length^#(X)) :35
     
     15: active^#(length(cons(N, L))) ->
         c_15(uLength^#(and(isNat(N), isNatList(L)), L))
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
     
     16: active^#(zeros()) -> c_16(cons^#(0(), zeros()))
     
     17: active^#(cons(X1, X2)) -> c_17(cons^#(active(X1), X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
     
     18: active^#(take(X1, X2)) -> c_18(take^#(X1, active(X2)))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     19: active^#(take(X1, X2)) -> c_19(take^#(active(X1), X2))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     20: active^#(take(0(), IL)) -> c_20(uTake1^#(isNatIList(IL)))
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
     
     21: active^#(take(s(M), cons(N, IL))) ->
         c_21(uTake2^#(and(isNat(M), and(isNat(N), isNatIList(IL))),
                       M,
                       N,
                       IL))
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
     
     22: active^#(uTake1(X)) -> c_22(uTake1^#(active(X)))
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
     
     23: active^#(uTake1(tt())) -> c_23()
     
     24: active^#(uTake2(X1, X2, X3, X4)) ->
         c_24(uTake2^#(active(X1), X2, X3, X4))
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
     
     25: active^#(uTake2(tt(), M, N, IL)) ->
         c_25(cons^#(N, take(M, IL)))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
     
     26: active^#(uLength(X1, X2)) -> c_26(uLength^#(active(X1), X2))
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
     
     27: active^#(uLength(tt(), L)) -> c_27(s^#(length(L)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     28: and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     29: and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     30: and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     31: isNatList^#(ok(X)) -> c_32(isNatList^#(X))
        -->_1 isNatList^#(ok(X)) -> c_32(isNatList^#(X)) :31
     
     32: isNat^#(ok(X)) -> c_33(isNat^#(X))
        -->_1 isNat^#(ok(X)) -> c_33(isNat^#(X)) :32
     
     33: s^#(mark(X)) -> c_34(s^#(X))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     34: s^#(ok(X)) -> c_35(s^#(X))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     35: length^#(mark(X)) -> c_36(length^#(X))
        -->_1 length^#(ok(X)) -> c_37(length^#(X)) :36
        -->_1 length^#(mark(X)) -> c_36(length^#(X)) :35
     
     36: length^#(ok(X)) -> c_37(length^#(X))
        -->_1 length^#(ok(X)) -> c_37(length^#(X)) :36
        -->_1 length^#(mark(X)) -> c_36(length^#(X)) :35
     
     37: uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
     
     38: uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
     
     39: cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
     
     40: cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
     
     41: take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     42: take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     43: take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     44: uTake1^#(mark(X)) -> c_43(uTake1^#(X))
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
     
     45: uTake1^#(ok(X)) -> c_44(uTake1^#(X))
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
     
     46: uTake2^#(mark(X1), X2, X3, X4) ->
         c_45(uTake2^#(X1, X2, X3, X4))
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
     
     47: uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
         c_46(uTake2^#(X1, X2, X3, X4))
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
     
     48: isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
        -->_1 isNatIList^#(ok(X)) -> c_31(isNatIList^#(X)) :48
     
     49: proper^#(and(X1, X2)) -> c_49(and^#(proper(X1), proper(X2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2)) :30
        -->_1 and^#(mark(X1), X2) -> c_29(and^#(X1, X2)) :29
        -->_1 and^#(X1, mark(X2)) -> c_28(and^#(X1, X2)) :28
     
     50: proper^#(tt()) -> c_50()
     
     51: proper^#(isNatIList(X)) -> c_51(isNatIList^#(proper(X)))
        -->_1 isNatIList^#(ok(X)) -> c_31(isNatIList^#(X)) :48
     
     52: proper^#(isNatList(X)) -> c_52(isNatList^#(proper(X)))
        -->_1 isNatList^#(ok(X)) -> c_32(isNatList^#(X)) :31
     
     53: proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
        -->_1 isNat^#(ok(X)) -> c_33(isNat^#(X)) :32
     
     54: proper^#(0()) -> c_54()
     
     55: proper^#(s(X)) -> c_55(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     56: proper^#(length(X)) -> c_56(length^#(proper(X)))
        -->_1 length^#(ok(X)) -> c_37(length^#(X)) :36
        -->_1 length^#(mark(X)) -> c_36(length^#(X)) :35
     
     57: proper^#(zeros()) -> c_57()
     
     58: proper^#(cons(X1, X2)) -> c_58(cons^#(proper(X1), proper(X2)))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2)) :40
        -->_1 cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2)) :39
     
     59: proper^#(nil()) -> c_59()
     
     60: proper^#(take(X1, X2)) -> c_60(take^#(proper(X1), proper(X2)))
        -->_1 take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2)) :43
        -->_1 take^#(mark(X1), X2) -> c_41(take^#(X1, X2)) :42
        -->_1 take^#(X1, mark(X2)) -> c_40(take^#(X1, X2)) :41
     
     61: proper^#(uTake1(X)) -> c_61(uTake1^#(proper(X)))
        -->_1 uTake1^#(ok(X)) -> c_44(uTake1^#(X)) :45
        -->_1 uTake1^#(mark(X)) -> c_43(uTake1^#(X)) :44
     
     62: proper^#(uTake2(X1, X2, X3, X4)) ->
         c_62(uTake2^#(proper(X1), proper(X2), proper(X3), proper(X4)))
        -->_1 uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
              c_46(uTake2^#(X1, X2, X3, X4)) :47
        -->_1 uTake2^#(mark(X1), X2, X3, X4) ->
              c_45(uTake2^#(X1, X2, X3, X4)) :46
     
     63: proper^#(uLength(X1, X2)) ->
         c_63(uLength^#(proper(X1), proper(X2)))
        -->_1 uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2)) :38
        -->_1 uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2)) :37
     
     64: top^#(mark(X)) -> c_64(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_65(top^#(active(X))) :65
        -->_1 top^#(mark(X)) -> c_64(top^#(proper(X))) :64
     
     65: top^#(ok(X)) -> c_65(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_65(top^#(active(X))) :65
        -->_1 top^#(mark(X)) -> c_64(top^#(proper(X))) :64
     
   
   Only the nodes
   {16,28,30,29,31,32,33,34,35,36,37,38,39,40,41,43,42,44,45,46,47,48,50,54,57,59,64,65}
   are reachable from nodes
   {16,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,50,54,57,59,64,65}
   that start derivation from marked basic terms. The nodes not
   reachable are removed from the problem.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { active^#(zeros()) -> c_16(cons^#(0(), zeros()))
     , and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
     , and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
     , isNatList^#(ok(X)) -> c_32(isNatList^#(X))
     , isNat^#(ok(X)) -> c_33(isNat^#(X))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , length^#(mark(X)) -> c_36(length^#(X))
     , length^#(ok(X)) -> c_37(length^#(X))
     , uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
     , uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
     , cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
     , take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
     , take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
     , take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
     , uTake1^#(mark(X)) -> c_43(uTake1^#(X))
     , uTake1^#(ok(X)) -> c_44(uTake1^#(X))
     , uTake2^#(mark(X1), X2, X3, X4) -> c_45(uTake2^#(X1, X2, X3, X4))
     , uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       c_46(uTake2^#(X1, X2, X3, X4))
     , isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
     , proper^#(tt()) -> c_50()
     , proper^#(0()) -> c_54()
     , proper^#(zeros()) -> c_57()
     , proper^#(nil()) -> c_59()
     , top^#(mark(X)) -> c_64(top^#(proper(X)))
     , top^#(ok(X)) -> c_65(top^#(active(X))) }
   Strict Trs:
     { active(and(X1, X2)) -> and(X1, active(X2))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), T)) -> mark(T)
     , active(isNatIList(IL)) -> mark(isNatList(IL))
     , active(isNatIList(zeros())) -> mark(tt())
     , active(isNatIList(cons(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNatList(cons(N, L))) ->
       mark(and(isNat(N), isNatList(L)))
     , active(isNatList(nil())) -> mark(tt())
     , active(isNatList(take(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(s(N))) -> mark(isNat(N))
     , active(isNat(length(L))) -> mark(isNatList(L))
     , active(s(X)) -> s(active(X))
     , active(length(X)) -> length(active(X))
     , active(length(cons(N, L))) ->
       mark(uLength(and(isNat(N), isNatList(L)), L))
     , active(zeros()) -> mark(cons(0(), zeros()))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(take(X1, X2)) -> take(X1, active(X2))
     , active(take(X1, X2)) -> take(active(X1), X2)
     , active(take(0(), IL)) -> mark(uTake1(isNatIList(IL)))
     , active(take(s(M), cons(N, IL))) ->
       mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))),
                   M,
                   N,
                   IL))
     , active(uTake1(X)) -> uTake1(active(X))
     , active(uTake1(tt())) -> mark(nil())
     , active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4)
     , active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL)))
     , active(uLength(X1, X2)) -> uLength(active(X1), X2)
     , active(uLength(tt(), L)) -> mark(s(length(L)))
     , and(X1, mark(X2)) -> mark(and(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatIList(ok(X)) -> ok(isNatIList(X))
     , isNatList(ok(X)) -> ok(isNatList(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , length(mark(X)) -> mark(length(X))
     , length(ok(X)) -> ok(length(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , take(X1, mark(X2)) -> mark(take(X1, X2))
     , take(mark(X1), X2) -> mark(take(X1, X2))
     , take(ok(X1), ok(X2)) -> ok(take(X1, X2))
     , uTake1(mark(X)) -> mark(uTake1(X))
     , uTake1(ok(X)) -> ok(uTake1(X))
     , uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4))
     , uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       ok(uTake2(X1, X2, X3, X4))
     , uLength(mark(X1), X2) -> mark(uLength(X1, X2))
     , uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(isNatIList(X)) -> isNatIList(proper(X))
     , proper(isNatList(X)) -> isNatList(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(0()) -> ok(0())
     , proper(s(X)) -> s(proper(X))
     , proper(length(X)) -> length(proper(X))
     , proper(zeros()) -> ok(zeros())
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , proper(take(X1, X2)) -> take(proper(X1), proper(X2))
     , proper(uTake1(X)) -> uTake1(proper(X))
     , proper(uTake2(X1, X2, X3, X4)) ->
       uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
     , proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,23,24,25,26} by
   applications of Pre({1,23,24,25,26}) = {}. Here rules are labeled
   as follows:
   
     DPs:
       { 1: active^#(zeros()) -> c_16(cons^#(0(), zeros()))
       , 2: and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
       , 3: and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
       , 4: and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
       , 5: isNatList^#(ok(X)) -> c_32(isNatList^#(X))
       , 6: isNat^#(ok(X)) -> c_33(isNat^#(X))
       , 7: s^#(mark(X)) -> c_34(s^#(X))
       , 8: s^#(ok(X)) -> c_35(s^#(X))
       , 9: length^#(mark(X)) -> c_36(length^#(X))
       , 10: length^#(ok(X)) -> c_37(length^#(X))
       , 11: uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
       , 12: uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
       , 13: cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
       , 14: cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
       , 15: take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
       , 16: take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
       , 17: take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
       , 18: uTake1^#(mark(X)) -> c_43(uTake1^#(X))
       , 19: uTake1^#(ok(X)) -> c_44(uTake1^#(X))
       , 20: uTake2^#(mark(X1), X2, X3, X4) ->
             c_45(uTake2^#(X1, X2, X3, X4))
       , 21: uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
             c_46(uTake2^#(X1, X2, X3, X4))
       , 22: isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
       , 23: proper^#(tt()) -> c_50()
       , 24: proper^#(0()) -> c_54()
       , 25: proper^#(zeros()) -> c_57()
       , 26: proper^#(nil()) -> c_59()
       , 27: top^#(mark(X)) -> c_64(top^#(proper(X)))
       , 28: top^#(ok(X)) -> c_65(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { and^#(X1, mark(X2)) -> c_28(and^#(X1, X2))
     , and^#(mark(X1), X2) -> c_29(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_30(and^#(X1, X2))
     , isNatList^#(ok(X)) -> c_32(isNatList^#(X))
     , isNat^#(ok(X)) -> c_33(isNat^#(X))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , length^#(mark(X)) -> c_36(length^#(X))
     , length^#(ok(X)) -> c_37(length^#(X))
     , uLength^#(mark(X1), X2) -> c_47(uLength^#(X1, X2))
     , uLength^#(ok(X1), ok(X2)) -> c_48(uLength^#(X1, X2))
     , cons^#(mark(X1), X2) -> c_38(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_39(cons^#(X1, X2))
     , take^#(X1, mark(X2)) -> c_40(take^#(X1, X2))
     , take^#(mark(X1), X2) -> c_41(take^#(X1, X2))
     , take^#(ok(X1), ok(X2)) -> c_42(take^#(X1, X2))
     , uTake1^#(mark(X)) -> c_43(uTake1^#(X))
     , uTake1^#(ok(X)) -> c_44(uTake1^#(X))
     , uTake2^#(mark(X1), X2, X3, X4) -> c_45(uTake2^#(X1, X2, X3, X4))
     , uTake2^#(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       c_46(uTake2^#(X1, X2, X3, X4))
     , isNatIList^#(ok(X)) -> c_31(isNatIList^#(X))
     , top^#(mark(X)) -> c_64(top^#(proper(X)))
     , top^#(ok(X)) -> c_65(top^#(active(X))) }
   Strict Trs:
     { active(and(X1, X2)) -> and(X1, active(X2))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), T)) -> mark(T)
     , active(isNatIList(IL)) -> mark(isNatList(IL))
     , active(isNatIList(zeros())) -> mark(tt())
     , active(isNatIList(cons(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNatList(cons(N, L))) ->
       mark(and(isNat(N), isNatList(L)))
     , active(isNatList(nil())) -> mark(tt())
     , active(isNatList(take(N, IL))) ->
       mark(and(isNat(N), isNatIList(IL)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(s(N))) -> mark(isNat(N))
     , active(isNat(length(L))) -> mark(isNatList(L))
     , active(s(X)) -> s(active(X))
     , active(length(X)) -> length(active(X))
     , active(length(cons(N, L))) ->
       mark(uLength(and(isNat(N), isNatList(L)), L))
     , active(zeros()) -> mark(cons(0(), zeros()))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(take(X1, X2)) -> take(X1, active(X2))
     , active(take(X1, X2)) -> take(active(X1), X2)
     , active(take(0(), IL)) -> mark(uTake1(isNatIList(IL)))
     , active(take(s(M), cons(N, IL))) ->
       mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))),
                   M,
                   N,
                   IL))
     , active(uTake1(X)) -> uTake1(active(X))
     , active(uTake1(tt())) -> mark(nil())
     , active(uTake2(X1, X2, X3, X4)) -> uTake2(active(X1), X2, X3, X4)
     , active(uTake2(tt(), M, N, IL)) -> mark(cons(N, take(M, IL)))
     , active(uLength(X1, X2)) -> uLength(active(X1), X2)
     , active(uLength(tt(), L)) -> mark(s(length(L)))
     , and(X1, mark(X2)) -> mark(and(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatIList(ok(X)) -> ok(isNatIList(X))
     , isNatList(ok(X)) -> ok(isNatList(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , length(mark(X)) -> mark(length(X))
     , length(ok(X)) -> ok(length(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , take(X1, mark(X2)) -> mark(take(X1, X2))
     , take(mark(X1), X2) -> mark(take(X1, X2))
     , take(ok(X1), ok(X2)) -> ok(take(X1, X2))
     , uTake1(mark(X)) -> mark(uTake1(X))
     , uTake1(ok(X)) -> ok(uTake1(X))
     , uTake2(mark(X1), X2, X3, X4) -> mark(uTake2(X1, X2, X3, X4))
     , uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) ->
       ok(uTake2(X1, X2, X3, X4))
     , uLength(mark(X1), X2) -> mark(uLength(X1, X2))
     , uLength(ok(X1), ok(X2)) -> ok(uLength(X1, X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(isNatIList(X)) -> isNatIList(proper(X))
     , proper(isNatList(X)) -> isNatList(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(0()) -> ok(0())
     , proper(s(X)) -> s(proper(X))
     , proper(length(X)) -> length(proper(X))
     , proper(zeros()) -> ok(zeros())
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , proper(take(X1, X2)) -> take(proper(X1), proper(X2))
     , proper(uTake1(X)) -> uTake1(proper(X))
     , proper(uTake2(X1, X2, X3, X4)) ->
       uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
     , proper(uLength(X1, X2)) -> uLength(proper(X1), proper(X2))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { active^#(zeros()) -> c_16(cons^#(0(), zeros()))
     , proper^#(tt()) -> c_50()
     , proper^#(0()) -> c_54()
     , proper^#(zeros()) -> c_57()
     , proper^#(nil()) -> c_59() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..