MAYBE

Problem:
 U11(tt(),V2) -> U12(isNat(activate(V2)))
 U12(tt()) -> tt()
 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)))
 U61(tt()) -> 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))
 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))
 plus(N,0()) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0()) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
 0() -> n__0()
 plus(X1,X2) -> n__plus(X1,X2)
 s(X) -> n__s(X)
 x(X1,X2) -> n__x(X1,X2)
 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)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   U11#(tt(),V2) -> activate#(V2)
   U11#(tt(),V2) -> isNat#(activate(V2))
   U11#(tt(),V2) -> U12#(isNat(activate(V2)))
   U31#(tt(),V2) -> activate#(V2)
   U31#(tt(),V2) -> isNat#(activate(V2))
   U31#(tt(),V2) -> U32#(isNat(activate(V2)))
   U41#(tt(),N) -> activate#(N)
   U51#(tt(),M,N) -> activate#(M)
   U51#(tt(),M,N) -> activate#(N)
   U51#(tt(),M,N) -> isNat#(activate(N))
   U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
   U52#(tt(),M,N) -> activate#(M)
   U52#(tt(),M,N) -> activate#(N)
   U52#(tt(),M,N) -> plus#(activate(N),activate(M))
   U52#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
   U61#(tt()) -> 0#()
   U71#(tt(),M,N) -> activate#(M)
   U71#(tt(),M,N) -> activate#(N)
   U71#(tt(),M,N) -> isNat#(activate(N))
   U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
   U72#(tt(),M,N) -> activate#(M)
   U72#(tt(),M,N) -> activate#(N)
   U72#(tt(),M,N) -> x#(activate(N),activate(M))
   U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N))
   isNat#(n__plus(V1,V2)) -> activate#(V2)
   isNat#(n__plus(V1,V2)) -> activate#(V1)
   isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
   isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
   isNat#(n__s(V1)) -> activate#(V1)
   isNat#(n__s(V1)) -> isNat#(activate(V1))
   isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
   isNat#(n__x(V1,V2)) -> activate#(V2)
   isNat#(n__x(V1,V2)) -> activate#(V1)
   isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
   isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
   plus#(N,0()) -> isNat#(N)
   plus#(N,0()) -> U41#(isNat(N),N)
   plus#(N,s(M)) -> isNat#(M)
   plus#(N,s(M)) -> U51#(isNat(M),M,N)
   x#(N,0()) -> isNat#(N)
   x#(N,0()) -> U61#(isNat(N))
   x#(N,s(M)) -> isNat#(M)
   x#(N,s(M)) -> U71#(isNat(M),M,N)
   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)
  TRS:
   U11(tt(),V2) -> U12(isNat(activate(V2)))
   U12(tt()) -> tt()
   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)))
   U61(tt()) -> 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))
   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))
   plus(N,0()) -> U41(isNat(N),N)
   plus(N,s(M)) -> U51(isNat(M),M,N)
   x(N,0()) -> U61(isNat(N))
   x(N,s(M)) -> U71(isNat(M),M,N)
   0() -> n__0()
   plus(X1,X2) -> n__plus(X1,X2)
   s(X) -> n__s(X)
   x(X1,X2) -> n__x(X1,X2)
   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)
   activate(X) -> X
  TDG Processor:
   DPs:
    U11#(tt(),V2) -> activate#(V2)
    U11#(tt(),V2) -> isNat#(activate(V2))
    U11#(tt(),V2) -> U12#(isNat(activate(V2)))
    U31#(tt(),V2) -> activate#(V2)
    U31#(tt(),V2) -> isNat#(activate(V2))
    U31#(tt(),V2) -> U32#(isNat(activate(V2)))
    U41#(tt(),N) -> activate#(N)
    U51#(tt(),M,N) -> activate#(M)
    U51#(tt(),M,N) -> activate#(N)
    U51#(tt(),M,N) -> isNat#(activate(N))
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
    U52#(tt(),M,N) -> activate#(M)
    U52#(tt(),M,N) -> activate#(N)
    U52#(tt(),M,N) -> plus#(activate(N),activate(M))
    U52#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
    U61#(tt()) -> 0#()
    U71#(tt(),M,N) -> activate#(M)
    U71#(tt(),M,N) -> activate#(N)
    U71#(tt(),M,N) -> isNat#(activate(N))
    U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
    U72#(tt(),M,N) -> activate#(M)
    U72#(tt(),M,N) -> activate#(N)
    U72#(tt(),M,N) -> x#(activate(N),activate(M))
    U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N))
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    isNat#(n__s(V1)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    isNat#(n__x(V1,V2)) -> activate#(V2)
    isNat#(n__x(V1,V2)) -> activate#(V1)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    plus#(N,0()) -> isNat#(N)
    plus#(N,0()) -> U41#(isNat(N),N)
    plus#(N,s(M)) -> isNat#(M)
    plus#(N,s(M)) -> U51#(isNat(M),M,N)
    x#(N,0()) -> isNat#(N)
    x#(N,0()) -> U61#(isNat(N))
    x#(N,s(M)) -> isNat#(M)
    x#(N,s(M)) -> U71#(isNat(M),M,N)
    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)
   TRS:
    U11(tt(),V2) -> U12(isNat(activate(V2)))
    U12(tt()) -> tt()
    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)))
    U61(tt()) -> 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))
    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))
    plus(N,0()) -> U41(isNat(N),N)
    plus(N,s(M)) -> U51(isNat(M),M,N)
    x(N,0()) -> U61(isNat(N))
    x(N,s(M)) -> U71(isNat(M),M,N)
    0() -> n__0()
    plus(X1,X2) -> n__plus(X1,X2)
    s(X) -> n__s(X)
    x(X1,X2) -> n__x(X1,X2)
    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)
    activate(X) -> X
   graph:
    x#(N,0()) -> U61#(isNat(N)) -> U61#(tt()) -> 0#()
    x#(N,0()) -> isNat#(N) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    x#(N,0()) -> isNat#(N) -> isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    x#(N,0()) -> isNat#(N) -> isNat#(n__x(V1,V2)) -> activate#(V1)
    x#(N,0()) -> isNat#(N) -> isNat#(n__x(V1,V2)) -> activate#(V2)
    x#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    x#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNat#(activate(V1))
    x#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1)
    x#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    x#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    x#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    x#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    x#(N,s(M)) -> U71#(isNat(M),M,N) ->
    U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
    x#(N,s(M)) -> U71#(isNat(M),M,N) ->
    U71#(tt(),M,N) -> isNat#(activate(N))
    x#(N,s(M)) -> U71#(isNat(M),M,N) ->
    U71#(tt(),M,N) -> activate#(N)
    x#(N,s(M)) -> U71#(isNat(M),M,N) -> U71#(tt(),M,N) -> activate#(M)
    x#(N,s(M)) -> isNat#(M) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__x(V1,V2)) -> activate#(V1)
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__x(V1,V2)) -> activate#(V2)
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNat#(activate(V1))
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1)
    x#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    x#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    x#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    x#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U72#(tt(),M,N) -> x#(activate(N),activate(M)) ->
    x#(N,s(M)) -> U71#(isNat(M),M,N)
    U72#(tt(),M,N) -> x#(activate(N),activate(M)) ->
    x#(N,s(M)) -> isNat#(M)
    U72#(tt(),M,N) -> x#(activate(N),activate(M)) ->
    x#(N,0()) -> U61#(isNat(N))
    U72#(tt(),M,N) -> x#(activate(N),activate(M)) ->
    x#(N,0()) -> isNat#(N)
    U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) ->
    plus#(N,s(M)) -> U51#(isNat(M),M,N)
    U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) ->
    plus#(N,s(M)) -> isNat#(M)
    U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) ->
    plus#(N,0()) -> U41#(isNat(N),N)
    U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) ->
    plus#(N,0()) -> isNat#(N)
    U72#(tt(),M,N) -> activate#(N) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U72#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U72#(tt(),M,N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U72#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U72#(tt(),M,N) -> activate#(M) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U72#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X)
    U72#(tt(),M,N) -> activate#(M) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U72#(tt(),M,N) -> activate#(M) ->
    activate#(n__0()) -> 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))
    U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) ->
    U72#(tt(),M,N) -> x#(activate(N),activate(M))
    U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) ->
    U72#(tt(),M,N) -> activate#(N)
    U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N)) ->
    U72#(tt(),M,N) -> activate#(M)
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U71#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U71#(tt(),M,N) -> activate#(N) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U71#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U71#(tt(),M,N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U71#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U71#(tt(),M,N) -> activate#(M) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U71#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X)
    U71#(tt(),M,N) -> activate#(M) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U71#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#()
    plus#(N,0()) -> U41#(isNat(N),N) -> U41#(tt(),N) -> activate#(N)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    plus#(N,0()) -> isNat#(N) -> isNat#(n__x(V1,V2)) -> activate#(V1)
    plus#(N,0()) -> isNat#(N) -> isNat#(n__x(V1,V2)) -> activate#(V2)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNat#(activate(V1))
    plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    plus#(N,s(M)) -> U51#(isNat(M),M,N) ->
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
    plus#(N,s(M)) -> U51#(isNat(M),M,N) ->
    U51#(tt(),M,N) -> isNat#(activate(N))
    plus#(N,s(M)) -> U51#(isNat(M),M,N) ->
    U51#(tt(),M,N) -> activate#(N)
    plus#(N,s(M)) -> U51#(isNat(M),M,N) -> U51#(tt(),M,N) -> activate#(M)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__x(V1,V2)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__x(V1,V2)) -> activate#(V2)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U52#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> U51#(isNat(M),M,N)
    U52#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> isNat#(M)
    U52#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,0()) -> U41#(isNat(N),N)
    U52#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,0()) -> isNat#(N)
    U52#(tt(),M,N) -> activate#(N) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U52#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U52#(tt(),M,N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U52#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U52#(tt(),M,N) -> activate#(M) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U52#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X)
    U52#(tt(),M,N) -> activate#(M) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U52#(tt(),M,N) -> activate#(M) ->
    activate#(n__0()) -> 0#()
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) ->
    U52#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) ->
    U52#(tt(),M,N) -> plus#(activate(N),activate(M))
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) ->
    U52#(tt(),M,N) -> activate#(N)
    U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N)) ->
    U52#(tt(),M,N) -> activate#(M)
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U51#(tt(),M,N) -> isNat#(activate(N)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U51#(tt(),M,N) -> activate#(N) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U51#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U51#(tt(),M,N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U51#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U51#(tt(),M,N) -> activate#(M) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U51#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X)
    U51#(tt(),M,N) -> activate#(M) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U51#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#()
    U41#(tt(),N) -> activate#(N) -> activate#(n__x(X1,X2)) -> x#(X1,X2)
    U41#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U41#(tt(),N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U41#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U31#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U31#(tt(),V2) -> activate#(V2) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U31#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X)
    U31#(tt(),V2) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U31#(tt(),V2) -> activate#(V2) ->
    activate#(n__0()) -> 0#()
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ->
    U31#(tt(),V2) -> U32#(isNat(activate(V2)))
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ->
    U31#(tt(),V2) -> isNat#(activate(V2))
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2)) ->
    U31#(tt(),V2) -> activate#(V2)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__x(V1,V2)) -> activate#(V2) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    isNat#(n__x(V1,V2)) -> activate#(V2) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__x(V1,V2)) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__x(V1,V2)) -> activate#(V2) ->
    activate#(n__0()) -> 0#()
    isNat#(n__x(V1,V2)) -> activate#(V1) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    isNat#(n__x(V1,V2)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__x(V1,V2)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__x(V1,V2)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__0()) -> 0#()
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ->
    U11#(tt(),V2) -> U12#(isNat(activate(V2)))
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ->
    U11#(tt(),V2) -> isNat#(activate(V2))
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) ->
    U11#(tt(),V2) -> activate#(V2)
    activate#(n__x(X1,X2)) -> x#(X1,X2) ->
    x#(N,s(M)) -> U71#(isNat(M),M,N)
    activate#(n__x(X1,X2)) -> x#(X1,X2) ->
    x#(N,s(M)) -> isNat#(M)
    activate#(n__x(X1,X2)) -> x#(X1,X2) ->
    x#(N,0()) -> U61#(isNat(N))
    activate#(n__x(X1,X2)) -> x#(X1,X2) ->
    x#(N,0()) -> isNat#(N)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,s(M)) -> U51#(isNat(M),M,N)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,s(M)) -> isNat#(M)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,0()) -> U41#(isNat(N),N)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,0()) -> isNat#(N)
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> activate#(V1)
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__x(V1,V2)) -> activate#(V2)
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> isNat#(activate(V1))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U11#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U11#(tt(),V2) -> activate#(V2) ->
    activate#(n__x(X1,X2)) -> x#(X1,X2)
    U11#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X)
    U11#(tt(),V2) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U11#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#()
   SCC Processor:
    #sccs: 1
    #rules: 39
    #arcs: 229/2209
    DPs:
     x#(N,0()) -> isNat#(N)
     isNat#(n__plus(V1,V2)) -> activate#(V2)
     activate#(n__plus(X1,X2)) -> plus#(X1,X2)
     plus#(N,0()) -> isNat#(N)
     isNat#(n__plus(V1,V2)) -> activate#(V1)
     activate#(n__x(X1,X2)) -> x#(X1,X2)
     x#(N,s(M)) -> isNat#(M)
     isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
     isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
     U11#(tt(),V2) -> activate#(V2)
     U11#(tt(),V2) -> isNat#(activate(V2))
     isNat#(n__s(V1)) -> activate#(V1)
     isNat#(n__s(V1)) -> isNat#(activate(V1))
     isNat#(n__x(V1,V2)) -> activate#(V2)
     isNat#(n__x(V1,V2)) -> activate#(V1)
     isNat#(n__x(V1,V2)) -> isNat#(activate(V1))
     isNat#(n__x(V1,V2)) -> U31#(isNat(activate(V1)),activate(V2))
     U31#(tt(),V2) -> activate#(V2)
     U31#(tt(),V2) -> isNat#(activate(V2))
     x#(N,s(M)) -> U71#(isNat(M),M,N)
     U71#(tt(),M,N) -> activate#(M)
     U71#(tt(),M,N) -> activate#(N)
     U71#(tt(),M,N) -> isNat#(activate(N))
     U71#(tt(),M,N) -> U72#(isNat(activate(N)),activate(M),activate(N))
     U72#(tt(),M,N) -> activate#(M)
     U72#(tt(),M,N) -> activate#(N)
     U72#(tt(),M,N) -> x#(activate(N),activate(M))
     U72#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N))
     plus#(N,0()) -> U41#(isNat(N),N)
     U41#(tt(),N) -> activate#(N)
     plus#(N,s(M)) -> isNat#(M)
     plus#(N,s(M)) -> U51#(isNat(M),M,N)
     U51#(tt(),M,N) -> activate#(M)
     U51#(tt(),M,N) -> activate#(N)
     U51#(tt(),M,N) -> isNat#(activate(N))
     U51#(tt(),M,N) -> U52#(isNat(activate(N)),activate(M),activate(N))
     U52#(tt(),M,N) -> activate#(M)
     U52#(tt(),M,N) -> activate#(N)
     U52#(tt(),M,N) -> plus#(activate(N),activate(M))
    TRS:
     U11(tt(),V2) -> U12(isNat(activate(V2)))
     U12(tt()) -> tt()
     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)))
     U61(tt()) -> 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))
     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))
     plus(N,0()) -> U41(isNat(N),N)
     plus(N,s(M)) -> U51(isNat(M),M,N)
     x(N,0()) -> U61(isNat(N))
     x(N,s(M)) -> U71(isNat(M),M,N)
     0() -> n__0()
     plus(X1,X2) -> n__plus(X1,X2)
     s(X) -> n__s(X)
     x(X1,X2) -> n__x(X1,X2)
     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)
     activate(X) -> X
    Open