MAYBE

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

Strict Trs:
  { active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
  , active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2))
  , active(U12(X1, X2)) -> U12(active(X1), X2)
  , active(U12(tt(), V2)) -> mark(U13(isNat(V2)))
  , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
  , active(isNat(plus(V1, V2))) ->
    mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
  , active(isNat(0())) -> mark(tt())
  , active(U13(X)) -> U13(active(X))
  , active(U13(tt())) -> mark(tt())
  , active(U21(X1, X2)) -> U21(active(X1), X2)
  , active(U21(tt(), V1)) -> mark(U22(isNat(V1)))
  , active(U22(X)) -> U22(active(X))
  , active(U22(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(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(and(and(isNat(M), isNatKind(M)),
                 and(isNat(N), isNatKind(N))),
             M,
             N))
  , active(plus(N, 0())) -> mark(U31(and(isNat(N), isNatKind(N)), N))
  , active(and(X1, X2)) -> and(active(X1), X2)
  , active(and(tt(), X)) -> mark(X)
  , active(isNatKind(s(V1))) -> mark(isNatKind(V1))
  , active(isNatKind(plus(V1, V2))) ->
    mark(and(isNatKind(V1), isNatKind(V2)))
  , active(isNatKind(0())) -> mark(tt())
  , U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
  , U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
  , U12(mark(X1), X2) -> mark(U12(X1, X2))
  , U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
  , isNat(ok(X)) -> ok(isNat(X))
  , U13(mark(X)) -> mark(U13(X))
  , U13(ok(X)) -> ok(U13(X))
  , U21(mark(X1), X2) -> mark(U21(X1, X2))
  , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
  , U22(mark(X)) -> mark(U22(X))
  , U22(ok(X)) -> ok(U22(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))
  , 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))
  , and(mark(X1), X2) -> mark(and(X1, X2))
  , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
  , isNatKind(ok(X)) -> ok(isNatKind(X))
  , proper(U11(X1, X2, X3)) ->
    U11(proper(X1), proper(X2), proper(X3))
  , proper(tt()) -> ok(tt())
  , proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
  , proper(isNat(X)) -> isNat(proper(X))
  , proper(U13(X)) -> U13(proper(X))
  , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
  , proper(U22(X)) -> U22(proper(X))
  , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
  , proper(U41(X1, X2, X3)) ->
    U41(proper(X1), proper(X2), proper(X3))
  , proper(s(X)) -> s(proper(X))
  , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
  , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
  , proper(0()) -> ok(0())
  , proper(isNatKind(X)) -> isNatKind(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) '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) '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) '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, X3)) -> c_1(U11^#(active(X1), X2, X3))
     , active^#(U11(tt(), V1, V2)) -> c_2(U12^#(isNat(V1), V2))
     , active^#(U12(X1, X2)) -> c_3(U12^#(active(X1), X2))
     , active^#(U12(tt(), V2)) -> c_4(U13^#(isNat(V2)))
     , active^#(isNat(s(V1))) -> c_5(U21^#(isNatKind(V1), V1))
     , active^#(isNat(plus(V1, V2))) ->
       c_6(U11^#(and(isNatKind(V1), isNatKind(V2)), V1, V2))
     , active^#(isNat(0())) -> c_7()
     , active^#(U13(X)) -> c_8(U13^#(active(X)))
     , active^#(U13(tt())) -> c_9()
     , active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2))
     , active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1)))
     , active^#(U22(X)) -> c_12(U22^#(active(X)))
     , active^#(U22(tt())) -> c_13()
     , active^#(U31(X1, X2)) -> c_14(U31^#(active(X1), X2))
     , active^#(U31(tt(), N)) -> c_15(N)
     , active^#(U41(X1, X2, X3)) -> c_16(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_17(s^#(plus(N, M)))
     , active^#(s(X)) -> c_18(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_19(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_20(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) ->
       c_21(U41^#(and(and(isNat(M), isNatKind(M)),
                      and(isNat(N), isNatKind(N))),
                  M,
                  N))
     , active^#(plus(N, 0())) ->
       c_22(U31^#(and(isNat(N), isNatKind(N)), N))
     , active^#(and(X1, X2)) -> c_23(and^#(active(X1), X2))
     , active^#(and(tt(), X)) -> c_24(X)
     , active^#(isNatKind(s(V1))) -> c_25(isNatKind^#(V1))
     , active^#(isNatKind(plus(V1, V2))) ->
       c_26(and^#(isNatKind(V1), isNatKind(V2)))
     , active^#(isNatKind(0())) -> c_27()
     , U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3))
     , U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
     , U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
     , U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
     , U13^#(mark(X)) -> c_33(U13^#(X))
     , U13^#(ok(X)) -> c_34(U13^#(X))
     , U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
     , U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
     , U22^#(mark(X)) -> c_37(U22^#(X))
     , U22^#(ok(X)) -> c_38(U22^#(X))
     , U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
     , s^#(mark(X)) -> c_43(s^#(X))
     , s^#(ok(X)) -> c_44(s^#(X))
     , plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
     , and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
     , isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
     , isNat^#(ok(X)) -> c_32(isNat^#(X))
     , proper^#(U11(X1, X2, X3)) ->
       c_51(U11^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(tt()) -> c_52()
     , proper^#(U12(X1, X2)) -> c_53(U12^#(proper(X1), proper(X2)))
     , proper^#(isNat(X)) -> c_54(isNat^#(proper(X)))
     , proper^#(U13(X)) -> c_55(U13^#(proper(X)))
     , proper^#(U21(X1, X2)) -> c_56(U21^#(proper(X1), proper(X2)))
     , proper^#(U22(X)) -> c_57(U22^#(proper(X)))
     , proper^#(U31(X1, X2)) -> c_58(U31^#(proper(X1), proper(X2)))
     , proper^#(U41(X1, X2, X3)) ->
       c_59(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_60(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_61(plus^#(proper(X1), proper(X2)))
     , proper^#(and(X1, X2)) -> c_62(and^#(proper(X1), proper(X2)))
     , proper^#(0()) -> c_63()
     , proper^#(isNatKind(X)) -> c_64(isNatKind^#(proper(X)))
     , top^#(mark(X)) -> c_65(top^#(proper(X)))
     , top^#(ok(X)) -> c_66(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, X3)) -> c_1(U11^#(active(X1), X2, X3))
     , active^#(U11(tt(), V1, V2)) -> c_2(U12^#(isNat(V1), V2))
     , active^#(U12(X1, X2)) -> c_3(U12^#(active(X1), X2))
     , active^#(U12(tt(), V2)) -> c_4(U13^#(isNat(V2)))
     , active^#(isNat(s(V1))) -> c_5(U21^#(isNatKind(V1), V1))
     , active^#(isNat(plus(V1, V2))) ->
       c_6(U11^#(and(isNatKind(V1), isNatKind(V2)), V1, V2))
     , active^#(isNat(0())) -> c_7()
     , active^#(U13(X)) -> c_8(U13^#(active(X)))
     , active^#(U13(tt())) -> c_9()
     , active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2))
     , active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1)))
     , active^#(U22(X)) -> c_12(U22^#(active(X)))
     , active^#(U22(tt())) -> c_13()
     , active^#(U31(X1, X2)) -> c_14(U31^#(active(X1), X2))
     , active^#(U31(tt(), N)) -> c_15(N)
     , active^#(U41(X1, X2, X3)) -> c_16(U41^#(active(X1), X2, X3))
     , active^#(U41(tt(), M, N)) -> c_17(s^#(plus(N, M)))
     , active^#(s(X)) -> c_18(s^#(active(X)))
     , active^#(plus(X1, X2)) -> c_19(plus^#(X1, active(X2)))
     , active^#(plus(X1, X2)) -> c_20(plus^#(active(X1), X2))
     , active^#(plus(N, s(M))) ->
       c_21(U41^#(and(and(isNat(M), isNatKind(M)),
                      and(isNat(N), isNatKind(N))),
                  M,
                  N))
     , active^#(plus(N, 0())) ->
       c_22(U31^#(and(isNat(N), isNatKind(N)), N))
     , active^#(and(X1, X2)) -> c_23(and^#(active(X1), X2))
     , active^#(and(tt(), X)) -> c_24(X)
     , active^#(isNatKind(s(V1))) -> c_25(isNatKind^#(V1))
     , active^#(isNatKind(plus(V1, V2))) ->
       c_26(and^#(isNatKind(V1), isNatKind(V2)))
     , active^#(isNatKind(0())) -> c_27()
     , U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3))
     , U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
     , U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
     , U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
     , U13^#(mark(X)) -> c_33(U13^#(X))
     , U13^#(ok(X)) -> c_34(U13^#(X))
     , U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
     , U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
     , U22^#(mark(X)) -> c_37(U22^#(X))
     , U22^#(ok(X)) -> c_38(U22^#(X))
     , U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
     , s^#(mark(X)) -> c_43(s^#(X))
     , s^#(ok(X)) -> c_44(s^#(X))
     , plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
     , and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
     , isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
     , isNat^#(ok(X)) -> c_32(isNat^#(X))
     , proper^#(U11(X1, X2, X3)) ->
       c_51(U11^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(tt()) -> c_52()
     , proper^#(U12(X1, X2)) -> c_53(U12^#(proper(X1), proper(X2)))
     , proper^#(isNat(X)) -> c_54(isNat^#(proper(X)))
     , proper^#(U13(X)) -> c_55(U13^#(proper(X)))
     , proper^#(U21(X1, X2)) -> c_56(U21^#(proper(X1), proper(X2)))
     , proper^#(U22(X)) -> c_57(U22^#(proper(X)))
     , proper^#(U31(X1, X2)) -> c_58(U31^#(proper(X1), proper(X2)))
     , proper^#(U41(X1, X2, X3)) ->
       c_59(U41^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(s(X)) -> c_60(s^#(proper(X)))
     , proper^#(plus(X1, X2)) -> c_61(plus^#(proper(X1), proper(X2)))
     , proper^#(and(X1, X2)) -> c_62(and^#(proper(X1), proper(X2)))
     , proper^#(0()) -> c_63()
     , proper^#(isNatKind(X)) -> c_64(isNatKind^#(proper(X)))
     , top^#(mark(X)) -> c_65(top^#(proper(X)))
     , top^#(ok(X)) -> c_66(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
     , active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2))
     , active(U12(X1, X2)) -> U12(active(X1), X2)
     , active(U12(tt(), V2)) -> mark(U13(isNat(V2)))
     , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
     , active(isNat(plus(V1, V2))) ->
       mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
     , active(isNat(0())) -> mark(tt())
     , active(U13(X)) -> U13(active(X))
     , active(U13(tt())) -> mark(tt())
     , active(U21(X1, X2)) -> U21(active(X1), X2)
     , active(U21(tt(), V1)) -> mark(U22(isNat(V1)))
     , active(U22(X)) -> U22(active(X))
     , active(U22(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(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(and(and(isNat(M), isNatKind(M)),
                    and(isNat(N), isNatKind(N))),
                M,
                N))
     , active(plus(N, 0())) -> mark(U31(and(isNat(N), isNatKind(N)), N))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     , active(isNatKind(plus(V1, V2))) ->
       mark(and(isNatKind(V1), isNatKind(V2)))
     , active(isNatKind(0())) -> mark(tt())
     , U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
     , U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
     , U12(mark(X1), X2) -> mark(U12(X1, X2))
     , U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , U13(mark(X)) -> mark(U13(X))
     , U13(ok(X)) -> ok(U13(X))
     , U21(mark(X1), X2) -> mark(U21(X1, X2))
     , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
     , U22(mark(X)) -> mark(U22(X))
     , U22(ok(X)) -> ok(U22(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))
     , 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))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatKind(ok(X)) -> ok(isNatKind(X))
     , proper(U11(X1, X2, X3)) ->
       U11(proper(X1), proper(X2), proper(X3))
     , proper(tt()) -> ok(tt())
     , proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U13(X)) -> U13(proper(X))
     , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
     , proper(U22(X)) -> U22(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , proper(isNatKind(X)) -> isNatKind(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, X3)) -> c_1(U11^#(active(X1), X2, X3))
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
     
     2: active^#(U11(tt(), V1, V2)) -> c_2(U12^#(isNat(V1), V2))
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
     
     3: active^#(U12(X1, X2)) -> c_3(U12^#(active(X1), X2))
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
     
     4: active^#(U12(tt(), V2)) -> c_4(U13^#(isNat(V2)))
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
     
     5: active^#(isNat(s(V1))) -> c_5(U21^#(isNatKind(V1), V1))
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
     
     6: active^#(isNat(plus(V1, V2))) ->
        c_6(U11^#(and(isNatKind(V1), isNatKind(V2)), V1, V2))
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
     
     7: active^#(isNat(0())) -> c_7()
     
     8: active^#(U13(X)) -> c_8(U13^#(active(X)))
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
     
     9: active^#(U13(tt())) -> c_9()
     
     10: active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2))
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
     
     11: active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1)))
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
     
     12: active^#(U22(X)) -> c_12(U22^#(active(X)))
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
     
     13: active^#(U22(tt())) -> c_13()
     
     14: active^#(U31(X1, X2)) -> c_14(U31^#(active(X1), X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
     
     15: active^#(U31(tt(), N)) -> c_15(N)
        -->_1 top^#(ok(X)) -> c_66(top^#(active(X))) :66
        -->_1 top^#(mark(X)) -> c_65(top^#(proper(X))) :65
        -->_1 proper^#(isNatKind(X)) -> c_64(isNatKind^#(proper(X))) :64
        -->_1 proper^#(and(X1, X2)) ->
              c_62(and^#(proper(X1), proper(X2))) :62
        -->_1 proper^#(plus(X1, X2)) ->
              c_61(plus^#(proper(X1), proper(X2))) :61
        -->_1 proper^#(s(X)) -> c_60(s^#(proper(X))) :60
        -->_1 proper^#(U41(X1, X2, X3)) ->
              c_59(U41^#(proper(X1), proper(X2), proper(X3))) :59
        -->_1 proper^#(U31(X1, X2)) ->
              c_58(U31^#(proper(X1), proper(X2))) :58
        -->_1 proper^#(U22(X)) -> c_57(U22^#(proper(X))) :57
        -->_1 proper^#(U21(X1, X2)) ->
              c_56(U21^#(proper(X1), proper(X2))) :56
        -->_1 proper^#(U13(X)) -> c_55(U13^#(proper(X))) :55
        -->_1 proper^#(isNat(X)) -> c_54(isNat^#(proper(X))) :54
        -->_1 proper^#(U12(X1, X2)) ->
              c_53(U12^#(proper(X1), proper(X2))) :53
        -->_1 proper^#(U11(X1, X2, X3)) ->
              c_51(U11^#(proper(X1), proper(X2), proper(X3))) :51
        -->_1 isNat^#(ok(X)) -> c_32(isNat^#(X)) :50
        -->_1 isNatKind^#(ok(X)) -> c_50(isNatKind^#(X)) :49
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
        -->_1 active^#(isNatKind(plus(V1, V2))) ->
              c_26(and^#(isNatKind(V1), isNatKind(V2))) :26
        -->_1 active^#(isNatKind(s(V1))) -> c_25(isNatKind^#(V1)) :25
        -->_1 active^#(and(tt(), X)) -> c_24(X) :24
        -->_1 active^#(and(X1, X2)) -> c_23(and^#(active(X1), X2)) :23
        -->_1 active^#(plus(N, 0())) ->
              c_22(U31^#(and(isNat(N), isNatKind(N)), N)) :22
        -->_1 active^#(plus(N, s(M))) ->
              c_21(U41^#(and(and(isNat(M), isNatKind(M)),
                             and(isNat(N), isNatKind(N))),
                         M,
                         N)) :21
        -->_1 active^#(plus(X1, X2)) -> c_20(plus^#(active(X1), X2)) :20
        -->_1 active^#(plus(X1, X2)) -> c_19(plus^#(X1, active(X2))) :19
        -->_1 active^#(s(X)) -> c_18(s^#(active(X))) :18
        -->_1 active^#(U41(tt(), M, N)) -> c_17(s^#(plus(N, M))) :17
        -->_1 active^#(U41(X1, X2, X3)) ->
              c_16(U41^#(active(X1), X2, X3)) :16
        -->_1 proper^#(0()) -> c_63() :63
        -->_1 proper^#(tt()) -> c_52() :52
        -->_1 active^#(isNatKind(0())) -> c_27() :27
        -->_1 active^#(U31(tt(), N)) -> c_15(N) :15
        -->_1 active^#(U31(X1, X2)) -> c_14(U31^#(active(X1), X2)) :14
        -->_1 active^#(U22(tt())) -> c_13() :13
        -->_1 active^#(U22(X)) -> c_12(U22^#(active(X))) :12
        -->_1 active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) :11
        -->_1 active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) :10
        -->_1 active^#(U13(tt())) -> c_9() :9
        -->_1 active^#(U13(X)) -> c_8(U13^#(active(X))) :8
        -->_1 active^#(isNat(0())) -> c_7() :7
        -->_1 active^#(isNat(plus(V1, V2))) ->
              c_6(U11^#(and(isNatKind(V1), isNatKind(V2)), V1, V2)) :6
        -->_1 active^#(isNat(s(V1))) -> c_5(U21^#(isNatKind(V1), V1)) :5
        -->_1 active^#(U12(tt(), V2)) -> c_4(U13^#(isNat(V2))) :4
        -->_1 active^#(U12(X1, X2)) -> c_3(U12^#(active(X1), X2)) :3
        -->_1 active^#(U11(tt(), V1, V2)) -> c_2(U12^#(isNat(V1), V2)) :2
        -->_1 active^#(U11(X1, X2, X3)) ->
              c_1(U11^#(active(X1), X2, X3)) :1
     
     16: active^#(U41(X1, X2, X3)) -> c_16(U41^#(active(X1), X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
     
     17: active^#(U41(tt(), M, N)) -> c_17(s^#(plus(N, M)))
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
     
     18: active^#(s(X)) -> c_18(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
     
     19: active^#(plus(X1, X2)) -> c_19(plus^#(X1, active(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     20: active^#(plus(X1, X2)) -> c_20(plus^#(active(X1), X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     21: active^#(plus(N, s(M))) ->
         c_21(U41^#(and(and(isNat(M), isNatKind(M)),
                        and(isNat(N), isNatKind(N))),
                    M,
                    N))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
     
     22: active^#(plus(N, 0())) ->
         c_22(U31^#(and(isNat(N), isNatKind(N)), N))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
     
     23: active^#(and(X1, X2)) -> c_23(and^#(active(X1), X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
     
     24: active^#(and(tt(), X)) -> c_24(X)
        -->_1 top^#(ok(X)) -> c_66(top^#(active(X))) :66
        -->_1 top^#(mark(X)) -> c_65(top^#(proper(X))) :65
        -->_1 proper^#(isNatKind(X)) -> c_64(isNatKind^#(proper(X))) :64
        -->_1 proper^#(and(X1, X2)) ->
              c_62(and^#(proper(X1), proper(X2))) :62
        -->_1 proper^#(plus(X1, X2)) ->
              c_61(plus^#(proper(X1), proper(X2))) :61
        -->_1 proper^#(s(X)) -> c_60(s^#(proper(X))) :60
        -->_1 proper^#(U41(X1, X2, X3)) ->
              c_59(U41^#(proper(X1), proper(X2), proper(X3))) :59
        -->_1 proper^#(U31(X1, X2)) ->
              c_58(U31^#(proper(X1), proper(X2))) :58
        -->_1 proper^#(U22(X)) -> c_57(U22^#(proper(X))) :57
        -->_1 proper^#(U21(X1, X2)) ->
              c_56(U21^#(proper(X1), proper(X2))) :56
        -->_1 proper^#(U13(X)) -> c_55(U13^#(proper(X))) :55
        -->_1 proper^#(isNat(X)) -> c_54(isNat^#(proper(X))) :54
        -->_1 proper^#(U12(X1, X2)) ->
              c_53(U12^#(proper(X1), proper(X2))) :53
        -->_1 proper^#(U11(X1, X2, X3)) ->
              c_51(U11^#(proper(X1), proper(X2), proper(X3))) :51
        -->_1 isNat^#(ok(X)) -> c_32(isNat^#(X)) :50
        -->_1 isNatKind^#(ok(X)) -> c_50(isNatKind^#(X)) :49
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
        -->_1 active^#(isNatKind(plus(V1, V2))) ->
              c_26(and^#(isNatKind(V1), isNatKind(V2))) :26
        -->_1 active^#(isNatKind(s(V1))) -> c_25(isNatKind^#(V1)) :25
        -->_1 proper^#(0()) -> c_63() :63
        -->_1 proper^#(tt()) -> c_52() :52
        -->_1 active^#(isNatKind(0())) -> c_27() :27
        -->_1 active^#(and(tt(), X)) -> c_24(X) :24
        -->_1 active^#(and(X1, X2)) -> c_23(and^#(active(X1), X2)) :23
        -->_1 active^#(plus(N, 0())) ->
              c_22(U31^#(and(isNat(N), isNatKind(N)), N)) :22
        -->_1 active^#(plus(N, s(M))) ->
              c_21(U41^#(and(and(isNat(M), isNatKind(M)),
                             and(isNat(N), isNatKind(N))),
                         M,
                         N)) :21
        -->_1 active^#(plus(X1, X2)) -> c_20(plus^#(active(X1), X2)) :20
        -->_1 active^#(plus(X1, X2)) -> c_19(plus^#(X1, active(X2))) :19
        -->_1 active^#(s(X)) -> c_18(s^#(active(X))) :18
        -->_1 active^#(U41(tt(), M, N)) -> c_17(s^#(plus(N, M))) :17
        -->_1 active^#(U41(X1, X2, X3)) ->
              c_16(U41^#(active(X1), X2, X3)) :16
        -->_1 active^#(U31(tt(), N)) -> c_15(N) :15
        -->_1 active^#(U31(X1, X2)) -> c_14(U31^#(active(X1), X2)) :14
        -->_1 active^#(U22(tt())) -> c_13() :13
        -->_1 active^#(U22(X)) -> c_12(U22^#(active(X))) :12
        -->_1 active^#(U21(tt(), V1)) -> c_11(U22^#(isNat(V1))) :11
        -->_1 active^#(U21(X1, X2)) -> c_10(U21^#(active(X1), X2)) :10
        -->_1 active^#(U13(tt())) -> c_9() :9
        -->_1 active^#(U13(X)) -> c_8(U13^#(active(X))) :8
        -->_1 active^#(isNat(0())) -> c_7() :7
        -->_1 active^#(isNat(plus(V1, V2))) ->
              c_6(U11^#(and(isNatKind(V1), isNatKind(V2)), V1, V2)) :6
        -->_1 active^#(isNat(s(V1))) -> c_5(U21^#(isNatKind(V1), V1)) :5
        -->_1 active^#(U12(tt(), V2)) -> c_4(U13^#(isNat(V2))) :4
        -->_1 active^#(U12(X1, X2)) -> c_3(U12^#(active(X1), X2)) :3
        -->_1 active^#(U11(tt(), V1, V2)) -> c_2(U12^#(isNat(V1), V2)) :2
        -->_1 active^#(U11(X1, X2, X3)) ->
              c_1(U11^#(active(X1), X2, X3)) :1
     
     25: active^#(isNatKind(s(V1))) -> c_25(isNatKind^#(V1))
        -->_1 isNatKind^#(ok(X)) -> c_50(isNatKind^#(X)) :49
     
     26: active^#(isNatKind(plus(V1, V2))) ->
         c_26(and^#(isNatKind(V1), isNatKind(V2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
     
     27: active^#(isNatKind(0())) -> c_27()
     
     28: U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3))
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
     
     29: U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
     
     30: U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
     
     31: U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
     
     32: U13^#(mark(X)) -> c_33(U13^#(X))
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
     
     33: U13^#(ok(X)) -> c_34(U13^#(X))
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
     
     34: U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
     
     35: U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
     
     36: U22^#(mark(X)) -> c_37(U22^#(X))
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
     
     37: U22^#(ok(X)) -> c_38(U22^#(X))
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
     
     38: U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
     
     39: U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
     
     40: U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
     
     41: U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
     
     42: s^#(mark(X)) -> c_43(s^#(X))
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
     
     43: s^#(ok(X)) -> c_44(s^#(X))
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
     
     44: plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     45: plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     46: plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     47: and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
     
     48: and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
     
     49: isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
        -->_1 isNatKind^#(ok(X)) -> c_50(isNatKind^#(X)) :49
     
     50: isNat^#(ok(X)) -> c_32(isNat^#(X))
        -->_1 isNat^#(ok(X)) -> c_32(isNat^#(X)) :50
     
     51: proper^#(U11(X1, X2, X3)) ->
         c_51(U11^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3)) :29
        -->_1 U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3)) :28
     
     52: proper^#(tt()) -> c_52()
     
     53: proper^#(U12(X1, X2)) -> c_53(U12^#(proper(X1), proper(X2)))
        -->_1 U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2)) :31
        -->_1 U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2)) :30
     
     54: proper^#(isNat(X)) -> c_54(isNat^#(proper(X)))
        -->_1 isNat^#(ok(X)) -> c_32(isNat^#(X)) :50
     
     55: proper^#(U13(X)) -> c_55(U13^#(proper(X)))
        -->_1 U13^#(ok(X)) -> c_34(U13^#(X)) :33
        -->_1 U13^#(mark(X)) -> c_33(U13^#(X)) :32
     
     56: proper^#(U21(X1, X2)) -> c_56(U21^#(proper(X1), proper(X2)))
        -->_1 U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2)) :35
        -->_1 U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2)) :34
     
     57: proper^#(U22(X)) -> c_57(U22^#(proper(X)))
        -->_1 U22^#(ok(X)) -> c_38(U22^#(X)) :37
        -->_1 U22^#(mark(X)) -> c_37(U22^#(X)) :36
     
     58: proper^#(U31(X1, X2)) -> c_58(U31^#(proper(X1), proper(X2)))
        -->_1 U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2)) :39
        -->_1 U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2)) :38
     
     59: proper^#(U41(X1, X2, X3)) ->
         c_59(U41^#(proper(X1), proper(X2), proper(X3)))
        -->_1 U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3)) :41
        -->_1 U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3)) :40
     
     60: proper^#(s(X)) -> c_60(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_44(s^#(X)) :43
        -->_1 s^#(mark(X)) -> c_43(s^#(X)) :42
     
     61: proper^#(plus(X1, X2)) -> c_61(plus^#(proper(X1), proper(X2)))
        -->_1 plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2)) :46
        -->_1 plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2)) :45
        -->_1 plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2)) :44
     
     62: proper^#(and(X1, X2)) -> c_62(and^#(proper(X1), proper(X2)))
        -->_1 and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2)) :48
        -->_1 and^#(mark(X1), X2) -> c_48(and^#(X1, X2)) :47
     
     63: proper^#(0()) -> c_63()
     
     64: proper^#(isNatKind(X)) -> c_64(isNatKind^#(proper(X)))
        -->_1 isNatKind^#(ok(X)) -> c_50(isNatKind^#(X)) :49
     
     65: top^#(mark(X)) -> c_65(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_66(top^#(active(X))) :66
        -->_1 top^#(mark(X)) -> c_65(top^#(proper(X))) :65
     
     66: top^#(ok(X)) -> c_66(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_66(top^#(active(X))) :66
        -->_1 top^#(mark(X)) -> c_65(top^#(proper(X))) :65
     
   
   Only the nodes
   {28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,46,45,47,48,49,50,52,63,65,66}
   are reachable from nodes
   {28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,52,63,65,66}
   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, X3) -> c_28(U11^#(X1, X2, X3))
     , U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
     , U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
     , U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
     , U13^#(mark(X)) -> c_33(U13^#(X))
     , U13^#(ok(X)) -> c_34(U13^#(X))
     , U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
     , U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
     , U22^#(mark(X)) -> c_37(U22^#(X))
     , U22^#(ok(X)) -> c_38(U22^#(X))
     , U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
     , s^#(mark(X)) -> c_43(s^#(X))
     , s^#(ok(X)) -> c_44(s^#(X))
     , plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
     , and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
     , isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
     , isNat^#(ok(X)) -> c_32(isNat^#(X))
     , proper^#(tt()) -> c_52()
     , proper^#(0()) -> c_63()
     , top^#(mark(X)) -> c_65(top^#(proper(X)))
     , top^#(ok(X)) -> c_66(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
     , active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2))
     , active(U12(X1, X2)) -> U12(active(X1), X2)
     , active(U12(tt(), V2)) -> mark(U13(isNat(V2)))
     , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
     , active(isNat(plus(V1, V2))) ->
       mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
     , active(isNat(0())) -> mark(tt())
     , active(U13(X)) -> U13(active(X))
     , active(U13(tt())) -> mark(tt())
     , active(U21(X1, X2)) -> U21(active(X1), X2)
     , active(U21(tt(), V1)) -> mark(U22(isNat(V1)))
     , active(U22(X)) -> U22(active(X))
     , active(U22(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(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(and(and(isNat(M), isNatKind(M)),
                    and(isNat(N), isNatKind(N))),
                M,
                N))
     , active(plus(N, 0())) -> mark(U31(and(isNat(N), isNatKind(N)), N))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     , active(isNatKind(plus(V1, V2))) ->
       mark(and(isNatKind(V1), isNatKind(V2)))
     , active(isNatKind(0())) -> mark(tt())
     , U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
     , U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
     , U12(mark(X1), X2) -> mark(U12(X1, X2))
     , U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , U13(mark(X)) -> mark(U13(X))
     , U13(ok(X)) -> ok(U13(X))
     , U21(mark(X1), X2) -> mark(U21(X1, X2))
     , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
     , U22(mark(X)) -> mark(U22(X))
     , U22(ok(X)) -> ok(U22(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))
     , 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))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatKind(ok(X)) -> ok(isNatKind(X))
     , proper(U11(X1, X2, X3)) ->
       U11(proper(X1), proper(X2), proper(X3))
     , proper(tt()) -> ok(tt())
     , proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U13(X)) -> U13(proper(X))
     , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
     , proper(U22(X)) -> U22(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , proper(isNatKind(X)) -> isNatKind(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 {24,25} by applications of
   Pre({24,25}) = {}. Here rules are labeled as follows:
   
     DPs:
       { 1: U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3))
       , 2: U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
       , 3: U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
       , 4: U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
       , 5: U13^#(mark(X)) -> c_33(U13^#(X))
       , 6: U13^#(ok(X)) -> c_34(U13^#(X))
       , 7: U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
       , 8: U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
       , 9: U22^#(mark(X)) -> c_37(U22^#(X))
       , 10: U22^#(ok(X)) -> c_38(U22^#(X))
       , 11: U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
       , 12: U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
       , 13: U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
       , 14: U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
       , 15: s^#(mark(X)) -> c_43(s^#(X))
       , 16: s^#(ok(X)) -> c_44(s^#(X))
       , 17: plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
       , 18: plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
       , 19: plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
       , 20: and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
       , 21: and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
       , 22: isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
       , 23: isNat^#(ok(X)) -> c_32(isNat^#(X))
       , 24: proper^#(tt()) -> c_52()
       , 25: proper^#(0()) -> c_63()
       , 26: top^#(mark(X)) -> c_65(top^#(proper(X)))
       , 27: top^#(ok(X)) -> c_66(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(mark(X1), X2, X3) -> c_28(U11^#(X1, X2, X3))
     , U11^#(ok(X1), ok(X2), ok(X3)) -> c_29(U11^#(X1, X2, X3))
     , U12^#(mark(X1), X2) -> c_30(U12^#(X1, X2))
     , U12^#(ok(X1), ok(X2)) -> c_31(U12^#(X1, X2))
     , U13^#(mark(X)) -> c_33(U13^#(X))
     , U13^#(ok(X)) -> c_34(U13^#(X))
     , U21^#(mark(X1), X2) -> c_35(U21^#(X1, X2))
     , U21^#(ok(X1), ok(X2)) -> c_36(U21^#(X1, X2))
     , U22^#(mark(X)) -> c_37(U22^#(X))
     , U22^#(ok(X)) -> c_38(U22^#(X))
     , U31^#(mark(X1), X2) -> c_39(U31^#(X1, X2))
     , U31^#(ok(X1), ok(X2)) -> c_40(U31^#(X1, X2))
     , U41^#(mark(X1), X2, X3) -> c_41(U41^#(X1, X2, X3))
     , U41^#(ok(X1), ok(X2), ok(X3)) -> c_42(U41^#(X1, X2, X3))
     , s^#(mark(X)) -> c_43(s^#(X))
     , s^#(ok(X)) -> c_44(s^#(X))
     , plus^#(X1, mark(X2)) -> c_45(plus^#(X1, X2))
     , plus^#(mark(X1), X2) -> c_46(plus^#(X1, X2))
     , plus^#(ok(X1), ok(X2)) -> c_47(plus^#(X1, X2))
     , and^#(mark(X1), X2) -> c_48(and^#(X1, X2))
     , and^#(ok(X1), ok(X2)) -> c_49(and^#(X1, X2))
     , isNatKind^#(ok(X)) -> c_50(isNatKind^#(X))
     , isNat^#(ok(X)) -> c_32(isNat^#(X))
     , top^#(mark(X)) -> c_65(top^#(proper(X)))
     , top^#(ok(X)) -> c_66(top^#(active(X))) }
   Strict Trs:
     { active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
     , active(U11(tt(), V1, V2)) -> mark(U12(isNat(V1), V2))
     , active(U12(X1, X2)) -> U12(active(X1), X2)
     , active(U12(tt(), V2)) -> mark(U13(isNat(V2)))
     , active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
     , active(isNat(plus(V1, V2))) ->
       mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
     , active(isNat(0())) -> mark(tt())
     , active(U13(X)) -> U13(active(X))
     , active(U13(tt())) -> mark(tt())
     , active(U21(X1, X2)) -> U21(active(X1), X2)
     , active(U21(tt(), V1)) -> mark(U22(isNat(V1)))
     , active(U22(X)) -> U22(active(X))
     , active(U22(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(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(and(and(isNat(M), isNatKind(M)),
                    and(isNat(N), isNatKind(N))),
                M,
                N))
     , active(plus(N, 0())) -> mark(U31(and(isNat(N), isNatKind(N)), N))
     , active(and(X1, X2)) -> and(active(X1), X2)
     , active(and(tt(), X)) -> mark(X)
     , active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     , active(isNatKind(plus(V1, V2))) ->
       mark(and(isNatKind(V1), isNatKind(V2)))
     , active(isNatKind(0())) -> mark(tt())
     , U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
     , U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
     , U12(mark(X1), X2) -> mark(U12(X1, X2))
     , U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
     , isNat(ok(X)) -> ok(isNat(X))
     , U13(mark(X)) -> mark(U13(X))
     , U13(ok(X)) -> ok(U13(X))
     , U21(mark(X1), X2) -> mark(U21(X1, X2))
     , U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
     , U22(mark(X)) -> mark(U22(X))
     , U22(ok(X)) -> ok(U22(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))
     , 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))
     , and(mark(X1), X2) -> mark(and(X1, X2))
     , and(ok(X1), ok(X2)) -> ok(and(X1, X2))
     , isNatKind(ok(X)) -> ok(isNatKind(X))
     , proper(U11(X1, X2, X3)) ->
       U11(proper(X1), proper(X2), proper(X3))
     , proper(tt()) -> ok(tt())
     , proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
     , proper(isNat(X)) -> isNat(proper(X))
     , proper(U13(X)) -> U13(proper(X))
     , proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
     , proper(U22(X)) -> U22(proper(X))
     , proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
     , proper(U41(X1, X2, X3)) ->
       U41(proper(X1), proper(X2), proper(X3))
     , proper(s(X)) -> s(proper(X))
     , proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
     , proper(and(X1, X2)) -> and(proper(X1), proper(X2))
     , proper(0()) -> ok(0())
     , proper(isNatKind(X)) -> isNatKind(proper(X))
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { proper^#(tt()) -> c_52()
     , proper^#(0()) -> c_63() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..