MAYBE

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

Strict Trs:
  { U101(tt(), M, N) ->
    U102(isNatKind(activate(M)), activate(M), activate(N))
  , U102(tt(), M, N) ->
    U103(isNat(activate(N)), activate(M), activate(N))
  , isNatKind(n__0()) -> tt()
  , isNatKind(n__plus(V1, V2)) ->
    U41(isNatKind(activate(V1)), activate(V2))
  , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
  , isNatKind(n__x(V1, V2)) ->
    U61(isNatKind(activate(V1)), activate(V2))
  , activate(X) -> X
  , activate(n__0()) -> 0()
  , activate(n__plus(X1, X2)) -> plus(X1, X2)
  , activate(n__s(X)) -> s(X)
  , activate(n__x(X1, X2)) -> x(X1, X2)
  , U103(tt(), M, N) ->
    U104(isNatKind(activate(N)), activate(M), activate(N))
  , isNat(n__0()) -> tt()
  , isNat(n__plus(V1, V2)) ->
    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
  , isNat(n__x(V1, V2)) ->
    U31(isNatKind(activate(V1)), activate(V1), activate(V2))
  , U104(tt(), M, N) ->
    plus(x(activate(N), activate(M)), activate(N))
  , plus(X1, X2) -> n__plus(X1, X2)
  , plus(N, s(M)) -> U81(isNat(M), M, N)
  , plus(N, 0()) -> U71(isNat(N), N)
  , x(X1, X2) -> n__x(X1, X2)
  , x(N, s(M)) -> U101(isNat(M), M, N)
  , x(N, 0()) -> U91(isNat(N), N)
  , U11(tt(), V1, V2) ->
    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
  , U12(tt(), V1, V2) ->
    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
  , U13(tt(), V1, V2) ->
    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
  , U15(tt(), V2) -> U16(isNat(activate(V2)))
  , U16(tt()) -> tt()
  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
  , U22(tt(), V1) -> U23(isNat(activate(V1)))
  , U23(tt()) -> tt()
  , U31(tt(), V1, V2) ->
    U32(isNatKind(activate(V1)), activate(V1), activate(V2))
  , U32(tt(), V1, V2) ->
    U33(isNatKind(activate(V2)), activate(V1), activate(V2))
  , U33(tt(), V1, V2) ->
    U34(isNatKind(activate(V2)), activate(V1), activate(V2))
  , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
  , U35(tt(), V2) -> U36(isNat(activate(V2)))
  , U36(tt()) -> tt()
  , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
  , U42(tt()) -> tt()
  , U51(tt()) -> tt()
  , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
  , U62(tt()) -> tt()
  , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
  , U72(tt(), N) -> activate(N)
  , U81(tt(), M, N) ->
    U82(isNatKind(activate(M)), activate(M), activate(N))
  , U82(tt(), M, N) ->
    U83(isNat(activate(N)), activate(M), activate(N))
  , U83(tt(), M, N) ->
    U84(isNatKind(activate(N)), activate(M), activate(N))
  , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
  , s(X) -> n__s(X)
  , U91(tt(), N) -> U92(isNatKind(activate(N)))
  , U92(tt()) -> 0()
  , 0() -> n__0() }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
   following reason:
   
   Computation stopped due to timeout after 60.0 seconds.

2) 'Best' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
      failed due to the following reason:
      
      Computation stopped due to timeout after 30.0 seconds.
   
   2) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
         following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
      2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
         to the following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
   
   3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed
      due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(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__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , 0^#() -> c_53()
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M))))
     , U92^#(tt()) -> c_52(0^#()) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(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__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , 0^#() -> c_53()
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M))))
     , U92^#(tt()) -> c_52(0^#()) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {4,9,16,25,39,41,46,47,48}
   by applications of Pre({4,9,16,25,39,41,46,47,48}) =
   {6,8,10,11,12,17,20,21,38,40,45,53}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: isNatKind^#(n__0()) -> c_3()
       , 5: isNatKind^#(n__plus(V1, V2)) ->
            c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 6: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 7: isNatKind^#(n__x(V1, V2)) ->
            c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 8: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 9: U51^#(tt()) -> c_41()
       , 10: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 11: activate^#(X) -> c_7(X)
       , 12: activate^#(n__0()) -> c_8(0^#())
       , 13: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 14: activate^#(n__s(X)) -> c_10(s^#(X))
       , 15: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 16: 0^#() -> c_53()
       , 17: plus^#(X1, X2) -> c_18(X1, X2)
       , 18: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 19: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 20: s^#(X) -> c_50(X)
       , 21: x^#(X1, X2) -> c_21(X1, X2)
       , 22: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 23: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 24: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 25: isNat^#(n__0()) -> c_13()
       , 26: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 27: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 28: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 29: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 30: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 31: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 32: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 33: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 34: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 35: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 36: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 37: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 38: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 39: U16^#(tt()) -> c_29()
       , 40: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 41: U23^#(tt()) -> c_32()
       , 42: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 43: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 44: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 45: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 46: U36^#(tt()) -> c_38()
       , 47: U42^#(tt()) -> c_40()
       , 48: U62^#(tt()) -> c_43()
       , 49: U72^#(tt(), N) -> c_45(activate^#(N))
       , 50: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 51: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 52: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(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__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M))))
     , U92^#(tt()) -> c_52(0^#()) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , U51^#(tt()) -> c_41()
     , 0^#() -> c_53()
     , isNat^#(n__0()) -> c_13()
     , U16^#(tt()) -> c_29()
     , U23^#(tt()) -> c_32()
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {5,7,8,10,34,35,39,44} by
   applications of Pre({5,7,8,10,34,35,39,44}) =
   {4,6,9,14,17,18,26,30,33,38,40}. Here rules are labeled as follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: isNatKind^#(n__plus(V1, V2)) ->
            c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 5: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 6: isNatKind^#(n__x(V1, V2)) ->
            c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 7: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 8: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 9: activate^#(X) -> c_7(X)
       , 10: activate^#(n__0()) -> c_8(0^#())
       , 11: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 12: activate^#(n__s(X)) -> c_10(s^#(X))
       , 13: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 14: plus^#(X1, X2) -> c_18(X1, X2)
       , 15: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 16: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 17: s^#(X) -> c_50(X)
       , 18: x^#(X1, X2) -> c_21(X1, X2)
       , 19: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 20: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 21: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 22: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 23: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 24: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 25: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 26: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 27: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 28: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 29: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 30: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 31: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 32: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 33: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 34: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 35: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 36: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 37: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 38: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 39: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 40: U72^#(tt(), N) -> c_45(activate^#(N))
       , 41: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 42: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 43: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 44: U92^#(tt()) -> c_52(0^#())
       , 45: isNatKind^#(n__0()) -> c_3()
       , 46: U51^#(tt()) -> c_41()
       , 47: 0^#() -> c_53()
       , 48: isNat^#(n__0()) -> c_13()
       , 49: U16^#(tt()) -> c_29()
       , 50: U23^#(tt()) -> c_32()
       , 51: U36^#(tt()) -> c_38()
       , 52: U42^#(tt()) -> c_40()
       , 53: U62^#(tt()) -> c_43() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , isNat^#(n__0()) -> c_13()
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {4,5,22,26,29,32} by
   applications of Pre({4,5,22,26,29,32}) = {6,10,13,14,16,19,28,31}.
   Here rules are labeled as follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: isNatKind^#(n__plus(V1, V2)) ->
            c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 5: isNatKind^#(n__x(V1, V2)) ->
            c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 6: activate^#(X) -> c_7(X)
       , 7: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 8: activate^#(n__s(X)) -> c_10(s^#(X))
       , 9: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 10: plus^#(X1, X2) -> c_18(X1, X2)
       , 11: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 12: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 13: s^#(X) -> c_50(X)
       , 14: x^#(X1, X2) -> c_21(X1, X2)
       , 15: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 16: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 17: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 18: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 19: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 20: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 21: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 22: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 23: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 24: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 25: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 26: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 27: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 28: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 29: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 30: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 31: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 32: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 33: U72^#(tt(), N) -> c_45(activate^#(N))
       , 34: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 35: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 36: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 37: isNatKind^#(n__0()) -> c_3()
       , 38: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 39: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 40: U51^#(tt()) -> c_41()
       , 41: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 42: activate^#(n__0()) -> c_8(0^#())
       , 43: 0^#() -> c_53()
       , 44: isNat^#(n__0()) -> c_13()
       , 45: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 46: U16^#(tt()) -> c_29()
       , 47: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 48: U23^#(tt()) -> c_32()
       , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 50: U36^#(tt()) -> c_38()
       , 51: U42^#(tt()) -> c_40()
       , 52: U62^#(tt()) -> c_43()
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , isNat^#(n__0()) -> c_13()
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {14,17,24,26} by
   applications of Pre({14,17,24,26}) = {4,7,8,11,12,23,25}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: activate^#(X) -> c_7(X)
       , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 6: activate^#(n__s(X)) -> c_10(s^#(X))
       , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 8: plus^#(X1, X2) -> c_18(X1, X2)
       , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 11: s^#(X) -> c_50(X)
       , 12: x^#(X1, X2) -> c_21(X1, X2)
       , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 14: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 15: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 16: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 17: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 18: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 19: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 20: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 21: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 22: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 23: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 24: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 25: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 26: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 27: U72^#(tt(), N) -> c_45(activate^#(N))
       , 28: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 29: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 30: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 31: isNatKind^#(n__0()) -> c_3()
       , 32: isNatKind^#(n__plus(V1, V2)) ->
             c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 33: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 34: isNatKind^#(n__x(V1, V2)) ->
             c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 35: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 36: U51^#(tt()) -> c_41()
       , 37: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 38: activate^#(n__0()) -> c_8(0^#())
       , 39: 0^#() -> c_53()
       , 40: isNat^#(n__0()) -> c_13()
       , 41: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 42: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 43: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 44: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 45: U16^#(tt()) -> c_29()
       , 46: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 47: U23^#(tt()) -> c_32()
       , 48: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 50: U36^#(tt()) -> c_38()
       , 51: U42^#(tt()) -> c_40()
       , 52: U62^#(tt()) -> c_43()
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {21,22} by applications of
   Pre({21,22}) = {4,8,11,12,17,18}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: activate^#(X) -> c_7(X)
       , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 6: activate^#(n__s(X)) -> c_10(s^#(X))
       , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 8: plus^#(X1, X2) -> c_18(X1, X2)
       , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 11: s^#(X) -> c_50(X)
       , 12: x^#(X1, X2) -> c_21(X1, X2)
       , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 14: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 15: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 16: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 17: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 18: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 19: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 20: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 21: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 22: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 23: U72^#(tt(), N) -> c_45(activate^#(N))
       , 24: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 25: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 26: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 27: isNatKind^#(n__0()) -> c_3()
       , 28: isNatKind^#(n__plus(V1, V2)) ->
             c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 29: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 30: isNatKind^#(n__x(V1, V2)) ->
             c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 31: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 32: U51^#(tt()) -> c_41()
       , 33: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 34: activate^#(n__0()) -> c_8(0^#())
       , 35: 0^#() -> c_53()
       , 36: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 37: isNat^#(n__0()) -> c_13()
       , 38: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 39: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 40: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 41: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 42: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 43: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 44: U16^#(tt()) -> c_29()
       , 45: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 46: U23^#(tt()) -> c_32()
       , 47: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 48: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 50: U36^#(tt()) -> c_38()
       , 51: U42^#(tt()) -> c_40()
       , 52: U62^#(tt()) -> c_43()
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {17,18} by applications of
   Pre({17,18}) = {4,8,11,12,15,16}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: activate^#(X) -> c_7(X)
       , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 6: activate^#(n__s(X)) -> c_10(s^#(X))
       , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 8: plus^#(X1, X2) -> c_18(X1, X2)
       , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 11: s^#(X) -> c_50(X)
       , 12: x^#(X1, X2) -> c_21(X1, X2)
       , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 14: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 15: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 16: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 17: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 18: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 19: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 20: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 21: U72^#(tt(), N) -> c_45(activate^#(N))
       , 22: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 23: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 24: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 25: isNatKind^#(n__0()) -> c_3()
       , 26: isNatKind^#(n__plus(V1, V2)) ->
             c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 27: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 28: isNatKind^#(n__x(V1, V2)) ->
             c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 29: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 30: U51^#(tt()) -> c_41()
       , 31: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 32: activate^#(n__0()) -> c_8(0^#())
       , 33: 0^#() -> c_53()
       , 34: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 35: isNat^#(n__0()) -> c_13()
       , 36: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 37: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 39: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 40: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 41: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 43: U16^#(tt()) -> c_29()
       , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 45: U23^#(tt()) -> c_32()
       , 46: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 47: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 48: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 50: U36^#(tt()) -> c_38()
       , 51: U42^#(tt()) -> c_40()
       , 52: U62^#(tt()) -> c_43()
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {15,16} by applications of
   Pre({15,16}) = {4,8,11,12}. Here rules are labeled as follows:
   
     DPs:
       { 1: U101^#(tt(), M, N) ->
            c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 2: U102^#(tt(), M, N) ->
            c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
       , 3: U103^#(tt(), M, N) ->
            c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 4: activate^#(X) -> c_7(X)
       , 5: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 6: activate^#(n__s(X)) -> c_10(s^#(X))
       , 7: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 8: plus^#(X1, X2) -> c_18(X1, X2)
       , 9: plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
       , 10: plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
       , 11: s^#(X) -> c_50(X)
       , 12: x^#(X1, X2) -> c_21(X1, X2)
       , 13: x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
       , 14: U104^#(tt(), M, N) ->
             c_17(plus^#(x(activate(N), activate(M)), activate(N)))
       , 15: isNat^#(n__plus(V1, V2)) ->
             c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 16: isNat^#(n__x(V1, V2)) ->
             c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 17: U81^#(tt(), M, N) ->
             c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
       , 18: U71^#(tt(), N) ->
             c_44(U72^#(isNatKind(activate(N)), activate(N)))
       , 19: U72^#(tt(), N) -> c_45(activate^#(N))
       , 20: U82^#(tt(), M, N) ->
             c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
       , 21: U83^#(tt(), M, N) ->
             c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
       , 22: U84^#(tt(), M, N) ->
             c_49(s^#(plus(activate(N), activate(M))))
       , 23: isNatKind^#(n__0()) -> c_3()
       , 24: isNatKind^#(n__plus(V1, V2)) ->
             c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
       , 25: isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
       , 26: isNatKind^#(n__x(V1, V2)) ->
             c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
       , 27: U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
       , 28: U51^#(tt()) -> c_41()
       , 29: U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
       , 30: activate^#(n__0()) -> c_8(0^#())
       , 31: 0^#() -> c_53()
       , 32: x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
       , 33: isNat^#(n__0()) -> c_13()
       , 34: isNat^#(n__s(V1)) ->
             c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
       , 35: U11^#(tt(), V1, V2) ->
             c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 36: U21^#(tt(), V1) ->
             c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
       , 37: U31^#(tt(), V1, V2) ->
             c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
       , 38: U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
       , 39: U12^#(tt(), V1, V2) ->
             c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 40: U13^#(tt(), V1, V2) ->
             c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 41: U14^#(tt(), V1, V2) ->
             c_27(U15^#(isNat(activate(V1)), activate(V2)))
       , 42: U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
       , 43: U16^#(tt()) -> c_29()
       , 44: U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
       , 45: U23^#(tt()) -> c_32()
       , 46: U32^#(tt(), V1, V2) ->
             c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 47: U33^#(tt(), V1, V2) ->
             c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
       , 48: U34^#(tt(), V1, V2) ->
             c_36(U35^#(isNat(activate(V1)), activate(V2)))
       , 49: U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
       , 50: U36^#(tt()) -> c_38()
       , 51: U42^#(tt()) -> c_40()
       , 52: U62^#(tt()) -> c_43()
       , 53: U92^#(tt()) -> c_52(0^#()) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U101^#(tt(), M, N) ->
       c_1(U102^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U102^#(tt(), M, N) ->
       c_2(U103^#(isNat(activate(N)), activate(M), activate(N)))
     , U103^#(tt(), M, N) ->
       c_12(U104^#(isNatKind(activate(N)), activate(M), activate(N)))
     , activate^#(X) -> c_7(X)
     , activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_18(X1, X2)
     , plus^#(N, s(M)) -> c_19(U81^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_20(U71^#(isNat(N), N))
     , s^#(X) -> c_50(X)
     , x^#(X1, X2) -> c_21(X1, X2)
     , x^#(N, s(M)) -> c_22(U101^#(isNat(M), M, N))
     , U104^#(tt(), M, N) ->
       c_17(plus^#(x(activate(N), activate(M)), activate(N)))
     , U81^#(tt(), M, N) ->
       c_46(U82^#(isNatKind(activate(M)), activate(M), activate(N)))
     , U71^#(tt(), N) ->
       c_44(U72^#(isNatKind(activate(N)), activate(N)))
     , U72^#(tt(), N) -> c_45(activate^#(N))
     , U82^#(tt(), M, N) ->
       c_47(U83^#(isNat(activate(N)), activate(M), activate(N)))
     , U83^#(tt(), M, N) ->
       c_48(U84^#(isNatKind(activate(N)), activate(M), activate(N)))
     , U84^#(tt(), M, N) -> c_49(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U101(tt(), M, N) ->
       U102(isNatKind(activate(M)), activate(M), activate(N))
     , U102(tt(), M, N) ->
       U103(isNat(activate(N)), activate(M), activate(N))
     , isNatKind(n__0()) -> tt()
     , isNatKind(n__plus(V1, V2)) ->
       U41(isNatKind(activate(V1)), activate(V2))
     , isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
     , isNatKind(n__x(V1, V2)) ->
       U61(isNatKind(activate(V1)), activate(V2))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__x(X1, X2)) -> x(X1, X2)
     , U103(tt(), M, N) ->
       U104(isNatKind(activate(N)), activate(M), activate(N))
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
     , isNat(n__x(V1, V2)) ->
       U31(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U104(tt(), M, N) ->
       plus(x(activate(N), activate(M)), activate(N))
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U81(isNat(M), M, N)
     , plus(N, 0()) -> U71(isNat(N), N)
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U101(isNat(M), M, N)
     , x(N, 0()) -> U91(isNat(N), N)
     , U11(tt(), V1, V2) ->
       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U12(tt(), V1, V2) ->
       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U13(tt(), V1, V2) ->
       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
     , U15(tt(), V2) -> U16(isNat(activate(V2)))
     , U16(tt()) -> tt()
     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
     , U22(tt(), V1) -> U23(isNat(activate(V1)))
     , U23(tt()) -> tt()
     , U31(tt(), V1, V2) ->
       U32(isNatKind(activate(V1)), activate(V1), activate(V2))
     , U32(tt(), V1, V2) ->
       U33(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U33(tt(), V1, V2) ->
       U34(isNatKind(activate(V2)), activate(V1), activate(V2))
     , U34(tt(), V1, V2) -> U35(isNat(activate(V1)), activate(V2))
     , U35(tt(), V2) -> U36(isNat(activate(V2)))
     , U36(tt()) -> tt()
     , U41(tt(), V2) -> U42(isNatKind(activate(V2)))
     , U42(tt()) -> tt()
     , U51(tt()) -> tt()
     , U61(tt(), V2) -> U62(isNatKind(activate(V2)))
     , U62(tt()) -> tt()
     , U71(tt(), N) -> U72(isNatKind(activate(N)), activate(N))
     , U72(tt(), N) -> activate(N)
     , U81(tt(), M, N) ->
       U82(isNatKind(activate(M)), activate(M), activate(N))
     , U82(tt(), M, N) ->
       U83(isNat(activate(N)), activate(M), activate(N))
     , U83(tt(), M, N) ->
       U84(isNatKind(activate(N)), activate(M), activate(N))
     , U84(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , U91(tt(), N) -> U92(isNatKind(activate(N)))
     , U92(tt()) -> 0()
     , 0() -> n__0() }
   Weak DPs:
     { isNatKind^#(n__0()) -> c_3()
     , isNatKind^#(n__plus(V1, V2)) ->
       c_4(U41^#(isNatKind(activate(V1)), activate(V2)))
     , isNatKind^#(n__s(V1)) -> c_5(U51^#(isNatKind(activate(V1))))
     , isNatKind^#(n__x(V1, V2)) ->
       c_6(U61^#(isNatKind(activate(V1)), activate(V2)))
     , U41^#(tt(), V2) -> c_39(U42^#(isNatKind(activate(V2))))
     , U51^#(tt()) -> c_41()
     , U61^#(tt(), V2) -> c_42(U62^#(isNatKind(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_53()
     , x^#(N, 0()) -> c_23(U91^#(isNat(N), N))
     , isNat^#(n__0()) -> c_13()
     , isNat^#(n__plus(V1, V2)) ->
       c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , isNat^#(n__s(V1)) ->
       c_15(U21^#(isNatKind(activate(V1)), activate(V1)))
     , isNat^#(n__x(V1, V2)) ->
       c_16(U31^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U11^#(tt(), V1, V2) ->
       c_24(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U21^#(tt(), V1) ->
       c_30(U22^#(isNatKind(activate(V1)), activate(V1)))
     , U31^#(tt(), V1, V2) ->
       c_33(U32^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
     , U91^#(tt(), N) -> c_51(U92^#(isNatKind(activate(N))))
     , U12^#(tt(), V1, V2) ->
       c_25(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U13^#(tt(), V1, V2) ->
       c_26(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U14^#(tt(), V1, V2) ->
       c_27(U15^#(isNat(activate(V1)), activate(V2)))
     , U15^#(tt(), V2) -> c_28(U16^#(isNat(activate(V2))))
     , U16^#(tt()) -> c_29()
     , U22^#(tt(), V1) -> c_31(U23^#(isNat(activate(V1))))
     , U23^#(tt()) -> c_32()
     , U32^#(tt(), V1, V2) ->
       c_34(U33^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U33^#(tt(), V1, V2) ->
       c_35(U34^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
     , U34^#(tt(), V1, V2) ->
       c_36(U35^#(isNat(activate(V1)), activate(V2)))
     , U35^#(tt(), V2) -> c_37(U36^#(isNat(activate(V2))))
     , U36^#(tt()) -> c_38()
     , U42^#(tt()) -> c_40()
     , U62^#(tt()) -> c_43()
     , U92^#(tt()) -> c_52(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..