MAYBE

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

Strict Trs:
  { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
  , U12(tt(), V2) -> U13(isNat(activate(V2)))
  , isNat(n__0()) -> tt()
  , isNat(n__plus(V1, V2)) ->
    U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
        activate(V1),
        activate(V2))
  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
  , isNat(n__x(V1, V2)) ->
    U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
        activate(V1),
        activate(V2))
  , activate(X) -> X
  , activate(n__0()) -> 0()
  , activate(n__plus(X1, X2)) -> plus(X1, X2)
  , activate(n__isNatKind(X)) -> isNatKind(X)
  , activate(n__s(X)) -> s(X)
  , activate(n__x(X1, X2)) -> x(X1, X2)
  , activate(n__and(X1, X2)) -> and(X1, X2)
  , U13(tt()) -> tt()
  , U21(tt(), V1) -> U22(isNat(activate(V1)))
  , U22(tt()) -> tt()
  , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
  , U32(tt(), V2) -> U33(isNat(activate(V2)))
  , U33(tt()) -> tt()
  , U41(tt(), N) -> activate(N)
  , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
  , s(X) -> n__s(X)
  , plus(X1, X2) -> n__plus(X1, X2)
  , plus(N, s(M)) ->
    U51(and(and(isNat(M), n__isNatKind(M)),
            n__and(isNat(N), n__isNatKind(N))),
        M,
        N)
  , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
  , U61(tt()) -> 0()
  , 0() -> n__0()
  , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
  , x(X1, X2) -> n__x(X1, X2)
  , x(N, s(M)) ->
    U71(and(and(isNat(M), n__isNatKind(M)),
            n__and(isNat(N), n__isNatKind(N))),
        M,
        N)
  , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
  , and(X1, X2) -> n__and(X1, X2)
  , and(tt(), X) -> activate(X)
  , isNatKind(X) -> n__isNatKind(X)
  , isNatKind(n__0()) -> tt()
  , isNatKind(n__plus(V1, V2)) ->
    and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
  , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
  , isNatKind(n__x(V1, V2)) ->
    and(isNatKind(activate(V1)), n__isNatKind(activate(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) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) '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
      
      2) '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
      
   
   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:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__0()) -> c_8(0^#())
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , 0^#() -> c_27()
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__0()) -> c_35()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U22^#(tt()) -> c_16()
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U33^#(tt()) -> c_19()
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_26(0^#())
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__0()) -> c_8(0^#())
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , 0^#() -> c_27()
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__0()) -> c_35()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U22^#(tt()) -> c_16()
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U33^#(tt()) -> c_19()
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_26(0^#())
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
     , U12(tt(), V2) -> U13(isNat(activate(V2)))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__isNatKind(X)) -> isNatKind(X)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , activate(n__and(X1, X2)) -> and(X1, X2)
     , U13(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNat(activate(V1)))
     , U22(tt()) -> tt()
     , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
     , U32(tt(), V2) -> U33(isNat(activate(V2)))
     , U33(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) ->
       U51(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) ->
       U71(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
     , and(X1, X2) -> n__and(X1, X2)
     , and(tt(), X) -> activate(X)
     , isNatKind(X) -> n__isNatKind(X)
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
     , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     , isNatKind(n__x(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3,4,17,22,32,34} by
   applications of Pre({3,4,17,22,32,34}) =
   {2,8,10,11,13,18,21,24,26,27,30,33,37}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U11^#(tt(), V1, V2) ->
            c_1(U12^#(isNat(activate(V1)), activate(V2)))
       , 2: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
       , 3: U13^#(tt()) -> c_14()
       , 4: isNat^#(n__0()) -> c_3()
       , 5: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 6: isNat^#(n__s(V1)) ->
            c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 7: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 8: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
       , 9: U31^#(tt(), V1, V2) ->
            c_17(U32^#(isNat(activate(V1)), activate(V2)))
       , 10: activate^#(X) -> c_7(X)
       , 11: activate^#(n__0()) -> c_8(0^#())
       , 12: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 13: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
       , 14: activate^#(n__s(X)) -> c_11(s^#(X))
       , 15: activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
       , 16: activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
       , 17: 0^#() -> c_27()
       , 18: plus^#(X1, X2) -> c_23(X1, X2)
       , 19: plus^#(N, s(M)) ->
             c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 20: plus^#(N, 0()) ->
             c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
       , 21: isNatKind^#(X) -> c_34(X)
       , 22: isNatKind^#(n__0()) -> c_35()
       , 23: isNatKind^#(n__plus(V1, V2)) ->
             c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 24: isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
       , 25: isNatKind^#(n__x(V1, V2)) ->
             c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 26: s^#(X) -> c_22(X)
       , 27: x^#(X1, X2) -> c_29(X1, X2)
       , 28: x^#(N, s(M)) ->
             c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 29: x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
       , 30: and^#(X1, X2) -> c_32(X1, X2)
       , 31: and^#(tt(), X) -> c_33(activate^#(X))
       , 32: U22^#(tt()) -> c_16()
       , 33: U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
       , 34: U33^#(tt()) -> c_19()
       , 35: U41^#(tt(), N) -> c_20(activate^#(N))
       , 36: U51^#(tt(), M, N) ->
             c_21(s^#(plus(activate(N), activate(M))))
       , 37: U61^#(tt()) -> c_26(0^#())
       , 38: U71^#(tt(), M, N) ->
             c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__0()) -> c_8(0^#())
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_26(0^#())
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
     , U12(tt(), V2) -> U13(isNat(activate(V2)))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__isNatKind(X)) -> isNatKind(X)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , activate(n__and(X1, X2)) -> and(X1, X2)
     , U13(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNat(activate(V1)))
     , U22(tt()) -> tt()
     , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
     , U32(tt(), V2) -> U33(isNat(activate(V2)))
     , U33(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) ->
       U51(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) ->
       U71(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
     , and(X1, X2) -> n__and(X1, X2)
     , and(tt(), X) -> activate(X)
     , isNatKind(X) -> n__isNatKind(X)
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
     , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     , isNatKind(n__x(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) }
   Weak DPs:
     { U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , 0^#() -> c_27()
     , isNatKind^#(n__0()) -> c_35()
     , U22^#(tt()) -> c_16()
     , U33^#(tt()) -> c_19() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,6,9,28,31} by
   applications of Pre({2,6,9,28,31}) =
   {1,4,7,8,15,18,22,23,25,26,27,29}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U11^#(tt(), V1, V2) ->
            c_1(U12^#(isNat(activate(V1)), activate(V2)))
       , 2: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
       , 3: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 4: isNat^#(n__s(V1)) ->
            c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 5: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 6: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
       , 7: U31^#(tt(), V1, V2) ->
            c_17(U32^#(isNat(activate(V1)), activate(V2)))
       , 8: activate^#(X) -> c_7(X)
       , 9: activate^#(n__0()) -> c_8(0^#())
       , 10: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 11: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
       , 12: activate^#(n__s(X)) -> c_11(s^#(X))
       , 13: activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
       , 14: activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
       , 15: plus^#(X1, X2) -> c_23(X1, X2)
       , 16: plus^#(N, s(M)) ->
             c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 17: plus^#(N, 0()) ->
             c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
       , 18: isNatKind^#(X) -> c_34(X)
       , 19: isNatKind^#(n__plus(V1, V2)) ->
             c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 20: isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
       , 21: isNatKind^#(n__x(V1, V2)) ->
             c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 22: s^#(X) -> c_22(X)
       , 23: x^#(X1, X2) -> c_29(X1, X2)
       , 24: x^#(N, s(M)) ->
             c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 25: x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
       , 26: and^#(X1, X2) -> c_32(X1, X2)
       , 27: and^#(tt(), X) -> c_33(activate^#(X))
       , 28: U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
       , 29: U41^#(tt(), N) -> c_20(activate^#(N))
       , 30: U51^#(tt(), M, N) ->
             c_21(s^#(plus(activate(N), activate(M))))
       , 31: U61^#(tt()) -> c_26(0^#())
       , 32: U71^#(tt(), M, N) ->
             c_28(plus^#(x(activate(N), activate(M)), activate(N)))
       , 33: U13^#(tt()) -> c_14()
       , 34: isNat^#(n__0()) -> c_3()
       , 35: 0^#() -> c_27()
       , 36: isNatKind^#(n__0()) -> c_35()
       , 37: U22^#(tt()) -> c_16()
       , 38: U33^#(tt()) -> c_19() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
     , U12(tt(), V2) -> U13(isNat(activate(V2)))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__isNatKind(X)) -> isNatKind(X)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , activate(n__and(X1, X2)) -> and(X1, X2)
     , U13(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNat(activate(V1)))
     , U22(tt()) -> tt()
     , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
     , U32(tt(), V2) -> U33(isNat(activate(V2)))
     , U33(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) ->
       U51(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) ->
       U71(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
     , and(X1, X2) -> n__and(X1, X2)
     , and(tt(), X) -> activate(X)
     , isNatKind(X) -> n__isNatKind(X)
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
     , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     , isNatKind(n__x(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) }
   Weak DPs:
     { U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_27()
     , isNatKind^#(n__0()) -> c_35()
     , U22^#(tt()) -> c_16()
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U33^#(tt()) -> c_19()
     , U61^#(tt()) -> c_26(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,3,5,22} by applications
   of Pre({1,3,5,22}) = {2,4,6,10,12,15,19,20,23}. Here rules are
   labeled as follows:
   
     DPs:
       { 1: U11^#(tt(), V1, V2) ->
            c_1(U12^#(isNat(activate(V1)), activate(V2)))
       , 2: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 3: isNat^#(n__s(V1)) ->
            c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 4: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 5: U31^#(tt(), V1, V2) ->
            c_17(U32^#(isNat(activate(V1)), activate(V2)))
       , 6: activate^#(X) -> c_7(X)
       , 7: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 8: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
       , 9: activate^#(n__s(X)) -> c_11(s^#(X))
       , 10: activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
       , 11: activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
       , 12: plus^#(X1, X2) -> c_23(X1, X2)
       , 13: plus^#(N, s(M)) ->
             c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 14: plus^#(N, 0()) ->
             c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
       , 15: isNatKind^#(X) -> c_34(X)
       , 16: isNatKind^#(n__plus(V1, V2)) ->
             c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 17: isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
       , 18: isNatKind^#(n__x(V1, V2)) ->
             c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 19: s^#(X) -> c_22(X)
       , 20: x^#(X1, X2) -> c_29(X1, X2)
       , 21: x^#(N, s(M)) ->
             c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 22: x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
       , 23: and^#(X1, X2) -> c_32(X1, X2)
       , 24: and^#(tt(), X) -> c_33(activate^#(X))
       , 25: U41^#(tt(), N) -> c_20(activate^#(N))
       , 26: U51^#(tt(), M, N) ->
             c_21(s^#(plus(activate(N), activate(M))))
       , 27: U71^#(tt(), M, N) ->
             c_28(plus^#(x(activate(N), activate(M)), activate(N)))
       , 28: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
       , 29: U13^#(tt()) -> c_14()
       , 30: isNat^#(n__0()) -> c_3()
       , 31: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
       , 32: activate^#(n__0()) -> c_8(0^#())
       , 33: 0^#() -> c_27()
       , 34: isNatKind^#(n__0()) -> c_35()
       , 35: U22^#(tt()) -> c_16()
       , 36: U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
       , 37: U33^#(tt()) -> c_19()
       , 38: U61^#(tt()) -> c_26(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
     , U12(tt(), V2) -> U13(isNat(activate(V2)))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__isNatKind(X)) -> isNatKind(X)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , activate(n__and(X1, X2)) -> and(X1, X2)
     , U13(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNat(activate(V1)))
     , U22(tt()) -> tt()
     , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
     , U32(tt(), V2) -> U33(isNat(activate(V2)))
     , U33(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) ->
       U51(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) ->
       U71(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
     , and(X1, X2) -> n__and(X1, X2)
     , and(tt(), X) -> activate(X)
     , isNatKind(X) -> n__isNatKind(X)
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
     , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     , isNatKind(n__x(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) }
   Weak DPs:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_27()
     , isNatKind^#(n__0()) -> c_35()
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , U22^#(tt()) -> c_16()
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U33^#(tt()) -> c_19()
     , U61^#(tt()) -> c_26(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2} by applications of
   Pre({1,2}) = {3,9,12,16,17,19}. Here rules are labeled as follows:
   
     DPs:
       { 1: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 2: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                      activate(V1),
                      activate(V2)))
       , 3: activate^#(X) -> c_7(X)
       , 4: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 5: activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
       , 6: activate^#(n__s(X)) -> c_11(s^#(X))
       , 7: activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
       , 8: activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
       , 9: plus^#(X1, X2) -> c_23(X1, X2)
       , 10: plus^#(N, s(M)) ->
             c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 11: plus^#(N, 0()) ->
             c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
       , 12: isNatKind^#(X) -> c_34(X)
       , 13: isNatKind^#(n__plus(V1, V2)) ->
             c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 14: isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
       , 15: isNatKind^#(n__x(V1, V2)) ->
             c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
       , 16: s^#(X) -> c_22(X)
       , 17: x^#(X1, X2) -> c_29(X1, X2)
       , 18: x^#(N, s(M)) ->
             c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                            n__and(isNat(N), n__isNatKind(N))),
                        M,
                        N))
       , 19: and^#(X1, X2) -> c_32(X1, X2)
       , 20: and^#(tt(), X) -> c_33(activate^#(X))
       , 21: U41^#(tt(), N) -> c_20(activate^#(N))
       , 22: U51^#(tt(), M, N) ->
             c_21(s^#(plus(activate(N), activate(M))))
       , 23: U71^#(tt(), M, N) ->
             c_28(plus^#(x(activate(N), activate(M)), activate(N)))
       , 24: U11^#(tt(), V1, V2) ->
             c_1(U12^#(isNat(activate(V1)), activate(V2)))
       , 25: U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
       , 26: U13^#(tt()) -> c_14()
       , 27: isNat^#(n__0()) -> c_3()
       , 28: isNat^#(n__s(V1)) ->
             c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 29: U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
       , 30: U31^#(tt(), V1, V2) ->
             c_17(U32^#(isNat(activate(V1)), activate(V2)))
       , 31: activate^#(n__0()) -> c_8(0^#())
       , 32: 0^#() -> c_27()
       , 33: isNatKind^#(n__0()) -> c_35()
       , 34: x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
       , 35: U22^#(tt()) -> c_16()
       , 36: U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
       , 37: U33^#(tt()) -> c_19()
       , 38: U61^#(tt()) -> c_26(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__isNatKind(X)) -> c_10(isNatKind^#(X))
     , activate^#(n__s(X)) -> c_11(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_12(x^#(X1, X2))
     , activate^#(n__and(X1, X2)) -> c_13(and^#(X1, X2))
     , plus^#(X1, X2) -> c_23(X1, X2)
     , plus^#(N, s(M)) ->
       c_24(U51^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , plus^#(N, 0()) -> c_25(U41^#(and(isNat(N), n__isNatKind(N)), N))
     , isNatKind^#(X) -> c_34(X)
     , isNatKind^#(n__plus(V1, V2)) ->
       c_36(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , isNatKind^#(n__s(V1)) -> c_37(isNatKind^#(activate(V1)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_38(and^#(isNatKind(activate(V1)), n__isNatKind(activate(V2))))
     , s^#(X) -> c_22(X)
     , x^#(X1, X2) -> c_29(X1, X2)
     , x^#(N, s(M)) ->
       c_30(U71^#(and(and(isNat(M), n__isNatKind(M)),
                      n__and(isNat(N), n__isNatKind(N))),
                  M,
                  N))
     , and^#(X1, X2) -> c_32(X1, X2)
     , and^#(tt(), X) -> c_33(activate^#(X))
     , U41^#(tt(), N) -> c_20(activate^#(N))
     , U51^#(tt(), M, N) -> c_21(s^#(plus(activate(N), activate(M))))
     , U71^#(tt(), M, N) ->
       c_28(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V1, V2) -> U12(isNat(activate(V1)), activate(V2))
     , U12(tt(), V2) -> U13(isNat(activate(V2)))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__isNatKind(X)) -> isNatKind(X)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , activate(n__and(X1, X2)) -> and(X1, X2)
     , U13(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNat(activate(V1)))
     , U22(tt()) -> tt()
     , U31(tt(), V1, V2) -> U32(isNat(activate(V1)), activate(V2))
     , U32(tt(), V2) -> U33(isNat(activate(V2)))
     , U33(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) ->
       U51(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , plus(N, 0()) -> U41(and(isNat(N), n__isNatKind(N)), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) ->
       U71(and(and(isNat(M), n__isNatKind(M)),
               n__and(isNat(N), n__isNatKind(N))),
           M,
           N)
     , x(N, 0()) -> U61(and(isNat(N), n__isNatKind(N)))
     , and(X1, X2) -> n__and(X1, X2)
     , and(tt(), X) -> activate(X)
     , isNatKind(X) -> n__isNatKind(X)
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
     , isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     , isNatKind(n__x(V1, V2)) ->
       and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) }
   Weak DPs:
     { U11^#(tt(), V1, V2) ->
       c_1(U12^#(isNat(activate(V1)), activate(V2)))
     , U12^#(tt(), V2) -> c_2(U13^#(isNat(activate(V2))))
     , U13^#(tt()) -> c_14()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_5(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))),
                 activate(V1),
                 activate(V2)))
     , U21^#(tt(), V1) -> c_15(U22^#(isNat(activate(V1))))
     , U31^#(tt(), V1, V2) ->
       c_17(U32^#(isNat(activate(V1)), activate(V2)))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_27()
     , isNatKind^#(n__0()) -> c_35()
     , x^#(N, 0()) -> c_31(U61^#(and(isNat(N), n__isNatKind(N))))
     , U22^#(tt()) -> c_16()
     , U32^#(tt(), V2) -> c_18(U33^#(isNat(activate(V2))))
     , U33^#(tt()) -> c_19()
     , U61^#(tt()) -> c_26(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..