MAYBE

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

Strict Trs:
  { active(U11(X1, X2)) -> U11(active(X1), X2)
  , active(U11(tt(), N)) -> mark(N)
  , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
  , active(U21(tt(), M, N)) -> mark(s(plus(N, M)))
  , active(s(X)) -> s(active(X))
  , active(plus(X1, X2)) -> plus(X1, active(X2))
  , active(plus(X1, X2)) -> plus(active(X1), X2)
  , active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
  , active(plus(N, 0())) -> mark(U11(isNat(N), N))
  , active(U31(X)) -> U31(active(X))
  , active(U31(tt())) -> mark(0())
  , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
  , active(U41(tt(), M, N)) -> mark(plus(x(N, M), N))
  , active(x(X1, X2)) -> x(X1, active(X2))
  , active(x(X1, X2)) -> x(active(X1), X2)
  , active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
  , active(x(N, 0())) -> mark(U31(isNat(N)))
  , active(and(X1, X2)) -> and(active(X1), X2)
  , active(and(tt(), X)) -> mark(X)
  , active(isNat(s(V1))) -> mark(isNat(V1))
  , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
  , active(isNat(0())) -> mark(tt())
  , active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
  , U11(mark(X1), X2) -> mark(U11(X1, X2))
  , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
  , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
  , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
  , s(mark(X)) -> mark(s(X))
  , s(ok(X)) -> ok(s(X))
  , plus(X1, mark(X2)) -> mark(plus(X1, X2))
  , plus(mark(X1), X2) -> mark(plus(X1, X2))
  , plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
  , U31(mark(X)) -> mark(U31(X))
  , U31(ok(X)) -> ok(U31(X))
  , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
  , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
  , x(X1, mark(X2)) -> mark(x(X1, X2))
  , x(mark(X1), X2) -> mark(x(X1, X2))
  , x(ok(X1), ok(X2)) -> ok(x(X1, X2))
  , and(mark(X1), X2) -> mark(and(X1, X2))
  , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
  , isNat(ok(X)) -> ok(isNat(X))
  , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
  , proper(tt()) -> ok(tt())
  , proper(U21(X1, X2, X3)) ->
    U21(proper(X1), proper(X2), proper(X3))
  , proper(s(X)) -> s(proper(X))
  , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
  , proper(U31(X)) -> U31(proper(X))
  , proper(0()) -> ok(0())
  , proper(U41(X1, X2, X3)) ->
    U41(proper(X1), proper(X2), proper(X3))
  , proper(x(X1, X2)) -> x(proper(X1), proper(X2))
  , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
  , proper(isNat(X)) -> isNat(proper(X))
  , 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^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2))
     , active^#(U11(tt(), N)) -> c_2(N)
     , active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3))
     , active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M)))
     , active^#(s(X)) -> c_5(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) ->
       c_8(U21^#(and(isNat(M), isNat(N)), M, N))
     , active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N))
     , active^#(U31(X)) -> c_10(U31^#(active(X)))
     , active^#(U31(tt())) -> c_11()
     , active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_13(plus^#(x(N, M), N))
     , active^#(x(X1, X2)) -> c_14(x^#(X1, active(X2)))
     , active^#(x(X1, X2)) -> c_15(x^#(active(X1), X2))
     , active^#(x(N, s(M))) ->
       c_16(U41^#(and(isNat(M), isNat(N)), M, N))
     , active^#(x(N, 0())) -> c_17(U31^#(isNat(N)))
     , active^#(and(X1, X2)) -> c_18(and^#(active(X1), X2))
     , active^#(and(tt(), X)) -> c_19(X)
     , active^#(isNat(s(V1))) -> c_20(isNat^#(V1))
     , active^#(isNat(plus(V1, V2))) ->
       c_21(and^#(isNat(V1), isNat(V2)))
     , active^#(isNat(0())) -> c_22()
     , active^#(isNat(x(V1, V2))) -> c_23(and^#(isNat(V1), isNat(V2)))
     , U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
     , U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
     , U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
     , U31^#(mark(X)) -> c_33(U31^#(X))
     , U31^#(ok(X)) -> c_34(U31^#(X))
     , U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
     , x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
     , x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
     , x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
     , and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
     , isNat^#(ok(X)) -> c_42(isNat^#(X))
     , proper^#(U11(X1, X2)) -> c_43(U11^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_44()
     , proper^#(U21(X1, X2, X3)) ->
       c_45(U21^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_46(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_47(plus^#(proper(X1), proper(X2)))
     , proper^#(U31(X)) -> c_48(U31^#(proper(X)))
     , proper^#(0()) -> c_49()
     , proper^#(U41(X1, X2, X3)) ->
       c_50(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(x(X1, X2)) -> c_51(x^#(proper(X1), proper(X2)))
     , proper^#(and(X1, X2)) -> c_52(and^#(proper(X1), proper(X2)))
     , proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
     , top^#(mark(X)) -> c_54(top^#(proper(X)))
     , top^#(ok(X)) -> c_55(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^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2))
     , active^#(U11(tt(), N)) -> c_2(N)
     , active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3))
     , active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M)))
     , active^#(s(X)) -> c_5(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) ->
       c_8(U21^#(and(isNat(M), isNat(N)), M, N))
     , active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N))
     , active^#(U31(X)) -> c_10(U31^#(active(X)))
     , active^#(U31(tt())) -> c_11()
     , active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_13(plus^#(x(N, M), N))
     , active^#(x(X1, X2)) -> c_14(x^#(X1, active(X2)))
     , active^#(x(X1, X2)) -> c_15(x^#(active(X1), X2))
     , active^#(x(N, s(M))) ->
       c_16(U41^#(and(isNat(M), isNat(N)), M, N))
     , active^#(x(N, 0())) -> c_17(U31^#(isNat(N)))
     , active^#(and(X1, X2)) -> c_18(and^#(active(X1), X2))
     , active^#(and(tt(), X)) -> c_19(X)
     , active^#(isNat(s(V1))) -> c_20(isNat^#(V1))
     , active^#(isNat(plus(V1, V2))) ->
       c_21(and^#(isNat(V1), isNat(V2)))
     , active^#(isNat(0())) -> c_22()
     , active^#(isNat(x(V1, V2))) -> c_23(and^#(isNat(V1), isNat(V2)))
     , U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
     , U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
     , U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
     , U31^#(mark(X)) -> c_33(U31^#(X))
     , U31^#(ok(X)) -> c_34(U31^#(X))
     , U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
     , x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
     , x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
     , x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
     , and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
     , isNat^#(ok(X)) -> c_42(isNat^#(X))
     , proper^#(U11(X1, X2)) -> c_43(U11^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_44()
     , proper^#(U21(X1, X2, X3)) ->
       c_45(U21^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_46(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_47(plus^#(proper(X1), proper(X2)))
     , proper^#(U31(X)) -> c_48(U31^#(proper(X)))
     , proper^#(0()) -> c_49()
     , proper^#(U41(X1, X2, X3)) ->
       c_50(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(x(X1, X2)) -> c_51(x^#(proper(X1), proper(X2)))
     , proper^#(and(X1, X2)) -> c_52(and^#(proper(X1), proper(X2)))
     , proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
     , top^#(mark(X)) -> c_54(top^#(proper(X)))
     , top^#(ok(X)) -> c_55(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), N)) -> mark(N)
     , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
     , active(U21(tt(), M, N)) -> mark(s(plus(N, M)))
     , active(s(X)) -> s(active(X))
     , active(plus(X1, X2)) -> plus(X1, active(X2))
     , active(plus(X1, X2)) -> plus(active(X1), X2)
     , active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
     , active(plus(N, 0())) -> mark(U11(isNat(N), N))
     , active(U31(X)) -> U31(active(X))
     , active(U31(tt())) -> mark(0())
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(plus(x(N, M), N))
     , active(x(X1, X2)) -> x(X1, active(X2))
     , active(x(X1, X2)) -> x(active(X1), X2)
     , active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
     , active(x(N, 0())) -> mark(U31(isNat(N)))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNat(s(V1))) -> mark(isNat(V1))
     , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
     , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , plus(X1, mark(X2)) -> mark(plus(X1, X2))
     , plus(mark(X1), X2) -> mark(plus(X1, X2))
     , plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
     , U31(mark(X)) -> mark(U31(X))
     , U31(ok(X)) -> ok(U31(X))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , x(X1, mark(X2)) -> mark(x(X1, X2))
     , x(mark(X1), X2) -> mark(x(X1, X2))
     , x(ok(X1), ok(X2)) -> ok(x(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U21(X1, X2, X3)) ->
       U21(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(U31(X)) -> U31(proper(X))
     , proper(0()) -> ok(0())
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(x(X1, X2)) -> x(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Consider the dependency graph:
   
     1: active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
     
     2: active^#(U11(tt(), N)) -> c_2(N)
        -->_1 top^#(ok(X)) -> c_55(top^#(active(X))) :55
        -->_1 top^#(mark(X)) -> c_54(top^#(proper(X))) :54
        -->_1 proper^#(isNat(X)) -> c_53(isNat^#(proper(X))) :53
        -->_1 proper^#(and(X1, X2)) ->
              c_52(and^#(proper(X1), proper(X2))) :52
        -->_1 proper^#(x(X1, X2)) -> c_51(x^#(proper(X1), proper(X2))) :51
        -->_1 proper^#(U41(X1, X2, X3)) ->
              c_50(U41^#(proper(X1), proper(X2), proper(X3))) :50
        -->_1 proper^#(U31(X)) -> c_48(U31^#(proper(X))) :48
        -->_1 proper^#(plus(X1, X2)) ->
              c_47(plus^#(proper(X1), proper(X2))) :47
        -->_1 proper^#(s(X)) -> c_46(s^#(proper(X))) :46
        -->_1 proper^#(U21(X1, X2, X3)) ->
              c_45(U21^#(proper(X1), proper(X2), proper(X3))) :45
        -->_1 proper^#(U11(X1, X2)) ->
              c_43(U11^#(proper(X1), proper(X2))) :43
        -->_1 isNat^#(ok(X)) -> c_42(isNat^#(X)) :42
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
        -->_1 active^#(isNat(x(V1, V2))) ->
              c_23(and^#(isNat(V1), isNat(V2))) :23
        -->_1 active^#(isNat(plus(V1, V2))) ->
              c_21(and^#(isNat(V1), isNat(V2))) :21
        -->_1 active^#(isNat(s(V1))) -> c_20(isNat^#(V1)) :20
        -->_1 active^#(and(tt(), X)) -> c_19(X) :19
        -->_1 active^#(and(X1, X2)) -> c_18(and^#(active(X1), X2)) :18
        -->_1 active^#(x(N, 0())) -> c_17(U31^#(isNat(N))) :17
        -->_1 active^#(x(N, s(M))) ->
              c_16(U41^#(and(isNat(M), isNat(N)), M, N)) :16
        -->_1 active^#(x(X1, X2)) -> c_15(x^#(active(X1), X2)) :15
        -->_1 active^#(x(X1, X2)) -> c_14(x^#(X1, active(X2))) :14
        -->_1 active^#(U41(tt(), M, N)) -> c_13(plus^#(x(N, M), N)) :13
        -->_1 active^#(U41(X1, X2, X3)) ->
              c_12(U41^#(active(X1), X2, X3)) :12
        -->_1 active^#(U31(X)) -> c_10(U31^#(active(X))) :10
        -->_1 active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) :9
        -->_1 active^#(plus(N, s(M))) ->
              c_8(U21^#(and(isNat(M), isNat(N)), M, N)) :8
        -->_1 active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) :7
        -->_1 active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) :6
        -->_1 active^#(s(X)) -> c_5(s^#(active(X))) :5
        -->_1 active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) :4
        -->_1 active^#(U21(X1, X2, X3)) ->
              c_3(U21^#(active(X1), X2, X3)) :3
        -->_1 proper^#(0()) -> c_49() :49
        -->_1 proper^#(tt()) -> c_44() :44
        -->_1 active^#(isNat(0())) -> c_22() :22
        -->_1 active^#(U31(tt())) -> c_11() :11
        -->_1 active^#(U11(tt(), N)) -> c_2(N) :2
        -->_1 active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) :1
     
     3: active^#(U21(X1, X2, X3)) -> c_3(U21^#(active(X1), X2, X3))
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
     
     4: active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     5: active^#(s(X)) -> c_5(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     6: active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     7: active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     8: active^#(plus(N, s(M))) ->
        c_8(U21^#(and(isNat(M), isNat(N)), M, N))
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
     
     9: active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
     
     10: active^#(U31(X)) -> c_10(U31^#(active(X)))
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
     
     11: active^#(U31(tt())) -> c_11()
     
     12: active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
     
     13: active^#(U41(tt(), M, N)) -> c_13(plus^#(x(N, M), N))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     14: active^#(x(X1, X2)) -> c_14(x^#(X1, active(X2)))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     15: active^#(x(X1, X2)) -> c_15(x^#(active(X1), X2))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     16: active^#(x(N, s(M))) ->
         c_16(U41^#(and(isNat(M), isNat(N)), M, N))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
     
     17: active^#(x(N, 0())) -> c_17(U31^#(isNat(N)))
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
     
     18: active^#(and(X1, X2)) -> c_18(and^#(active(X1), X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     19: active^#(and(tt(), X)) -> c_19(X)
        -->_1 top^#(ok(X)) -> c_55(top^#(active(X))) :55
        -->_1 top^#(mark(X)) -> c_54(top^#(proper(X))) :54
        -->_1 proper^#(isNat(X)) -> c_53(isNat^#(proper(X))) :53
        -->_1 proper^#(and(X1, X2)) ->
              c_52(and^#(proper(X1), proper(X2))) :52
        -->_1 proper^#(x(X1, X2)) -> c_51(x^#(proper(X1), proper(X2))) :51
        -->_1 proper^#(U41(X1, X2, X3)) ->
              c_50(U41^#(proper(X1), proper(X2), proper(X3))) :50
        -->_1 proper^#(U31(X)) -> c_48(U31^#(proper(X))) :48
        -->_1 proper^#(plus(X1, X2)) ->
              c_47(plus^#(proper(X1), proper(X2))) :47
        -->_1 proper^#(s(X)) -> c_46(s^#(proper(X))) :46
        -->_1 proper^#(U21(X1, X2, X3)) ->
              c_45(U21^#(proper(X1), proper(X2), proper(X3))) :45
        -->_1 proper^#(U11(X1, X2)) ->
              c_43(U11^#(proper(X1), proper(X2))) :43
        -->_1 isNat^#(ok(X)) -> c_42(isNat^#(X)) :42
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
        -->_1 active^#(isNat(x(V1, V2))) ->
              c_23(and^#(isNat(V1), isNat(V2))) :23
        -->_1 active^#(isNat(plus(V1, V2))) ->
              c_21(and^#(isNat(V1), isNat(V2))) :21
        -->_1 active^#(isNat(s(V1))) -> c_20(isNat^#(V1)) :20
        -->_1 proper^#(0()) -> c_49() :49
        -->_1 proper^#(tt()) -> c_44() :44
        -->_1 active^#(isNat(0())) -> c_22() :22
        -->_1 active^#(and(tt(), X)) -> c_19(X) :19
        -->_1 active^#(and(X1, X2)) -> c_18(and^#(active(X1), X2)) :18
        -->_1 active^#(x(N, 0())) -> c_17(U31^#(isNat(N))) :17
        -->_1 active^#(x(N, s(M))) ->
              c_16(U41^#(and(isNat(M), isNat(N)), M, N)) :16
        -->_1 active^#(x(X1, X2)) -> c_15(x^#(active(X1), X2)) :15
        -->_1 active^#(x(X1, X2)) -> c_14(x^#(X1, active(X2))) :14
        -->_1 active^#(U41(tt(), M, N)) -> c_13(plus^#(x(N, M), N)) :13
        -->_1 active^#(U41(X1, X2, X3)) ->
              c_12(U41^#(active(X1), X2, X3)) :12
        -->_1 active^#(U31(tt())) -> c_11() :11
        -->_1 active^#(U31(X)) -> c_10(U31^#(active(X))) :10
        -->_1 active^#(plus(N, 0())) -> c_9(U11^#(isNat(N), N)) :9
        -->_1 active^#(plus(N, s(M))) ->
              c_8(U21^#(and(isNat(M), isNat(N)), M, N)) :8
        -->_1 active^#(plus(X1, X2)) -> c_7(plus^#(active(X1), X2)) :7
        -->_1 active^#(plus(X1, X2)) -> c_6(plus^#(X1, active(X2))) :6
        -->_1 active^#(s(X)) -> c_5(s^#(active(X))) :5
        -->_1 active^#(U21(tt(), M, N)) -> c_4(s^#(plus(N, M))) :4
        -->_1 active^#(U21(X1, X2, X3)) ->
              c_3(U21^#(active(X1), X2, X3)) :3
        -->_1 active^#(U11(tt(), N)) -> c_2(N) :2
        -->_1 active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) :1
     
     20: active^#(isNat(s(V1))) -> c_20(isNat^#(V1))
        -->_1 isNat^#(ok(X)) -> c_42(isNat^#(X)) :42
     
     21: active^#(isNat(plus(V1, V2))) ->
         c_21(and^#(isNat(V1), isNat(V2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     22: active^#(isNat(0())) -> c_22()
     
     23: active^#(isNat(x(V1, V2))) -> c_23(and^#(isNat(V1), isNat(V2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     24: U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
     
     25: U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
     
     26: U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
     
     27: U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
     
     28: s^#(mark(X)) -> c_28(s^#(X))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     29: s^#(ok(X)) -> c_29(s^#(X))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     30: plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     31: plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     32: plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     33: U31^#(mark(X)) -> c_33(U31^#(X))
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
     
     34: U31^#(ok(X)) -> c_34(U31^#(X))
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
     
     35: U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
     
     36: U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
     
     37: x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     38: x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     39: x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     40: and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     41: and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     42: isNat^#(ok(X)) -> c_42(isNat^#(X))
        -->_1 isNat^#(ok(X)) -> c_42(isNat^#(X)) :42
     
     43: proper^#(U11(X1, X2)) -> c_43(U11^#(proper(X1), proper(X2)))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2)) :25
        -->_1 U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2)) :24
     
     44: proper^#(tt()) -> c_44()
     
     45: proper^#(U21(X1, X2, X3)) ->
         c_45(U21^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3)) :27
        -->_1 U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3)) :26
     
     46: proper^#(s(X)) -> c_46(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     47: proper^#(plus(X1, X2)) -> c_47(plus^#(proper(X1), proper(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2)) :32
        -->_1 plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2)) :31
        -->_1 plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2)) :30
     
     48: proper^#(U31(X)) -> c_48(U31^#(proper(X)))
        -->_1 U31^#(ok(X)) -> c_34(U31^#(X)) :34
        -->_1 U31^#(mark(X)) -> c_33(U31^#(X)) :33
     
     49: proper^#(0()) -> c_49()
     
     50: proper^#(U41(X1, X2, X3)) ->
         c_50(U41^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3)) :36
        -->_1 U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3)) :35
     
     51: proper^#(x(X1, X2)) -> c_51(x^#(proper(X1), proper(X2)))
        -->_1 x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2)) :39
        -->_1 x^#(mark(X1), X2) -> c_38(x^#(X1, X2)) :38
        -->_1 x^#(X1, mark(X2)) -> c_37(x^#(X1, X2)) :37
     
     52: proper^#(and(X1, X2)) -> c_52(and^#(proper(X1), proper(X2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2)) :41
        -->_1 and^#(mark(X1), X2) -> c_40(and^#(X1, X2)) :40
     
     53: proper^#(isNat(X)) -> c_53(isNat^#(proper(X)))
        -->_1 isNat^#(ok(X)) -> c_42(isNat^#(X)) :42
     
     54: top^#(mark(X)) -> c_54(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_55(top^#(active(X))) :55
        -->_1 top^#(mark(X)) -> c_54(top^#(proper(X))) :54
     
     55: top^#(ok(X)) -> c_55(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_55(top^#(active(X))) :55
        -->_1 top^#(mark(X)) -> c_54(top^#(proper(X))) :54
     
   
   Only the nodes
   {24,25,26,27,28,29,30,32,31,33,34,35,36,37,39,38,40,41,42,44,49,54,55}
   are reachable from nodes
   {24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,44,49,54,55}
   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:
     { U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
     , U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
     , U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
     , U31^#(mark(X)) -> c_33(U31^#(X))
     , U31^#(ok(X)) -> c_34(U31^#(X))
     , U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
     , x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
     , x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
     , x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
     , and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
     , isNat^#(ok(X)) -> c_42(isNat^#(X))
     , proper^#(tt()) -> c_44()
     , proper^#(0()) -> c_49()
     , top^#(mark(X)) -> c_54(top^#(proper(X)))
     , top^#(ok(X)) -> c_55(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), N)) -> mark(N)
     , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
     , active(U21(tt(), M, N)) -> mark(s(plus(N, M)))
     , active(s(X)) -> s(active(X))
     , active(plus(X1, X2)) -> plus(X1, active(X2))
     , active(plus(X1, X2)) -> plus(active(X1), X2)
     , active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
     , active(plus(N, 0())) -> mark(U11(isNat(N), N))
     , active(U31(X)) -> U31(active(X))
     , active(U31(tt())) -> mark(0())
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(plus(x(N, M), N))
     , active(x(X1, X2)) -> x(X1, active(X2))
     , active(x(X1, X2)) -> x(active(X1), X2)
     , active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
     , active(x(N, 0())) -> mark(U31(isNat(N)))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNat(s(V1))) -> mark(isNat(V1))
     , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
     , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , plus(X1, mark(X2)) -> mark(plus(X1, X2))
     , plus(mark(X1), X2) -> mark(plus(X1, X2))
     , plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
     , U31(mark(X)) -> mark(U31(X))
     , U31(ok(X)) -> ok(U31(X))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , x(X1, mark(X2)) -> mark(x(X1, X2))
     , x(mark(X1), X2) -> mark(x(X1, X2))
     , x(ok(X1), ok(X2)) -> ok(x(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U21(X1, X2, X3)) ->
       U21(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(U31(X)) -> U31(proper(X))
     , proper(0()) -> ok(0())
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(x(X1, X2)) -> x(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {20,21} by applications of
   Pre({20,21}) = {}. Here rules are labeled as follows:
   
     DPs:
       { 1: U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
       , 2: U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
       , 3: U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
       , 4: U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
       , 5: s^#(mark(X)) -> c_28(s^#(X))
       , 6: s^#(ok(X)) -> c_29(s^#(X))
       , 7: plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
       , 8: plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
       , 9: plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
       , 10: U31^#(mark(X)) -> c_33(U31^#(X))
       , 11: U31^#(ok(X)) -> c_34(U31^#(X))
       , 12: U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
       , 13: U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
       , 14: x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
       , 15: x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
       , 16: x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
       , 17: and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
       , 18: and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
       , 19: isNat^#(ok(X)) -> c_42(isNat^#(X))
       , 20: proper^#(tt()) -> c_44()
       , 21: proper^#(0()) -> c_49()
       , 22: top^#(mark(X)) -> c_54(top^#(proper(X)))
       , 23: top^#(ok(X)) -> c_55(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(mark(X1), X2) -> c_24(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_25(U11^#(X1, X2))
     , U21^#(mark(X1), X2, X3) -> c_26(U21^#(X1, X2, X3))
     , U21^#(ok(X1), ok(X2), ok(X3)) -> c_27(U21^#(X1, X2, X3))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , plus^#(X1, mark(X2)) -> c_30(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_31(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_32(plus^#(X1, X2))
     , U31^#(mark(X)) -> c_33(U31^#(X))
     , U31^#(ok(X)) -> c_34(U31^#(X))
     , U41^#(mark(X1), X2, X3) -> c_35(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_36(U41^#(X1, X2, X3))
     , x^#(X1, mark(X2)) -> c_37(x^#(X1, X2))
     , x^#(mark(X1), X2) -> c_38(x^#(X1, X2))
     , x^#(ok(X1), ok(X2)) -> c_39(x^#(X1, X2))
     , and^#(mark(X1), X2) -> c_40(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_41(and^#(X1, X2))
     , isNat^#(ok(X)) -> c_42(isNat^#(X))
     , top^#(mark(X)) -> c_54(top^#(proper(X)))
     , top^#(ok(X)) -> c_55(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), N)) -> mark(N)
     , active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
     , active(U21(tt(), M, N)) -> mark(s(plus(N, M)))
     , active(s(X)) -> s(active(X))
     , active(plus(X1, X2)) -> plus(X1, active(X2))
     , active(plus(X1, X2)) -> plus(active(X1), X2)
     , active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
     , active(plus(N, 0())) -> mark(U11(isNat(N), N))
     , active(U31(X)) -> U31(active(X))
     , active(U31(tt())) -> mark(0())
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(plus(x(N, M), N))
     , active(x(X1, X2)) -> x(X1, active(X2))
     , active(x(X1, X2)) -> x(active(X1), X2)
     , active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
     , active(x(N, 0())) -> mark(U31(isNat(N)))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNat(s(V1))) -> mark(isNat(V1))
     , active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , active(isNat(0())) -> mark(tt())
     , active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
     , U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , plus(X1, mark(X2)) -> mark(plus(X1, X2))
     , plus(mark(X1), X2) -> mark(plus(X1, X2))
     , plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
     , U31(mark(X)) -> mark(U31(X))
     , U31(ok(X)) -> ok(U31(X))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , x(X1, mark(X2)) -> mark(x(X1, X2))
     , x(mark(X1), X2) -> mark(x(X1, X2))
     , x(ok(X1), ok(X2)) -> ok(x(X1, X2))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U21(X1, X2, X3)) ->
       U21(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(U31(X)) -> U31(proper(X))
     , proper(0()) -> ok(0())
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(x(X1, X2)) -> x(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { proper^#(tt()) -> c_44()
     , proper^#(0()) -> c_49() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..