MAYBE

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

Strict Trs:
  { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
  , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
  , a__U12(X1, X2) -> U12(X1, X2)
  , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
  , a__isNat(X) -> isNat(X)
  , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
  , a__isNat(0()) -> tt()
  , a__isNat(plus(V1, V2)) ->
    a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__isNat(x(V1, V2)) ->
    a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
  , a__U13(X) -> U13(X)
  , a__U13(tt()) -> tt()
  , a__U21(X1, X2) -> U21(X1, X2)
  , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
  , a__U22(X) -> U22(X)
  , a__U22(tt()) -> tt()
  , a__U31(X1, X2, X3) -> U31(X1, X2, X3)
  , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
  , a__U32(X1, X2) -> U32(X1, X2)
  , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
  , a__U33(X) -> U33(X)
  , a__U33(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), N) -> mark(N)
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(0()) -> 0()
  , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
  , mark(isNatKind(X)) -> a__isNatKind(X)
  , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
  , mark(and(X1, X2)) -> a__and(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
  , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
  , mark(U13(X)) -> a__U13(mark(X))
  , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
  , mark(U22(X)) -> a__U22(mark(X))
  , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
  , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
  , mark(U33(X)) -> a__U33(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
  , mark(U61(X)) -> a__U61(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
  , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
  , a__plus(X1, X2) -> plus(X1, X2)
  , a__plus(N, s(M)) ->
    a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
  , a__U61(X) -> U61(X)
  , a__U61(tt()) -> 0()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
  , a__x(X1, X2) -> x(X1, X2)
  , a__x(N, s(M)) ->
    a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                  and(isNat(N), isNatKind(N))),
           M,
           N)
  , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
  , a__and(X1, X2) -> and(X1, X2)
  , a__and(tt(), X) -> mark(X)
  , a__isNatKind(X) -> isNatKind(X)
  , a__isNatKind(s(V1)) -> a__isNatKind(V1)
  , a__isNatKind(0()) -> tt()
  , a__isNatKind(plus(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2))
  , a__isNatKind(x(V1, V2)) ->
    a__and(a__isNatKind(V1), isNatKind(V2)) }
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 minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   
   3) '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:
     { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3)
     , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2))
     , a__U12^#(X1, X2) -> c_3(X1, X2)
     , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)))
     , a__U13^#(X) -> c_10(X)
     , a__U13^#(tt()) -> c_11()
     , a__isNat^#(X) -> c_5(X)
     , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1))
     , a__isNat^#(0()) -> c_7()
     , a__isNat^#(plus(V1, V2)) ->
       c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__isNat^#(x(V1, V2)) ->
       c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__U21^#(X1, X2) -> c_12(X1, X2)
     , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1)))
     , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3)
     , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2))
     , a__U22^#(X) -> c_14(X)
     , a__U22^#(tt()) -> c_15()
     , a__U32^#(X1, X2) -> c_18(X1, X2)
     , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2)))
     , a__U33^#(X) -> c_20(X)
     , a__U33^#(tt()) -> c_21()
     , a__U41^#(X1, X2) -> c_22(X1, X2)
     , a__U41^#(tt(), N) -> c_23(mark^#(N))
     , mark^#(tt()) -> c_24()
     , mark^#(s(X)) -> c_25(mark^#(X))
     , mark^#(0()) -> c_26()
     , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2)))
     , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
     , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2)))
     , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2))
     , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
     , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3))
     , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2))
     , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)))
     , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2))
     , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)))
     , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3))
     , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2))
     , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)))
     , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2))
     , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3))
     , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)))
     , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3))
     , a__plus^#(X1, X2) -> c_46(X1, X2)
     , a__plus^#(N, s(M)) ->
       c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__plus^#(N, 0()) ->
       c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N))
     , a__isNatKind^#(X) -> c_58(X)
     , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
     , a__isNatKind^#(0()) -> c_60()
     , a__isNatKind^#(plus(V1, V2)) ->
       c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__isNatKind^#(x(V1, V2)) ->
       c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__x^#(X1, X2) -> c_53(X1, X2)
     , a__x^#(N, s(M)) ->
       c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__x^#(N, 0()) ->
       c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))))
     , a__and^#(X1, X2) -> c_56(X1, X2)
     , a__and^#(tt(), X) -> c_57(mark^#(X))
     , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3)
     , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M)))
     , a__U61^#(X) -> c_49(X)
     , a__U61^#(tt()) -> c_50()
     , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3)
     , a__U71^#(tt(), M, N) ->
       c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3)
     , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2))
     , a__U12^#(X1, X2) -> c_3(X1, X2)
     , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)))
     , a__U13^#(X) -> c_10(X)
     , a__U13^#(tt()) -> c_11()
     , a__isNat^#(X) -> c_5(X)
     , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1))
     , a__isNat^#(0()) -> c_7()
     , a__isNat^#(plus(V1, V2)) ->
       c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__isNat^#(x(V1, V2)) ->
       c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__U21^#(X1, X2) -> c_12(X1, X2)
     , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1)))
     , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3)
     , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2))
     , a__U22^#(X) -> c_14(X)
     , a__U22^#(tt()) -> c_15()
     , a__U32^#(X1, X2) -> c_18(X1, X2)
     , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2)))
     , a__U33^#(X) -> c_20(X)
     , a__U33^#(tt()) -> c_21()
     , a__U41^#(X1, X2) -> c_22(X1, X2)
     , a__U41^#(tt(), N) -> c_23(mark^#(N))
     , mark^#(tt()) -> c_24()
     , mark^#(s(X)) -> c_25(mark^#(X))
     , mark^#(0()) -> c_26()
     , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2)))
     , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
     , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2)))
     , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2))
     , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
     , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3))
     , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2))
     , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)))
     , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2))
     , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)))
     , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3))
     , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2))
     , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)))
     , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2))
     , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3))
     , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)))
     , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3))
     , a__plus^#(X1, X2) -> c_46(X1, X2)
     , a__plus^#(N, s(M)) ->
       c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__plus^#(N, 0()) ->
       c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N))
     , a__isNatKind^#(X) -> c_58(X)
     , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
     , a__isNatKind^#(0()) -> c_60()
     , a__isNatKind^#(plus(V1, V2)) ->
       c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__isNatKind^#(x(V1, V2)) ->
       c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__x^#(X1, X2) -> c_53(X1, X2)
     , a__x^#(N, s(M)) ->
       c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__x^#(N, 0()) ->
       c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))))
     , a__and^#(X1, X2) -> c_56(X1, X2)
     , a__and^#(tt(), X) -> c_57(mark^#(X))
     , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3)
     , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M)))
     , a__U61^#(X) -> c_49(X)
     , a__U61^#(tt()) -> c_50()
     , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3)
     , a__U71^#(tt(), M, N) ->
       c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) }
   Strict Trs:
     { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__isNat(X) -> isNat(X)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(x(V1, V2)) ->
       a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__U13(X) -> U13(X)
     , a__U13(tt()) -> tt()
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(X) -> U22(X)
     , a__U22(tt()) -> tt()
     , a__U31(X1, X2, X3) -> U31(X1, X2, X3)
     , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
     , a__U32(X1, X2) -> U32(X1, X2)
     , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
     , a__U33(X) -> U33(X)
     , a__U33(tt()) -> tt()
     , a__U41(X1, X2) -> U41(X1, X2)
     , a__U41(tt(), N) -> mark(N)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
     , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
     , mark(U33(X)) -> a__U33(mark(X))
     , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
     , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
     , mark(U61(X)) -> a__U61(mark(X))
     , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
     , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
     , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__plus(N, s(M)) ->
       a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
     , a__U61(X) -> U61(X)
     , a__U61(tt()) -> 0()
     , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
     , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
     , a__x(X1, X2) -> x(X1, X2)
     , a__x(N, s(M)) ->
       a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
     , a__and(X1, X2) -> and(X1, X2)
     , a__and(tt(), X) -> mark(X)
     , a__isNatKind(X) -> isNatKind(X)
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(x(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {6,9,17,21,24,26,49,60} by
   applications of Pre({6,9,17,21,24,26,49,60}) =
   {1,3,4,5,7,12,13,14,16,18,19,20,22,23,25,28,31,34,36,39,42,44,47,48,52,54,55,56,57,59,61}.
   Here rules are labeled as follows:
   
     DPs:
       { 1: a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3)
       , 2: a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2))
       , 3: a__U12^#(X1, X2) -> c_3(X1, X2)
       , 4: a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)))
       , 5: a__U13^#(X) -> c_10(X)
       , 6: a__U13^#(tt()) -> c_11()
       , 7: a__isNat^#(X) -> c_5(X)
       , 8: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1))
       , 9: a__isNat^#(0()) -> c_7()
       , 10: a__isNat^#(plus(V1, V2)) ->
             c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
       , 11: a__isNat^#(x(V1, V2)) ->
             c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
       , 12: a__U21^#(X1, X2) -> c_12(X1, X2)
       , 13: a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1)))
       , 14: a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3)
       , 15: a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2))
       , 16: a__U22^#(X) -> c_14(X)
       , 17: a__U22^#(tt()) -> c_15()
       , 18: a__U32^#(X1, X2) -> c_18(X1, X2)
       , 19: a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2)))
       , 20: a__U33^#(X) -> c_20(X)
       , 21: a__U33^#(tt()) -> c_21()
       , 22: a__U41^#(X1, X2) -> c_22(X1, X2)
       , 23: a__U41^#(tt(), N) -> c_23(mark^#(N))
       , 24: mark^#(tt()) -> c_24()
       , 25: mark^#(s(X)) -> c_25(mark^#(X))
       , 26: mark^#(0()) -> c_26()
       , 27: mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2)))
       , 28: mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
       , 29: mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2)))
       , 30: mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2))
       , 31: mark^#(isNat(X)) -> c_31(a__isNat^#(X))
       , 32: mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3))
       , 33: mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2))
       , 34: mark^#(U13(X)) -> c_34(a__U13^#(mark(X)))
       , 35: mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2))
       , 36: mark^#(U22(X)) -> c_36(a__U22^#(mark(X)))
       , 37: mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3))
       , 38: mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2))
       , 39: mark^#(U33(X)) -> c_39(a__U33^#(mark(X)))
       , 40: mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2))
       , 41: mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3))
       , 42: mark^#(U61(X)) -> c_42(a__U61^#(mark(X)))
       , 43: mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3))
       , 44: a__plus^#(X1, X2) -> c_46(X1, X2)
       , 45: a__plus^#(N, s(M)) ->
             c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                                  and(isNat(N), isNatKind(N))),
                           M,
                           N))
       , 46: a__plus^#(N, 0()) ->
             c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N))
       , 47: a__isNatKind^#(X) -> c_58(X)
       , 48: a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
       , 49: a__isNatKind^#(0()) -> c_60()
       , 50: a__isNatKind^#(plus(V1, V2)) ->
             c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)))
       , 51: a__isNatKind^#(x(V1, V2)) ->
             c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)))
       , 52: a__x^#(X1, X2) -> c_53(X1, X2)
       , 53: a__x^#(N, s(M)) ->
             c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                                  and(isNat(N), isNatKind(N))),
                           M,
                           N))
       , 54: a__x^#(N, 0()) ->
             c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))))
       , 55: a__and^#(X1, X2) -> c_56(X1, X2)
       , 56: a__and^#(tt(), X) -> c_57(mark^#(X))
       , 57: a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3)
       , 58: a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M)))
       , 59: a__U61^#(X) -> c_49(X)
       , 60: a__U61^#(tt()) -> c_50()
       , 61: a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3)
       , 62: a__U71^#(tt(), M, N) ->
             c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3)
     , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2))
     , a__U12^#(X1, X2) -> c_3(X1, X2)
     , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2)))
     , a__U13^#(X) -> c_10(X)
     , a__isNat^#(X) -> c_5(X)
     , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1))
     , a__isNat^#(plus(V1, V2)) ->
       c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__isNat^#(x(V1, V2)) ->
       c_9(a__U31^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2))
     , a__U21^#(X1, X2) -> c_12(X1, X2)
     , a__U21^#(tt(), V1) -> c_13(a__U22^#(a__isNat(V1)))
     , a__U31^#(X1, X2, X3) -> c_16(X1, X2, X3)
     , a__U31^#(tt(), V1, V2) -> c_17(a__U32^#(a__isNat(V1), V2))
     , a__U22^#(X) -> c_14(X)
     , a__U32^#(X1, X2) -> c_18(X1, X2)
     , a__U32^#(tt(), V2) -> c_19(a__U33^#(a__isNat(V2)))
     , a__U33^#(X) -> c_20(X)
     , a__U41^#(X1, X2) -> c_22(X1, X2)
     , a__U41^#(tt(), N) -> c_23(mark^#(N))
     , mark^#(s(X)) -> c_25(mark^#(X))
     , mark^#(plus(X1, X2)) -> c_27(a__plus^#(mark(X1), mark(X2)))
     , mark^#(isNatKind(X)) -> c_28(a__isNatKind^#(X))
     , mark^#(x(X1, X2)) -> c_29(a__x^#(mark(X1), mark(X2)))
     , mark^#(and(X1, X2)) -> c_30(a__and^#(mark(X1), X2))
     , mark^#(isNat(X)) -> c_31(a__isNat^#(X))
     , mark^#(U11(X1, X2, X3)) -> c_32(a__U11^#(mark(X1), X2, X3))
     , mark^#(U12(X1, X2)) -> c_33(a__U12^#(mark(X1), X2))
     , mark^#(U13(X)) -> c_34(a__U13^#(mark(X)))
     , mark^#(U21(X1, X2)) -> c_35(a__U21^#(mark(X1), X2))
     , mark^#(U22(X)) -> c_36(a__U22^#(mark(X)))
     , mark^#(U31(X1, X2, X3)) -> c_37(a__U31^#(mark(X1), X2, X3))
     , mark^#(U32(X1, X2)) -> c_38(a__U32^#(mark(X1), X2))
     , mark^#(U33(X)) -> c_39(a__U33^#(mark(X)))
     , mark^#(U41(X1, X2)) -> c_40(a__U41^#(mark(X1), X2))
     , mark^#(U51(X1, X2, X3)) -> c_41(a__U51^#(mark(X1), X2, X3))
     , mark^#(U61(X)) -> c_42(a__U61^#(mark(X)))
     , mark^#(U71(X1, X2, X3)) -> c_43(a__U71^#(mark(X1), X2, X3))
     , a__plus^#(X1, X2) -> c_46(X1, X2)
     , a__plus^#(N, s(M)) ->
       c_47(a__U51^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__plus^#(N, 0()) ->
       c_48(a__U41^#(a__and(a__isNat(N), isNatKind(N)), N))
     , a__isNatKind^#(X) -> c_58(X)
     , a__isNatKind^#(s(V1)) -> c_59(a__isNatKind^#(V1))
     , a__isNatKind^#(plus(V1, V2)) ->
       c_61(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__isNatKind^#(x(V1, V2)) ->
       c_62(a__and^#(a__isNatKind(V1), isNatKind(V2)))
     , a__x^#(X1, X2) -> c_53(X1, X2)
     , a__x^#(N, s(M)) ->
       c_54(a__U71^#(a__and(a__and(a__isNat(M), isNatKind(M)),
                            and(isNat(N), isNatKind(N))),
                     M,
                     N))
     , a__x^#(N, 0()) ->
       c_55(a__U61^#(a__and(a__isNat(N), isNatKind(N))))
     , a__and^#(X1, X2) -> c_56(X1, X2)
     , a__and^#(tt(), X) -> c_57(mark^#(X))
     , a__U51^#(X1, X2, X3) -> c_44(X1, X2, X3)
     , a__U51^#(tt(), M, N) -> c_45(a__plus^#(mark(N), mark(M)))
     , a__U61^#(X) -> c_49(X)
     , a__U71^#(X1, X2, X3) -> c_51(X1, X2, X3)
     , a__U71^#(tt(), M, N) ->
       c_52(a__plus^#(a__x(mark(N), mark(M)), mark(N))) }
   Strict Trs:
     { a__U11(X1, X2, X3) -> U11(X1, X2, X3)
     , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2)
     , a__U12(X1, X2) -> U12(X1, X2)
     , a__U12(tt(), V2) -> a__U13(a__isNat(V2))
     , a__isNat(X) -> isNat(X)
     , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
     , a__isNat(0()) -> tt()
     , a__isNat(plus(V1, V2)) ->
       a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__isNat(x(V1, V2)) ->
       a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
     , a__U13(X) -> U13(X)
     , a__U13(tt()) -> tt()
     , a__U21(X1, X2) -> U21(X1, X2)
     , a__U21(tt(), V1) -> a__U22(a__isNat(V1))
     , a__U22(X) -> U22(X)
     , a__U22(tt()) -> tt()
     , a__U31(X1, X2, X3) -> U31(X1, X2, X3)
     , a__U31(tt(), V1, V2) -> a__U32(a__isNat(V1), V2)
     , a__U32(X1, X2) -> U32(X1, X2)
     , a__U32(tt(), V2) -> a__U33(a__isNat(V2))
     , a__U33(X) -> U33(X)
     , a__U33(tt()) -> tt()
     , a__U41(X1, X2) -> U41(X1, X2)
     , a__U41(tt(), N) -> mark(N)
     , mark(tt()) -> tt()
     , mark(s(X)) -> s(mark(X))
     , mark(0()) -> 0()
     , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
     , mark(isNatKind(X)) -> a__isNatKind(X)
     , mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
     , mark(and(X1, X2)) -> a__and(mark(X1), X2)
     , mark(isNat(X)) -> a__isNat(X)
     , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
     , mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
     , mark(U13(X)) -> a__U13(mark(X))
     , mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
     , mark(U22(X)) -> a__U22(mark(X))
     , mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
     , mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
     , mark(U33(X)) -> a__U33(mark(X))
     , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
     , mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
     , mark(U61(X)) -> a__U61(mark(X))
     , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
     , a__U51(X1, X2, X3) -> U51(X1, X2, X3)
     , a__U51(tt(), M, N) -> s(a__plus(mark(N), mark(M)))
     , a__plus(X1, X2) -> plus(X1, X2)
     , a__plus(N, s(M)) ->
       a__U51(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , a__plus(N, 0()) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
     , a__U61(X) -> U61(X)
     , a__U61(tt()) -> 0()
     , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
     , a__U71(tt(), M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
     , a__x(X1, X2) -> x(X1, X2)
     , a__x(N, s(M)) ->
       a__U71(a__and(a__and(a__isNat(M), isNatKind(M)),
                     and(isNat(N), isNatKind(N))),
              M,
              N)
     , a__x(N, 0()) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
     , a__and(X1, X2) -> and(X1, X2)
     , a__and(tt(), X) -> mark(X)
     , a__isNatKind(X) -> isNatKind(X)
     , a__isNatKind(s(V1)) -> a__isNatKind(V1)
     , a__isNatKind(0()) -> tt()
     , a__isNatKind(plus(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2))
     , a__isNatKind(x(V1, V2)) ->
       a__and(a__isNatKind(V1), isNatKind(V2)) }
   Weak DPs:
     { a__U13^#(tt()) -> c_11()
     , a__isNat^#(0()) -> c_7()
     , a__U22^#(tt()) -> c_15()
     , a__U33^#(tt()) -> c_21()
     , mark^#(tt()) -> c_24()
     , mark^#(0()) -> c_26()
     , a__isNatKind^#(0()) -> c_60()
     , a__U61^#(tt()) -> c_50() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..