MAYBE

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

Strict Trs:
  { U11(tt(), V2) -> U12(isNat(activate(V2)))
  , U12(tt()) -> tt()
  , isNat(n__0()) -> tt()
  , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
  , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
  , isNat(n__x(V1, V2)) -> U31(isNat(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)
  , U21(tt()) -> tt()
  , U31(tt(), V2) -> U32(isNat(activate(V2)))
  , U32(tt()) -> tt()
  , U41(tt(), N) -> activate(N)
  , U51(tt(), M, N) ->
    U52(isNat(activate(N)), activate(M), activate(N))
  , U52(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(isNat(M), M, N)
  , plus(N, 0()) -> U41(isNat(N), N)
  , U61(tt()) -> 0()
  , 0() -> n__0()
  , U71(tt(), M, N) ->
    U72(isNat(activate(N)), activate(M), activate(N))
  , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
  , x(X1, X2) -> n__x(X1, X2)
  , x(N, s(M)) -> U71(isNat(M), M, N)
  , x(N, 0()) -> U61(isNat(N)) }
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 minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

3) '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(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(isNat(activate(V1)), activate(V2)))
     , U21^#(tt()) -> c_12()
     , U31^#(tt(), V2) -> c_13(U32^#(isNat(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_23()
     , plus^#(X1, X2) -> c_19(X1, X2)
     , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
     , s^#(X) -> c_18(X)
     , x^#(X1, X2) -> c_26(X1, X2)
     , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_28(U61^#(isNat(N)))
     , U32^#(tt()) -> c_14()
     , U41^#(tt(), N) -> c_15(activate^#(N))
     , U51^#(tt(), M, N) ->
       c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
     , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_22(0^#())
     , U71^#(tt(), M, N) ->
       c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
     , U72^#(tt(), M, N) ->
       c_25(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(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(isNat(activate(V1)), activate(V2)))
     , U21^#(tt()) -> c_12()
     , U31^#(tt(), V2) -> c_13(U32^#(isNat(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_23()
     , plus^#(X1, X2) -> c_19(X1, X2)
     , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
     , s^#(X) -> c_18(X)
     , x^#(X1, X2) -> c_26(X1, X2)
     , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_28(U61^#(isNat(N)))
     , U32^#(tt()) -> c_14()
     , U41^#(tt(), N) -> c_15(activate^#(N))
     , U51^#(tt(), M, N) ->
       c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
     , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_22(0^#())
     , U71^#(tt(), M, N) ->
       c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
     , U72^#(tt(), M, N) ->
       c_25(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , isNat(n__x(V1, V2)) -> U31(isNat(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)
     , U21(tt()) -> tt()
     , U31(tt(), V2) -> U32(isNat(activate(V2)))
     , U32(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) ->
       U52(isNat(activate(N)), activate(M), activate(N))
     , U52(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(isNat(M), M, N)
     , plus(N, 0()) -> U41(isNat(N), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) ->
       U72(isNat(activate(N)), activate(M), activate(N))
     , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U71(isNat(M), M, N)
     , x(N, 0()) -> U61(isNat(N)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,3,7,14,22} by
   applications of Pre({2,3,7,14,22}) = {1,5,8,9,10,15,18,19,26}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 2: U12^#(tt()) -> c_2()
       , 3: isNat^#(n__0()) -> c_3()
       , 4: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 5: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 6: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(isNat(activate(V1)), activate(V2)))
       , 7: U21^#(tt()) -> c_12()
       , 8: U31^#(tt(), V2) -> c_13(U32^#(isNat(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: 0^#() -> c_23()
       , 15: plus^#(X1, X2) -> c_19(X1, X2)
       , 16: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
       , 17: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
       , 18: s^#(X) -> c_18(X)
       , 19: x^#(X1, X2) -> c_26(X1, X2)
       , 20: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
       , 21: x^#(N, 0()) -> c_28(U61^#(isNat(N)))
       , 22: U32^#(tt()) -> c_14()
       , 23: U41^#(tt(), N) -> c_15(activate^#(N))
       , 24: U51^#(tt(), M, N) ->
             c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
       , 25: U52^#(tt(), M, N) ->
             c_17(s^#(plus(activate(N), activate(M))))
       , 26: U61^#(tt()) -> c_22(0^#())
       , 27: U71^#(tt(), M, N) ->
             c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
       , 28: U72^#(tt(), M, N) ->
             c_25(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(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(isNat(activate(V1)), activate(V2)))
     , U31^#(tt(), V2) -> c_13(U32^#(isNat(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_19(X1, X2)
     , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
     , s^#(X) -> c_18(X)
     , x^#(X1, X2) -> c_26(X1, X2)
     , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_28(U61^#(isNat(N)))
     , U41^#(tt(), N) -> c_15(activate^#(N))
     , U51^#(tt(), M, N) ->
       c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
     , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M))))
     , U61^#(tt()) -> c_22(0^#())
     , U71^#(tt(), M, N) ->
       c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
     , U72^#(tt(), M, N) ->
       c_25(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , isNat(n__x(V1, V2)) -> U31(isNat(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)
     , U21(tt()) -> tt()
     , U31(tt(), V2) -> U32(isNat(activate(V2)))
     , U32(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) ->
       U52(isNat(activate(N)), activate(M), activate(N))
     , U52(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(isNat(M), M, N)
     , plus(N, 0()) -> U41(isNat(N), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) ->
       U72(isNat(activate(N)), activate(M), activate(N))
     , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U71(isNat(M), M, N)
     , x(N, 0()) -> U61(isNat(N)) }
   Weak DPs:
     { U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , U21^#(tt()) -> c_12()
     , 0^#() -> c_23()
     , U32^#(tt()) -> c_14() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,3,5,7,21} by
   applications of Pre({1,3,5,7,21}) = {2,4,6,11,14,15,17,18}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 2: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 3: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 4: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(isNat(activate(V1)), activate(V2)))
       , 5: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2))))
       , 6: activate^#(X) -> c_7(X)
       , 7: activate^#(n__0()) -> c_8(0^#())
       , 8: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 9: activate^#(n__s(X)) -> c_10(s^#(X))
       , 10: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 11: plus^#(X1, X2) -> c_19(X1, X2)
       , 12: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
       , 13: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
       , 14: s^#(X) -> c_18(X)
       , 15: x^#(X1, X2) -> c_26(X1, X2)
       , 16: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
       , 17: x^#(N, 0()) -> c_28(U61^#(isNat(N)))
       , 18: U41^#(tt(), N) -> c_15(activate^#(N))
       , 19: U51^#(tt(), M, N) ->
             c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
       , 20: U52^#(tt(), M, N) ->
             c_17(s^#(plus(activate(N), activate(M))))
       , 21: U61^#(tt()) -> c_22(0^#())
       , 22: U71^#(tt(), M, N) ->
             c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
       , 23: U72^#(tt(), M, N) ->
             c_25(plus^#(x(activate(N), activate(M)), activate(N)))
       , 24: U12^#(tt()) -> c_2()
       , 25: isNat^#(n__0()) -> c_3()
       , 26: U21^#(tt()) -> c_12()
       , 27: 0^#() -> c_23()
       , 28: U32^#(tt()) -> c_14() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(isNat(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_19(X1, X2)
     , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
     , s^#(X) -> c_18(X)
     , x^#(X1, X2) -> c_26(X1, X2)
     , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
     , x^#(N, 0()) -> c_28(U61^#(isNat(N)))
     , U41^#(tt(), N) -> c_15(activate^#(N))
     , U51^#(tt(), M, N) ->
       c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
     , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M))))
     , U71^#(tt(), M, N) ->
       c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
     , U72^#(tt(), M, N) ->
       c_25(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , isNat(n__x(V1, V2)) -> U31(isNat(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)
     , U21(tt()) -> tt()
     , U31(tt(), V2) -> U32(isNat(activate(V2)))
     , U32(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) ->
       U52(isNat(activate(N)), activate(M), activate(N))
     , U52(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(isNat(M), M, N)
     , plus(N, 0()) -> U41(isNat(N), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) ->
       U72(isNat(activate(N)), activate(M), activate(N))
     , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U71(isNat(M), M, N)
     , x(N, 0()) -> U61(isNat(N)) }
   Weak DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , U21^#(tt()) -> c_12()
     , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_23()
     , U32^#(tt()) -> c_14()
     , U61^#(tt()) -> c_22(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,2,13} by applications
   of Pre({1,2,13}) = {3,6,7,10,11}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 2: isNat^#(n__x(V1, V2)) ->
            c_6(U31^#(isNat(activate(V1)), activate(V2)))
       , 3: activate^#(X) -> c_7(X)
       , 4: activate^#(n__plus(X1, X2)) -> c_9(plus^#(X1, X2))
       , 5: activate^#(n__s(X)) -> c_10(s^#(X))
       , 6: activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
       , 7: plus^#(X1, X2) -> c_19(X1, X2)
       , 8: plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
       , 9: plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
       , 10: s^#(X) -> c_18(X)
       , 11: x^#(X1, X2) -> c_26(X1, X2)
       , 12: x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
       , 13: x^#(N, 0()) -> c_28(U61^#(isNat(N)))
       , 14: U41^#(tt(), N) -> c_15(activate^#(N))
       , 15: U51^#(tt(), M, N) ->
             c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
       , 16: U52^#(tt(), M, N) ->
             c_17(s^#(plus(activate(N), activate(M))))
       , 17: U71^#(tt(), M, N) ->
             c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
       , 18: U72^#(tt(), M, N) ->
             c_25(plus^#(x(activate(N), activate(M)), activate(N)))
       , 19: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 20: U12^#(tt()) -> c_2()
       , 21: isNat^#(n__0()) -> c_3()
       , 22: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 23: U21^#(tt()) -> c_12()
       , 24: U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2))))
       , 25: activate^#(n__0()) -> c_8(0^#())
       , 26: 0^#() -> c_23()
       , 27: U32^#(tt()) -> c_14()
       , 28: U61^#(tt()) -> c_22(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__s(X)) -> c_10(s^#(X))
     , activate^#(n__x(X1, X2)) -> c_11(x^#(X1, X2))
     , plus^#(X1, X2) -> c_19(X1, X2)
     , plus^#(N, s(M)) -> c_20(U51^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_21(U41^#(isNat(N), N))
     , s^#(X) -> c_18(X)
     , x^#(X1, X2) -> c_26(X1, X2)
     , x^#(N, s(M)) -> c_27(U71^#(isNat(M), M, N))
     , U41^#(tt(), N) -> c_15(activate^#(N))
     , U51^#(tt(), M, N) ->
       c_16(U52^#(isNat(activate(N)), activate(M), activate(N)))
     , U52^#(tt(), M, N) -> c_17(s^#(plus(activate(N), activate(M))))
     , U71^#(tt(), M, N) ->
       c_24(U72^#(isNat(activate(N)), activate(M), activate(N)))
     , U72^#(tt(), M, N) ->
       c_25(plus^#(x(activate(N), activate(M)), activate(N))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , isNat(n__x(V1, V2)) -> U31(isNat(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)
     , U21(tt()) -> tt()
     , U31(tt(), V2) -> U32(isNat(activate(V2)))
     , U32(tt()) -> tt()
     , U41(tt(), N) -> activate(N)
     , U51(tt(), M, N) ->
       U52(isNat(activate(N)), activate(M), activate(N))
     , U52(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(isNat(M), M, N)
     , plus(N, 0()) -> U41(isNat(N), N)
     , U61(tt()) -> 0()
     , 0() -> n__0()
     , U71(tt(), M, N) ->
       U72(isNat(activate(N)), activate(M), activate(N))
     , U72(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N))
     , x(X1, X2) -> n__x(X1, X2)
     , x(N, s(M)) -> U71(isNat(M), M, N)
     , x(N, 0()) -> U61(isNat(N)) }
   Weak DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , isNat^#(n__x(V1, V2)) ->
       c_6(U31^#(isNat(activate(V1)), activate(V2)))
     , U21^#(tt()) -> c_12()
     , U31^#(tt(), V2) -> c_13(U32^#(isNat(activate(V2))))
     , activate^#(n__0()) -> c_8(0^#())
     , 0^#() -> c_23()
     , x^#(N, 0()) -> c_28(U61^#(isNat(N)))
     , U32^#(tt()) -> c_14()
     , U61^#(tt()) -> c_22(0^#()) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..