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(), V2)) -> mark(U12(isNat(V2)))
  , active(U12(X)) -> U12(active(X))
  , active(U12(tt())) -> mark(tt())
  , active(isNat(s(V1))) -> mark(U21(isNat(V1)))
  , active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
  , active(isNat(0())) -> mark(tt())
  , active(U21(X)) -> U21(active(X))
  , active(U21(tt())) -> mark(tt())
  , active(U31(X1, X2)) -> U31(active(X1), X2)
  , active(U31(tt(), N)) -> mark(N)
  , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
  , active(U41(tt(), M, N)) -> mark(U42(isNat(N), M, N))
  , active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
  , active(U42(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(U41(isNat(M), M, N))
  , active(plus(N, 0())) -> mark(U31(isNat(N), N))
  , U11(mark(X1), X2) -> mark(U11(X1, X2))
  , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
  , U12(mark(X)) -> mark(U12(X))
  , U12(ok(X)) -> ok(U12(X))
  , isNat(ok(X)) -> ok(isNat(X))
  , U21(mark(X)) -> mark(U21(X))
  , U21(ok(X)) -> ok(U21(X))
  , U31(mark(X1), X2) -> mark(U31(X1, X2))
  , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
  , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
  , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
  , U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
  , U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(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))
  , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
  , proper(tt()) -> ok(tt())
  , proper(U12(X)) -> U12(proper(X))
  , proper(isNat(X)) -> isNat(proper(X))
  , proper(U21(X)) -> U21(proper(X))
  , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
  , proper(U41(X1, X2, X3)) ->
    U41(proper(X1), proper(X2), proper(X3))
  , proper(U42(X1, X2, X3)) ->
    U42(proper(X1), proper(X2), proper(X3))
  , proper(s(X)) -> s(proper(X))
  , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
  , proper(0()) -> ok(0())
  , 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(), V2)) -> c_2(U12^#(isNat(V2)))
     , active^#(U12(X)) -> c_3(U12^#(active(X)))
     , active^#(U12(tt())) -> c_4()
     , active^#(isNat(s(V1))) -> c_5(U21^#(isNat(V1)))
     , active^#(isNat(plus(V1, V2))) -> c_6(U11^#(isNat(V1), V2))
     , active^#(isNat(0())) -> c_7()
     , active^#(U21(X)) -> c_8(U21^#(active(X)))
     , active^#(U21(tt())) -> c_9()
     , active^#(U31(X1, X2)) -> c_10(U31^#(active(X1), X2))
     , active^#(U31(tt(), N)) -> c_11(N)
     , active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_13(U42^#(isNat(N), M, N))
     , active^#(U42(X1, X2, X3)) -> c_14(U42^#(active(X1), X2, X3))
     , active^#(U42(tt(), M, N)) -> c_15(s^#(plus(N, M)))
     , active^#(s(X)) -> c_16(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_17(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_18(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) -> c_19(U41^#(isNat(M), M, N))
     , active^#(plus(N, 0())) -> c_20(U31^#(isNat(N), N))
     , U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
     , U12^#(mark(X)) -> c_23(U12^#(X))
     , U12^#(ok(X)) -> c_24(U12^#(X))
     , U21^#(mark(X)) -> c_26(U21^#(X))
     , U21^#(ok(X)) -> c_27(U21^#(X))
     , U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
     , U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
     , U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
     , isNat^#(ok(X)) -> c_25(isNat^#(X))
     , proper^#(U11(X1, X2)) -> c_39(U11^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_40()
     , proper^#(U12(X)) -> c_41(U12^#(proper(X)))
     , proper^#(isNat(X)) -> c_42(isNat^#(proper(X)))
     , proper^#(U21(X)) -> c_43(U21^#(proper(X)))
     , proper^#(U31(X1, X2)) -> c_44(U31^#(proper(X1), proper(X2)))
     , proper^#(U41(X1, X2, X3)) ->
       c_45(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(U42(X1, X2, X3)) ->
       c_46(U42^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_47(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_48(plus^#(proper(X1), proper(X2)))
     , proper^#(0()) -> c_49()
     , top^#(mark(X)) -> c_50(top^#(proper(X)))
     , top^#(ok(X)) -> c_51(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(), V2)) -> c_2(U12^#(isNat(V2)))
     , active^#(U12(X)) -> c_3(U12^#(active(X)))
     , active^#(U12(tt())) -> c_4()
     , active^#(isNat(s(V1))) -> c_5(U21^#(isNat(V1)))
     , active^#(isNat(plus(V1, V2))) -> c_6(U11^#(isNat(V1), V2))
     , active^#(isNat(0())) -> c_7()
     , active^#(U21(X)) -> c_8(U21^#(active(X)))
     , active^#(U21(tt())) -> c_9()
     , active^#(U31(X1, X2)) -> c_10(U31^#(active(X1), X2))
     , active^#(U31(tt(), N)) -> c_11(N)
     , active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_13(U42^#(isNat(N), M, N))
     , active^#(U42(X1, X2, X3)) -> c_14(U42^#(active(X1), X2, X3))
     , active^#(U42(tt(), M, N)) -> c_15(s^#(plus(N, M)))
     , active^#(s(X)) -> c_16(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_17(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_18(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) -> c_19(U41^#(isNat(M), M, N))
     , active^#(plus(N, 0())) -> c_20(U31^#(isNat(N), N))
     , U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
     , U12^#(mark(X)) -> c_23(U12^#(X))
     , U12^#(ok(X)) -> c_24(U12^#(X))
     , U21^#(mark(X)) -> c_26(U21^#(X))
     , U21^#(ok(X)) -> c_27(U21^#(X))
     , U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
     , U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
     , U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
     , isNat^#(ok(X)) -> c_25(isNat^#(X))
     , proper^#(U11(X1, X2)) -> c_39(U11^#(proper(X1), proper(X2)))
     , proper^#(tt()) -> c_40()
     , proper^#(U12(X)) -> c_41(U12^#(proper(X)))
     , proper^#(isNat(X)) -> c_42(isNat^#(proper(X)))
     , proper^#(U21(X)) -> c_43(U21^#(proper(X)))
     , proper^#(U31(X1, X2)) -> c_44(U31^#(proper(X1), proper(X2)))
     , proper^#(U41(X1, X2, X3)) ->
       c_45(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(U42(X1, X2, X3)) ->
       c_46(U42^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_47(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_48(plus^#(proper(X1), proper(X2)))
     , proper^#(0()) -> c_49()
     , top^#(mark(X)) -> c_50(top^#(proper(X)))
     , top^#(ok(X)) -> c_51(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), V2)) -> mark(U12(isNat(V2)))
     , active(U12(X)) -> U12(active(X))
     , active(U12(tt())) -> mark(tt())
     , active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     , active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
     , active(isNat(0())) -> mark(tt())
     , active(U21(X)) -> U21(active(X))
     , active(U21(tt())) -> mark(tt())
     , active(U31(X1, X2)) -> U31(active(X1), X2)
     , active(U31(tt(), N)) -> mark(N)
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(U42(isNat(N), M, N))
     , active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
     , active(U42(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(U41(isNat(M), M, N))
     , active(plus(N, 0())) -> mark(U31(isNat(N), N))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U12(mark(X)) -> mark(U12(X))
     , U12(ok(X)) -> ok(U12(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , U21(mark(X)) -> mark(U21(X))
     , U21(ok(X)) -> ok(U21(X))
     , U31(mark(X1), X2) -> mark(U31(X1, X2))
     , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
     , U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(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))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U12(X)) -> U12(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U21(X)) -> U21(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(U42(X1, X2, X3)) ->
       U42(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , 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_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
     
     2: active^#(U11(tt(), V2)) -> c_2(U12^#(isNat(V2)))
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
     
     3: active^#(U12(X)) -> c_3(U12^#(active(X)))
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
     
     4: active^#(U12(tt())) -> c_4()
     
     5: active^#(isNat(s(V1))) -> c_5(U21^#(isNat(V1)))
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
     
     6: active^#(isNat(plus(V1, V2))) -> c_6(U11^#(isNat(V1), V2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
     
     7: active^#(isNat(0())) -> c_7()
     
     8: active^#(U21(X)) -> c_8(U21^#(active(X)))
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
     
     9: active^#(U21(tt())) -> c_9()
     
     10: active^#(U31(X1, X2)) -> c_10(U31^#(active(X1), X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
     
     11: active^#(U31(tt(), N)) -> c_11(N)
        -->_1 top^#(ok(X)) -> c_51(top^#(active(X))) :51
        -->_1 top^#(mark(X)) -> c_50(top^#(proper(X))) :50
        -->_1 proper^#(plus(X1, X2)) ->
              c_48(plus^#(proper(X1), proper(X2))) :48
        -->_1 proper^#(s(X)) -> c_47(s^#(proper(X))) :47
        -->_1 proper^#(U42(X1, X2, X3)) ->
              c_46(U42^#(proper(X1), proper(X2), proper(X3))) :46
        -->_1 proper^#(U41(X1, X2, X3)) ->
              c_45(U41^#(proper(X1), proper(X2), proper(X3))) :45
        -->_1 proper^#(U31(X1, X2)) ->
              c_44(U31^#(proper(X1), proper(X2))) :44
        -->_1 proper^#(U21(X)) -> c_43(U21^#(proper(X))) :43
        -->_1 proper^#(isNat(X)) -> c_42(isNat^#(proper(X))) :42
        -->_1 proper^#(U12(X)) -> c_41(U12^#(proper(X))) :41
        -->_1 proper^#(U11(X1, X2)) ->
              c_39(U11^#(proper(X1), proper(X2))) :39
        -->_1 isNat^#(ok(X)) -> c_25(isNat^#(X)) :38
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
        -->_1 U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
        -->_1 active^#(plus(N, 0())) -> c_20(U31^#(isNat(N), N)) :20
        -->_1 active^#(plus(N, s(M))) -> c_19(U41^#(isNat(M), M, N)) :19
        -->_1 active^#(plus(X1, X2)) -> c_18(plus^#(active(X1), X2)) :18
        -->_1 active^#(plus(X1, X2)) -> c_17(plus^#(X1, active(X2))) :17
        -->_1 active^#(s(X)) -> c_16(s^#(active(X))) :16
        -->_1 active^#(U42(tt(), M, N)) -> c_15(s^#(plus(N, M))) :15
        -->_1 active^#(U42(X1, X2, X3)) ->
              c_14(U42^#(active(X1), X2, X3)) :14
        -->_1 active^#(U41(tt(), M, N)) -> c_13(U42^#(isNat(N), M, N)) :13
        -->_1 active^#(U41(X1, X2, X3)) ->
              c_12(U41^#(active(X1), X2, X3)) :12
        -->_1 proper^#(0()) -> c_49() :49
        -->_1 proper^#(tt()) -> c_40() :40
        -->_1 active^#(U31(tt(), N)) -> c_11(N) :11
        -->_1 active^#(U31(X1, X2)) -> c_10(U31^#(active(X1), X2)) :10
        -->_1 active^#(U21(tt())) -> c_9() :9
        -->_1 active^#(U21(X)) -> c_8(U21^#(active(X))) :8
        -->_1 active^#(isNat(0())) -> c_7() :7
        -->_1 active^#(isNat(plus(V1, V2))) -> c_6(U11^#(isNat(V1), V2)) :6
        -->_1 active^#(isNat(s(V1))) -> c_5(U21^#(isNat(V1))) :5
        -->_1 active^#(U12(tt())) -> c_4() :4
        -->_1 active^#(U12(X)) -> c_3(U12^#(active(X))) :3
        -->_1 active^#(U11(tt(), V2)) -> c_2(U12^#(isNat(V2))) :2
        -->_1 active^#(U11(X1, X2)) -> c_1(U11^#(active(X1), X2)) :1
     
     12: active^#(U41(X1, X2, X3)) -> c_12(U41^#(active(X1), X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
     
     13: active^#(U41(tt(), M, N)) -> c_13(U42^#(isNat(N), M, N))
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
     
     14: active^#(U42(X1, X2, X3)) -> c_14(U42^#(active(X1), X2, X3))
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
     
     15: active^#(U42(tt(), M, N)) -> c_15(s^#(plus(N, M)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     16: active^#(s(X)) -> c_16(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     17: active^#(plus(X1, X2)) -> c_17(plus^#(X1, active(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     18: active^#(plus(X1, X2)) -> c_18(plus^#(active(X1), X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     19: active^#(plus(N, s(M))) -> c_19(U41^#(isNat(M), M, N))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
     
     20: active^#(plus(N, 0())) -> c_20(U31^#(isNat(N), N))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
     
     21: U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
     
     22: U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
     
     23: U12^#(mark(X)) -> c_23(U12^#(X))
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
     
     24: U12^#(ok(X)) -> c_24(U12^#(X))
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
     
     25: U21^#(mark(X)) -> c_26(U21^#(X))
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
     
     26: U21^#(ok(X)) -> c_27(U21^#(X))
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
     
     27: U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
     
     28: U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
     
     29: U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
     
     30: U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
     
     31: U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
     
     32: U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
     
     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: plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     36: plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     37: plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     38: isNat^#(ok(X)) -> c_25(isNat^#(X))
        -->_1 isNat^#(ok(X)) -> c_25(isNat^#(X)) :38
     
     39: proper^#(U11(X1, X2)) -> c_39(U11^#(proper(X1), proper(X2)))
        -->_1 U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2)) :22
        -->_1 U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2)) :21
     
     40: proper^#(tt()) -> c_40()
     
     41: proper^#(U12(X)) -> c_41(U12^#(proper(X)))
        -->_1 U12^#(ok(X)) -> c_24(U12^#(X)) :24
        -->_1 U12^#(mark(X)) -> c_23(U12^#(X)) :23
     
     42: proper^#(isNat(X)) -> c_42(isNat^#(proper(X)))
        -->_1 isNat^#(ok(X)) -> c_25(isNat^#(X)) :38
     
     43: proper^#(U21(X)) -> c_43(U21^#(proper(X)))
        -->_1 U21^#(ok(X)) -> c_27(U21^#(X)) :26
        -->_1 U21^#(mark(X)) -> c_26(U21^#(X)) :25
     
     44: proper^#(U31(X1, X2)) -> c_44(U31^#(proper(X1), proper(X2)))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2)) :28
        -->_1 U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2)) :27
     
     45: proper^#(U41(X1, X2, X3)) ->
         c_45(U41^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3)) :30
        -->_1 U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3)) :29
     
     46: proper^#(U42(X1, X2, X3)) ->
         c_46(U42^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3)) :32
        -->_1 U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3)) :31
     
     47: proper^#(s(X)) -> c_47(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_35(s^#(X)) :34
        -->_1 s^#(mark(X)) -> c_34(s^#(X)) :33
     
     48: proper^#(plus(X1, X2)) -> c_48(plus^#(proper(X1), proper(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2)) :37
        -->_1 plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2)) :36
        -->_1 plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2)) :35
     
     49: proper^#(0()) -> c_49()
     
     50: top^#(mark(X)) -> c_50(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_51(top^#(active(X))) :51
        -->_1 top^#(mark(X)) -> c_50(top^#(proper(X))) :50
     
     51: top^#(ok(X)) -> c_51(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_51(top^#(active(X))) :51
        -->_1 top^#(mark(X)) -> c_50(top^#(proper(X))) :50
     
   
   Only the nodes
   {21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,37,36,38,40,49,50,51}
   are reachable from nodes
   {21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,40,49,50,51}
   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_21(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
     , U12^#(mark(X)) -> c_23(U12^#(X))
     , U12^#(ok(X)) -> c_24(U12^#(X))
     , U21^#(mark(X)) -> c_26(U21^#(X))
     , U21^#(ok(X)) -> c_27(U21^#(X))
     , U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
     , U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
     , U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
     , isNat^#(ok(X)) -> c_25(isNat^#(X))
     , proper^#(tt()) -> c_40()
     , proper^#(0()) -> c_49()
     , top^#(mark(X)) -> c_50(top^#(proper(X)))
     , top^#(ok(X)) -> c_51(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), V2)) -> mark(U12(isNat(V2)))
     , active(U12(X)) -> U12(active(X))
     , active(U12(tt())) -> mark(tt())
     , active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     , active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
     , active(isNat(0())) -> mark(tt())
     , active(U21(X)) -> U21(active(X))
     , active(U21(tt())) -> mark(tt())
     , active(U31(X1, X2)) -> U31(active(X1), X2)
     , active(U31(tt(), N)) -> mark(N)
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(U42(isNat(N), M, N))
     , active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
     , active(U42(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(U41(isNat(M), M, N))
     , active(plus(N, 0())) -> mark(U31(isNat(N), N))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U12(mark(X)) -> mark(U12(X))
     , U12(ok(X)) -> ok(U12(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , U21(mark(X)) -> mark(U21(X))
     , U21(ok(X)) -> ok(U21(X))
     , U31(mark(X1), X2) -> mark(U31(X1, X2))
     , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
     , U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(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))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U12(X)) -> U12(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U21(X)) -> U21(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(U42(X1, X2, X3)) ->
       U42(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {19,20} by applications of
   Pre({19,20}) = {}. Here rules are labeled as follows:
   
     DPs:
       { 1: U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2))
       , 2: U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
       , 3: U12^#(mark(X)) -> c_23(U12^#(X))
       , 4: U12^#(ok(X)) -> c_24(U12^#(X))
       , 5: U21^#(mark(X)) -> c_26(U21^#(X))
       , 6: U21^#(ok(X)) -> c_27(U21^#(X))
       , 7: U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
       , 8: U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
       , 9: U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
       , 10: U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
       , 11: U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
       , 12: U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
       , 13: s^#(mark(X)) -> c_34(s^#(X))
       , 14: s^#(ok(X)) -> c_35(s^#(X))
       , 15: plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
       , 16: plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
       , 17: plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
       , 18: isNat^#(ok(X)) -> c_25(isNat^#(X))
       , 19: proper^#(tt()) -> c_40()
       , 20: proper^#(0()) -> c_49()
       , 21: top^#(mark(X)) -> c_50(top^#(proper(X)))
       , 22: top^#(ok(X)) -> c_51(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(mark(X1), X2) -> c_21(U11^#(X1, X2))
     , U11^#(ok(X1), ok(X2)) -> c_22(U11^#(X1, X2))
     , U12^#(mark(X)) -> c_23(U12^#(X))
     , U12^#(ok(X)) -> c_24(U12^#(X))
     , U21^#(mark(X)) -> c_26(U21^#(X))
     , U21^#(ok(X)) -> c_27(U21^#(X))
     , U31^#(mark(X1), X2) -> c_28(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_29(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_30(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_31(U41^#(X1, X2, X3))
     , U42^#(mark(X1), X2, X3) -> c_32(U42^#(X1, X2, X3))
     , U42^#(ok(X1), ok(X2), ok(X3)) -> c_33(U42^#(X1, X2, X3))
     , s^#(mark(X)) -> c_34(s^#(X))
     , s^#(ok(X)) -> c_35(s^#(X))
     , plus^#(X1, mark(X2)) -> c_36(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_37(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_38(plus^#(X1, X2))
     , isNat^#(ok(X)) -> c_25(isNat^#(X))
     , top^#(mark(X)) -> c_50(top^#(proper(X)))
     , top^#(ok(X)) -> c_51(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2)) -> U11(active(X1), X2)
     , active(U11(tt(), V2)) -> mark(U12(isNat(V2)))
     , active(U12(X)) -> U12(active(X))
     , active(U12(tt())) -> mark(tt())
     , active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     , active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
     , active(isNat(0())) -> mark(tt())
     , active(U21(X)) -> U21(active(X))
     , active(U21(tt())) -> mark(tt())
     , active(U31(X1, X2)) -> U31(active(X1), X2)
     , active(U31(tt(), N)) -> mark(N)
     , active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
     , active(U41(tt(), M, N)) -> mark(U42(isNat(N), M, N))
     , active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
     , active(U42(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(U41(isNat(M), M, N))
     , active(plus(N, 0())) -> mark(U31(isNat(N), N))
     , U11(mark(X1), X2) -> mark(U11(X1, X2))
     , U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
     , U12(mark(X)) -> mark(U12(X))
     , U12(ok(X)) -> ok(U12(X))
     , isNat(ok(X)) -> ok(isNat(X))
     , U21(mark(X)) -> mark(U21(X))
     , U21(ok(X)) -> ok(U21(X))
     , U31(mark(X1), X2) -> mark(U31(X1, X2))
     , U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
     , U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
     , U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
     , U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
     , U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(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))
     , proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
     , proper(tt()) -> ok(tt())
     , proper(U12(X)) -> U12(proper(X))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U21(X)) -> U21(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(U42(X1, X2, X3)) ->
       U42(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { proper^#(tt()) -> c_40()
     , proper^#(0()) -> c_49() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..