MAYBE

Problem:
 U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
 U12(tt(),V2) -> U13(isNat(activate(V2)))
 U13(tt()) -> tt()
 U21(tt(),V1) -> U22(isNat(activate(V1)))
 U22(tt()) -> tt()
 U31(tt(),N) -> activate(N)
 U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
 and(tt(),X) -> activate(X)
 isNat(n__0()) -> tt()
 isNat(n__plus(V1,V2)) ->
 U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
 isNatKind(n__0()) -> tt()
 isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
 isNatKind(n__s(V1)) -> isNatKind(activate(V1))
 plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
 0() -> n__0()
 plus(X1,X2) -> n__plus(X1,X2)
 isNatKind(X) -> n__isNatKind(X)
 s(X) -> n__s(X)
 and(X1,X2) -> n__and(X1,X2)
 activate(n__0()) -> 0()
 activate(n__plus(X1,X2)) -> plus(X1,X2)
 activate(n__isNatKind(X)) -> isNatKind(X)
 activate(n__s(X)) -> s(X)
 activate(n__and(X1,X2)) -> and(X1,X2)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   U11#(tt(),V1,V2) -> activate#(V2)
   U11#(tt(),V1,V2) -> activate#(V1)
   U11#(tt(),V1,V2) -> isNat#(activate(V1))
   U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
   U12#(tt(),V2) -> activate#(V2)
   U12#(tt(),V2) -> isNat#(activate(V2))
   U12#(tt(),V2) -> U13#(isNat(activate(V2)))
   U21#(tt(),V1) -> activate#(V1)
   U21#(tt(),V1) -> isNat#(activate(V1))
   U21#(tt(),V1) -> U22#(isNat(activate(V1)))
   U31#(tt(),N) -> activate#(N)
   U41#(tt(),M,N) -> activate#(M)
   U41#(tt(),M,N) -> activate#(N)
   U41#(tt(),M,N) -> plus#(activate(N),activate(M))
   U41#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
   and#(tt(),X) -> activate#(X)
   isNat#(n__plus(V1,V2)) -> activate#(V2)
   isNat#(n__plus(V1,V2)) -> activate#(V1)
   isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
   isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   isNat#(n__plus(V1,V2)) ->
   U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   isNat#(n__s(V1)) -> activate#(V1)
   isNat#(n__s(V1)) -> isNatKind#(activate(V1))
   isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
   isNatKind#(n__plus(V1,V2)) -> activate#(V2)
   isNatKind#(n__plus(V1,V2)) -> activate#(V1)
   isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
   isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   isNatKind#(n__s(V1)) -> activate#(V1)
   isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
   plus#(N,0()) -> isNat#(N)
   plus#(N,0()) -> and#(isNat(N),n__isNatKind(N))
   plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
   plus#(N,s(M)) -> isNat#(N)
   plus#(N,s(M)) -> isNat#(M)
   plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
   plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
   plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
   activate#(n__0()) -> 0#()
   activate#(n__plus(X1,X2)) -> plus#(X1,X2)
   activate#(n__isNatKind(X)) -> isNatKind#(X)
   activate#(n__s(X)) -> s#(X)
   activate#(n__and(X1,X2)) -> and#(X1,X2)
  TRS:
   U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
   U12(tt(),V2) -> U13(isNat(activate(V2)))
   U13(tt()) -> tt()
   U21(tt(),V1) -> U22(isNat(activate(V1)))
   U22(tt()) -> tt()
   U31(tt(),N) -> activate(N)
   U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
   and(tt(),X) -> activate(X)
   isNat(n__0()) -> tt()
   isNat(n__plus(V1,V2)) ->
   U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
   isNatKind(n__0()) -> tt()
   isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   isNatKind(n__s(V1)) -> isNatKind(activate(V1))
   plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N)
   plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
   0() -> n__0()
   plus(X1,X2) -> n__plus(X1,X2)
   isNatKind(X) -> n__isNatKind(X)
   s(X) -> n__s(X)
   and(X1,X2) -> n__and(X1,X2)
   activate(n__0()) -> 0()
   activate(n__plus(X1,X2)) -> plus(X1,X2)
   activate(n__isNatKind(X)) -> isNatKind(X)
   activate(n__s(X)) -> s(X)
   activate(n__and(X1,X2)) -> and(X1,X2)
   activate(X) -> X
  ADG Processor:
   DPs:
    U11#(tt(),V1,V2) -> activate#(V2)
    U11#(tt(),V1,V2) -> activate#(V1)
    U11#(tt(),V1,V2) -> isNat#(activate(V1))
    U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
    U12#(tt(),V2) -> activate#(V2)
    U12#(tt(),V2) -> isNat#(activate(V2))
    U12#(tt(),V2) -> U13#(isNat(activate(V2)))
    U21#(tt(),V1) -> activate#(V1)
    U21#(tt(),V1) -> isNat#(activate(V1))
    U21#(tt(),V1) -> U22#(isNat(activate(V1)))
    U31#(tt(),N) -> activate#(N)
    U41#(tt(),M,N) -> activate#(M)
    U41#(tt(),M,N) -> activate#(N)
    U41#(tt(),M,N) -> plus#(activate(N),activate(M))
    U41#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
    and#(tt(),X) -> activate#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    isNat#(n__s(V1)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNatKind#(n__s(V1)) -> activate#(V1)
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    plus#(N,0()) -> isNat#(N)
    plus#(N,0()) -> and#(isNat(N),n__isNatKind(N))
    plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
    plus#(N,s(M)) -> isNat#(N)
    plus#(N,s(M)) -> isNat#(M)
    plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
    plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    activate#(n__0()) -> 0#()
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    activate#(n__s(X)) -> s#(X)
    activate#(n__and(X1,X2)) -> and#(X1,X2)
   TRS:
    U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
    U12(tt(),V2) -> U13(isNat(activate(V2)))
    U13(tt()) -> tt()
    U21(tt(),V1) -> U22(isNat(activate(V1)))
    U22(tt()) -> tt()
    U31(tt(),N) -> activate(N)
    U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
    and(tt(),X) -> activate(X)
    isNat(n__0()) -> tt()
    isNat(n__plus(V1,V2)) ->
    U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
    isNatKind(n__0()) -> tt()
    isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNatKind(n__s(V1)) -> isNatKind(activate(V1))
    plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N)
    plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    0() -> n__0()
    plus(X1,X2) -> n__plus(X1,X2)
    isNatKind(X) -> n__isNatKind(X)
    s(X) -> n__s(X)
    and(X1,X2) -> n__and(X1,X2)
    activate(n__0()) -> 0()
    activate(n__plus(X1,X2)) -> plus(X1,X2)
    activate(n__isNatKind(X)) -> isNatKind(X)
    activate(n__s(X)) -> s(X)
    activate(n__and(X1,X2)) -> and(X1,X2)
    activate(X) -> X
   graph:
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> activate#(V1)
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    isNatKind#(n__s(V1)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNatKind#(n__s(V1)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNatKind#(n__s(V1)) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNatKind#(n__s(V1)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNatKind#(n__s(V1)) -> activate#(V1) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> activate#(V1)
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) ->
    and#(tt(),X) -> activate#(X)
    isNatKind#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__0()) -> 0#()
    isNatKind#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNatKind#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNatKind#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__s(X)) -> s#(X)
    isNatKind#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNatKind#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNatKind#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNatKind#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNatKind#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNatKind#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    and#(tt(),X) -> activate#(X) -> activate#(n__0()) -> 0#()
    and#(tt(),X) -> activate#(X) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    and#(tt(),X) -> activate#(X) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    and#(tt(),X) -> activate#(X) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) ->
    and#(tt(),X) -> activate#(X)
    plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) ->
    U31#(tt(),N) -> activate#(N)
    plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2)
    plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1)
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    plus#(N,0()) -> isNat#(N) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) ->
    and#(tt(),X) -> activate#(X)
    plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) ->
    and#(tt(),X) -> activate#(X)
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    ->
    U41#(tt(),M,N) -> activate#(M)
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    ->
    U41#(tt(),M,N) -> activate#(N)
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    ->
    U41#(tt(),M,N) -> plus#(activate(N),activate(M))
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    -> U41#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2)
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    plus#(N,s(M)) -> isNat#(M) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2)
    plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    plus#(N,s(M)) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    plus#(N,s(M)) -> isNat#(N) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1)
    plus#(N,s(M)) -> isNat#(N) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    plus#(N,s(M)) -> isNat#(N) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,0()) -> isNat#(N)
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,0()) -> and#(isNat(N),n__isNatKind(N))
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> isNat#(N)
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> isNat#(M)
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
    U41#(tt(),M,N) -> plus#(activate(N),activate(M)) ->
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    U41#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#()
    U41#(tt(),M,N) -> activate#(M) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U41#(tt(),M,N) -> activate#(M) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U41#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X)
    U41#(tt(),M,N) -> activate#(M) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    U41#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U41#(tt(),M,N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U41#(tt(),M,N) -> activate#(N) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U41#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U41#(tt(),M,N) -> activate#(N) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    U31#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#()
    U31#(tt(),N) -> activate#(N) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U31#(tt(),N) -> activate#(N) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U31#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X)
    U31#(tt(),N) -> activate#(N) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    U21#(tt(),V1) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#()
    U21#(tt(),V1) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U21#(tt(),V1) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(X)
    U21#(tt(),V1) -> activate#(V1) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    U12#(tt(),V2) -> isNat#(activate(V2)) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    U12#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#()
    U12#(tt(),V2) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U12#(tt(),V2) -> activate#(V2) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U12#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X)
    U12#(tt(),V2) -> activate#(V2) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> activate#(V1)
    isNat#(n__s(V1)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) ->
    U21#(tt(),V1) -> activate#(V1)
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) ->
    U21#(tt(),V1) -> isNat#(activate(V1))
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) ->
    U21#(tt(),V1) -> U22#(isNat(activate(V1)))
    isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#()
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__s(V1)) -> activate#(V1) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> activate#(V1)
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) ->
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) ->
    and#(tt(),X) -> activate#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__0()) -> 0#()
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V2) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__0()) -> 0#()
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    isNat#(n__plus(V1,V2)) -> activate#(V1) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) ->
    U11#(tt(),V1,V2) -> activate#(V2)
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) ->
    U11#(tt(),V1,V2) -> activate#(V1)
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) ->
    U11#(tt(),V1,V2) -> isNat#(activate(V1))
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) ->
    U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
    activate#(n__and(X1,X2)) -> and#(X1,X2) ->
    and#(tt(),X) -> activate#(X)
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V2)
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__plus(V1,V2)) -> activate#(V1)
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__s(V1)) -> activate#(V1)
    activate#(n__isNatKind(X)) -> isNatKind#(X) ->
    isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,0()) -> isNat#(N)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,0()) -> and#(isNat(N),n__isNatKind(N))
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,s(M)) -> isNat#(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,s(M)) -> and#(isNat(M),n__isNatKind(M))
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
    activate#(n__plus(X1,X2)) -> plus#(X1,X2) ->
    plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) ->
    U12#(tt(),V2) -> activate#(V2)
    U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) ->
    U12#(tt(),V2) -> isNat#(activate(V2))
    U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) ->
    U12#(tt(),V2) -> U13#(isNat(activate(V2)))
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V2)
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> activate#(V1)
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__plus(V1,V2)) ->
    U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> activate#(V1)
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> isNatKind#(activate(V1))
    U11#(tt(),V1,V2) -> isNat#(activate(V1)) ->
    isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
    U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#()
    U11#(tt(),V1,V2) -> activate#(V2) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U11#(tt(),V1,V2) -> activate#(V2) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U11#(tt(),V1,V2) -> activate#(V2) ->
    activate#(n__s(X)) -> s#(X)
    U11#(tt(),V1,V2) -> activate#(V2) ->
    activate#(n__and(X1,X2)) -> and#(X1,X2)
    U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#()
    U11#(tt(),V1,V2) -> activate#(V1) ->
    activate#(n__plus(X1,X2)) -> plus#(X1,X2)
    U11#(tt(),V1,V2) -> activate#(V1) ->
    activate#(n__isNatKind(X)) -> isNatKind#(X)
    U11#(tt(),V1,V2) -> activate#(V1) ->
    activate#(n__s(X)) -> s#(X)
    U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2)
   SCC Processor:
    #sccs: 1
    #rules: 38
    #arcs: 185/1849
    DPs:
     isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
     isNatKind#(n__s(V1)) -> activate#(V1)
     activate#(n__and(X1,X2)) -> and#(X1,X2)
     and#(tt(),X) -> activate#(X)
     activate#(n__isNatKind(X)) -> isNatKind#(X)
     isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
     isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
     isNatKind#(n__plus(V1,V2)) -> activate#(V1)
     activate#(n__plus(X1,X2)) -> plus#(X1,X2)
     plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
     U41#(tt(),M,N) -> plus#(activate(N),activate(M))
     plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
     plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
     plus#(N,s(M)) -> isNat#(M)
     isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
     U21#(tt(),V1) -> isNat#(activate(V1))
     isNat#(n__s(V1)) -> isNatKind#(activate(V1))
     isNatKind#(n__plus(V1,V2)) -> activate#(V2)
     isNat#(n__s(V1)) -> activate#(V1)
     isNat#(n__plus(V1,V2)) ->
     U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
     U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
     U12#(tt(),V2) -> isNat#(activate(V2))
     isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
     isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
     isNat#(n__plus(V1,V2)) -> activate#(V1)
     isNat#(n__plus(V1,V2)) -> activate#(V2)
     U12#(tt(),V2) -> activate#(V2)
     U11#(tt(),V1,V2) -> isNat#(activate(V1))
     U11#(tt(),V1,V2) -> activate#(V1)
     U11#(tt(),V1,V2) -> activate#(V2)
     U21#(tt(),V1) -> activate#(V1)
     plus#(N,s(M)) -> isNat#(N)
     plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
     U31#(tt(),N) -> activate#(N)
     plus#(N,0()) -> and#(isNat(N),n__isNatKind(N))
     plus#(N,0()) -> isNat#(N)
     U41#(tt(),M,N) -> activate#(N)
     U41#(tt(),M,N) -> activate#(M)
    TRS:
     U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
     U12(tt(),V2) -> U13(isNat(activate(V2)))
     U13(tt()) -> tt()
     U21(tt(),V1) -> U22(isNat(activate(V1)))
     U22(tt()) -> tt()
     U31(tt(),N) -> activate(N)
     U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
     and(tt(),X) -> activate(X)
     isNat(n__0()) -> tt()
     isNat(n__plus(V1,V2)) ->
     U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
     isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
     isNatKind(n__0()) -> tt()
     isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
     isNatKind(n__s(V1)) -> isNatKind(activate(V1))
     plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N)
     plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
     0() -> n__0()
     plus(X1,X2) -> n__plus(X1,X2)
     isNatKind(X) -> n__isNatKind(X)
     s(X) -> n__s(X)
     and(X1,X2) -> n__and(X1,X2)
     activate(n__0()) -> 0()
     activate(n__plus(X1,X2)) -> plus(X1,X2)
     activate(n__isNatKind(X)) -> isNatKind(X)
     activate(n__s(X)) -> s(X)
     activate(n__and(X1,X2)) -> and(X1,X2)
     activate(X) -> X
    Open