MAYBE

Problem:
 active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
 active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
 active(U13(tt())) -> mark(tt())
 active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
 active(U22(tt())) -> mark(tt())
 active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
 active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
 active(U33(tt())) -> mark(tt())
 active(U41(tt(),N)) -> mark(N)
 active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
 active(U61(tt())) -> mark(0())
 active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
 active(and(tt(),X)) -> mark(X)
 active(isNat(0())) -> mark(tt())
 active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
 active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
 active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
 active(isNatKind(0())) -> mark(tt())
 active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
 active(isNatKind(s(V1))) -> mark(isNatKind(V1))
 active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
 active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
 active(plus(N,s(M))) -> mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
 active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
 active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(X1,X2)) -> U12(active(X1),X2)
 active(U13(X)) -> U13(active(X))
 active(U21(X1,X2)) -> U21(active(X1),X2)
 active(U22(X)) -> U22(active(X))
 active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
 active(U32(X1,X2)) -> U32(active(X1),X2)
 active(U33(X)) -> U33(active(X))
 active(U41(X1,X2)) -> U41(active(X1),X2)
 active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
 active(s(X)) -> s(active(X))
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(U61(X)) -> U61(active(X))
 active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
 active(x(X1,X2)) -> x(active(X1),X2)
 active(x(X1,X2)) -> x(X1,active(X2))
 active(and(X1,X2)) -> and(active(X1),X2)
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U12(mark(X1),X2) -> mark(U12(X1,X2))
 U13(mark(X)) -> mark(U13(X))
 U21(mark(X1),X2) -> mark(U21(X1,X2))
 U22(mark(X)) -> mark(U22(X))
 U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
 U32(mark(X1),X2) -> mark(U32(X1,X2))
 U33(mark(X)) -> mark(U33(X))
 U41(mark(X1),X2) -> mark(U41(X1,X2))
 U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
 s(mark(X)) -> mark(s(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 U61(mark(X)) -> mark(U61(X))
 U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
 x(mark(X1),X2) -> mark(x(X1,X2))
 x(X1,mark(X2)) -> mark(x(X1,X2))
 and(mark(X1),X2) -> mark(and(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(tt()) -> ok(tt())
 proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
 proper(isNat(X)) -> isNat(proper(X))
 proper(U13(X)) -> U13(proper(X))
 proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
 proper(U22(X)) -> U22(proper(X))
 proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
 proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
 proper(U33(X)) -> U33(proper(X))
 proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
 proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
 proper(s(X)) -> s(proper(X))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(U61(X)) -> U61(proper(X))
 proper(0()) -> ok(0())
 proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
 proper(x(X1,X2)) -> x(proper(X1),proper(X2))
 proper(and(X1,X2)) -> and(proper(X1),proper(X2))
 proper(isNatKind(X)) -> isNatKind(proper(X))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
 isNat(ok(X)) -> ok(isNat(X))
 U13(ok(X)) -> ok(U13(X))
 U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
 U22(ok(X)) -> ok(U22(X))
 U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
 U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
 U33(ok(X)) -> ok(U33(X))
 U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
 U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
 s(ok(X)) -> ok(s(X))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 U61(ok(X)) -> ok(U61(X))
 U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
 x(ok(X1),ok(X2)) -> ok(x(X1,X2))
 and(ok(X1),ok(X2)) -> ok(and(X1,X2))
 isNatKind(ok(X)) -> ok(isNatKind(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(U11(tt(),V1,V2)) -> isNat#(V1)
   active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2)
   active#(U12(tt(),V2)) -> isNat#(V2)
   active#(U12(tt(),V2)) -> U13#(isNat(V2))
   active#(U21(tt(),V1)) -> isNat#(V1)
   active#(U21(tt(),V1)) -> U22#(isNat(V1))
   active#(U31(tt(),V1,V2)) -> isNat#(V1)
   active#(U31(tt(),V1,V2)) -> U32#(isNat(V1),V2)
   active#(U32(tt(),V2)) -> isNat#(V2)
   active#(U32(tt(),V2)) -> U33#(isNat(V2))
   active#(U51(tt(),M,N)) -> plus#(N,M)
   active#(U51(tt(),M,N)) -> s#(plus(N,M))
   active#(U71(tt(),M,N)) -> x#(N,M)
   active#(U71(tt(),M,N)) -> plus#(x(N,M),N)
   active#(isNat(plus(V1,V2))) -> isNatKind#(V2)
   active#(isNat(plus(V1,V2))) -> isNatKind#(V1)
   active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
   active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
   active#(isNat(s(V1))) -> isNatKind#(V1)
   active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1)
   active#(isNat(x(V1,V2))) -> isNatKind#(V2)
   active#(isNat(x(V1,V2))) -> isNatKind#(V1)
   active#(isNat(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
   active#(isNat(x(V1,V2))) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
   active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2)
   active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1)
   active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
   active#(isNatKind(s(V1))) -> isNatKind#(V1)
   active#(isNatKind(x(V1,V2))) -> isNatKind#(V2)
   active#(isNatKind(x(V1,V2))) -> isNatKind#(V1)
   active#(isNatKind(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
   active#(plus(N,0())) -> isNatKind#(N)
   active#(plus(N,0())) -> isNat#(N)
   active#(plus(N,0())) -> and#(isNat(N),isNatKind(N))
   active#(plus(N,0())) -> U41#(and(isNat(N),isNatKind(N)),N)
   active#(plus(N,s(M))) -> isNatKind#(N)
   active#(plus(N,s(M))) -> isNat#(N)
   active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N))
   active#(plus(N,s(M))) -> isNatKind#(M)
   active#(plus(N,s(M))) -> isNat#(M)
   active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M))
   active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
   active#(plus(N,s(M))) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
   active#(x(N,0())) -> isNatKind#(N)
   active#(x(N,0())) -> isNat#(N)
   active#(x(N,0())) -> and#(isNat(N),isNatKind(N))
   active#(x(N,0())) -> U61#(and(isNat(N),isNatKind(N)))
   active#(x(N,s(M))) -> isNatKind#(N)
   active#(x(N,s(M))) -> isNat#(N)
   active#(x(N,s(M))) -> and#(isNat(N),isNatKind(N))
   active#(x(N,s(M))) -> isNatKind#(M)
   active#(x(N,s(M))) -> isNat#(M)
   active#(x(N,s(M))) -> and#(isNat(M),isNatKind(M))
   active#(x(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
   active#(x(N,s(M))) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
   active#(U11(X1,X2,X3)) -> active#(X1)
   active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
   active#(U12(X1,X2)) -> active#(X1)
   active#(U12(X1,X2)) -> U12#(active(X1),X2)
   active#(U13(X)) -> active#(X)
   active#(U13(X)) -> U13#(active(X))
   active#(U21(X1,X2)) -> active#(X1)
   active#(U21(X1,X2)) -> U21#(active(X1),X2)
   active#(U22(X)) -> active#(X)
   active#(U22(X)) -> U22#(active(X))
   active#(U31(X1,X2,X3)) -> active#(X1)
   active#(U31(X1,X2,X3)) -> U31#(active(X1),X2,X3)
   active#(U32(X1,X2)) -> active#(X1)
   active#(U32(X1,X2)) -> U32#(active(X1),X2)
   active#(U33(X)) -> active#(X)
   active#(U33(X)) -> U33#(active(X))
   active#(U41(X1,X2)) -> active#(X1)
   active#(U41(X1,X2)) -> U41#(active(X1),X2)
   active#(U51(X1,X2,X3)) -> active#(X1)
   active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(plus(X1,X2)) -> active#(X1)
   active#(plus(X1,X2)) -> plus#(active(X1),X2)
   active#(plus(X1,X2)) -> active#(X2)
   active#(plus(X1,X2)) -> plus#(X1,active(X2))
   active#(U61(X)) -> active#(X)
   active#(U61(X)) -> U61#(active(X))
   active#(U71(X1,X2,X3)) -> active#(X1)
   active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
   active#(x(X1,X2)) -> active#(X1)
   active#(x(X1,X2)) -> x#(active(X1),X2)
   active#(x(X1,X2)) -> active#(X2)
   active#(x(X1,X2)) -> x#(X1,active(X2))
   active#(and(X1,X2)) -> active#(X1)
   active#(and(X1,X2)) -> and#(active(X1),X2)
   U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
   U12#(mark(X1),X2) -> U12#(X1,X2)
   U13#(mark(X)) -> U13#(X)
   U21#(mark(X1),X2) -> U21#(X1,X2)
   U22#(mark(X)) -> U22#(X)
   U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3)
   U32#(mark(X1),X2) -> U32#(X1,X2)
   U33#(mark(X)) -> U33#(X)
   U41#(mark(X1),X2) -> U41#(X1,X2)
   U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
   s#(mark(X)) -> s#(X)
   plus#(mark(X1),X2) -> plus#(X1,X2)
   plus#(X1,mark(X2)) -> plus#(X1,X2)
   U61#(mark(X)) -> U61#(X)
   U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
   x#(mark(X1),X2) -> x#(X1,X2)
   x#(X1,mark(X2)) -> x#(X1,X2)
   and#(mark(X1),X2) -> and#(X1,X2)
   proper#(U11(X1,X2,X3)) -> proper#(X3)
   proper#(U11(X1,X2,X3)) -> proper#(X2)
   proper#(U11(X1,X2,X3)) -> proper#(X1)
   proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
   proper#(U12(X1,X2)) -> proper#(X2)
   proper#(U12(X1,X2)) -> proper#(X1)
   proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2))
   proper#(isNat(X)) -> proper#(X)
   proper#(isNat(X)) -> isNat#(proper(X))
   proper#(U13(X)) -> proper#(X)
   proper#(U13(X)) -> U13#(proper(X))
   proper#(U21(X1,X2)) -> proper#(X2)
   proper#(U21(X1,X2)) -> proper#(X1)
   proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
   proper#(U22(X)) -> proper#(X)
   proper#(U22(X)) -> U22#(proper(X))
   proper#(U31(X1,X2,X3)) -> proper#(X3)
   proper#(U31(X1,X2,X3)) -> proper#(X2)
   proper#(U31(X1,X2,X3)) -> proper#(X1)
   proper#(U31(X1,X2,X3)) -> U31#(proper(X1),proper(X2),proper(X3))
   proper#(U32(X1,X2)) -> proper#(X2)
   proper#(U32(X1,X2)) -> proper#(X1)
   proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
   proper#(U33(X)) -> proper#(X)
   proper#(U33(X)) -> U33#(proper(X))
   proper#(U41(X1,X2)) -> proper#(X2)
   proper#(U41(X1,X2)) -> proper#(X1)
   proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
   proper#(U51(X1,X2,X3)) -> proper#(X3)
   proper#(U51(X1,X2,X3)) -> proper#(X2)
   proper#(U51(X1,X2,X3)) -> proper#(X1)
   proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(plus(X1,X2)) -> proper#(X2)
   proper#(plus(X1,X2)) -> proper#(X1)
   proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
   proper#(U61(X)) -> proper#(X)
   proper#(U61(X)) -> U61#(proper(X))
   proper#(U71(X1,X2,X3)) -> proper#(X3)
   proper#(U71(X1,X2,X3)) -> proper#(X2)
   proper#(U71(X1,X2,X3)) -> proper#(X1)
   proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
   proper#(x(X1,X2)) -> proper#(X2)
   proper#(x(X1,X2)) -> proper#(X1)
   proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
   proper#(and(X1,X2)) -> proper#(X2)
   proper#(and(X1,X2)) -> proper#(X1)
   proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
   proper#(isNatKind(X)) -> proper#(X)
   proper#(isNatKind(X)) -> isNatKind#(proper(X))
   U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
   U12#(ok(X1),ok(X2)) -> U12#(X1,X2)
   isNat#(ok(X)) -> isNat#(X)
   U13#(ok(X)) -> U13#(X)
   U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
   U22#(ok(X)) -> U22#(X)
   U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3)
   U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
   U33#(ok(X)) -> U33#(X)
   U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
   U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
   s#(ok(X)) -> s#(X)
   plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
   U61#(ok(X)) -> U61#(X)
   U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
   x#(ok(X1),ok(X2)) -> x#(X1,X2)
   and#(ok(X1),ok(X2)) -> and#(X1,X2)
   isNatKind#(ok(X)) -> isNatKind#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
   active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
   active(U13(tt())) -> mark(tt())
   active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
   active(U22(tt())) -> mark(tt())
   active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
   active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
   active(U33(tt())) -> mark(tt())
   active(U41(tt(),N)) -> mark(N)
   active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
   active(U61(tt())) -> mark(0())
   active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
   active(and(tt(),X)) -> mark(X)
   active(isNat(0())) -> mark(tt())
   active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
   active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
   active(isNatKind(0())) -> mark(tt())
   active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
   active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
   active(plus(N,s(M))) ->
   mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
   active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
   active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
   active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
   active(U12(X1,X2)) -> U12(active(X1),X2)
   active(U13(X)) -> U13(active(X))
   active(U21(X1,X2)) -> U21(active(X1),X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
   active(U32(X1,X2)) -> U32(active(X1),X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1,X2)) -> U41(active(X1),X2)
   active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
   active(s(X)) -> s(active(X))
   active(plus(X1,X2)) -> plus(active(X1),X2)
   active(plus(X1,X2)) -> plus(X1,active(X2))
   active(U61(X)) -> U61(active(X))
   active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
   active(x(X1,X2)) -> x(active(X1),X2)
   active(x(X1,X2)) -> x(X1,active(X2))
   active(and(X1,X2)) -> and(active(X1),X2)
   U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
   U12(mark(X1),X2) -> mark(U12(X1,X2))
   U13(mark(X)) -> mark(U13(X))
   U21(mark(X1),X2) -> mark(U21(X1,X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
   U32(mark(X1),X2) -> mark(U32(X1,X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1),X2) -> mark(U41(X1,X2))
   U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1),X2) -> mark(plus(X1,X2))
   plus(X1,mark(X2)) -> mark(plus(X1,X2))
   U61(mark(X)) -> mark(U61(X))
   U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
   x(mark(X1),X2) -> mark(x(X1,X2))
   x(X1,mark(X2)) -> mark(x(X1,X2))
   and(mark(X1),X2) -> mark(and(X1,X2))
   proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
   proper(tt()) -> ok(tt())
   proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
   proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
   proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
   proper(U61(X)) -> U61(proper(X))
   proper(0()) -> ok(0())
   proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
   proper(x(X1,X2)) -> x(proper(X1),proper(X2))
   proper(and(X1,X2)) -> and(proper(X1),proper(X2))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
   U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
   isNat(ok(X)) -> ok(isNat(X))
   U13(ok(X)) -> ok(U13(X))
   U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
   U22(ok(X)) -> ok(U22(X))
   U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
   U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
   U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
   U61(ok(X)) -> ok(U61(X))
   U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
   x(ok(X1),ok(X2)) -> ok(x(X1,X2))
   and(ok(X1),ok(X2)) -> ok(and(X1,X2))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(U11(tt(),V1,V2)) -> isNat#(V1)
    active#(U11(tt(),V1,V2)) -> U12#(isNat(V1),V2)
    active#(U12(tt(),V2)) -> isNat#(V2)
    active#(U12(tt(),V2)) -> U13#(isNat(V2))
    active#(U21(tt(),V1)) -> isNat#(V1)
    active#(U21(tt(),V1)) -> U22#(isNat(V1))
    active#(U31(tt(),V1,V2)) -> isNat#(V1)
    active#(U31(tt(),V1,V2)) -> U32#(isNat(V1),V2)
    active#(U32(tt(),V2)) -> isNat#(V2)
    active#(U32(tt(),V2)) -> U33#(isNat(V2))
    active#(U51(tt(),M,N)) -> plus#(N,M)
    active#(U51(tt(),M,N)) -> s#(plus(N,M))
    active#(U71(tt(),M,N)) -> x#(N,M)
    active#(U71(tt(),M,N)) -> plus#(x(N,M),N)
    active#(isNat(plus(V1,V2))) -> isNatKind#(V2)
    active#(isNat(plus(V1,V2))) -> isNatKind#(V1)
    active#(isNat(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
    active#(isNat(plus(V1,V2))) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
    active#(isNat(s(V1))) -> isNatKind#(V1)
    active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1)
    active#(isNat(x(V1,V2))) -> isNatKind#(V2)
    active#(isNat(x(V1,V2))) -> isNatKind#(V1)
    active#(isNat(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
    active#(isNat(x(V1,V2))) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
    active#(isNatKind(plus(V1,V2))) -> isNatKind#(V2)
    active#(isNatKind(plus(V1,V2))) -> isNatKind#(V1)
    active#(isNatKind(plus(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
    active#(isNatKind(s(V1))) -> isNatKind#(V1)
    active#(isNatKind(x(V1,V2))) -> isNatKind#(V2)
    active#(isNatKind(x(V1,V2))) -> isNatKind#(V1)
    active#(isNatKind(x(V1,V2))) -> and#(isNatKind(V1),isNatKind(V2))
    active#(plus(N,0())) -> isNatKind#(N)
    active#(plus(N,0())) -> isNat#(N)
    active#(plus(N,0())) -> and#(isNat(N),isNatKind(N))
    active#(plus(N,0())) -> U41#(and(isNat(N),isNatKind(N)),N)
    active#(plus(N,s(M))) -> isNatKind#(N)
    active#(plus(N,s(M))) -> isNat#(N)
    active#(plus(N,s(M))) -> and#(isNat(N),isNatKind(N))
    active#(plus(N,s(M))) -> isNatKind#(M)
    active#(plus(N,s(M))) -> isNat#(M)
    active#(plus(N,s(M))) -> and#(isNat(M),isNatKind(M))
    active#(plus(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
    active#(plus(N,s(M))) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
    active#(x(N,0())) -> isNatKind#(N)
    active#(x(N,0())) -> isNat#(N)
    active#(x(N,0())) -> and#(isNat(N),isNatKind(N))
    active#(x(N,0())) -> U61#(and(isNat(N),isNatKind(N)))
    active#(x(N,s(M))) -> isNatKind#(N)
    active#(x(N,s(M))) -> isNat#(N)
    active#(x(N,s(M))) -> and#(isNat(N),isNatKind(N))
    active#(x(N,s(M))) -> isNatKind#(M)
    active#(x(N,s(M))) -> isNat#(M)
    active#(x(N,s(M))) -> and#(isNat(M),isNatKind(M))
    active#(x(N,s(M))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
    active#(x(N,s(M))) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
    active#(U11(X1,X2,X3)) -> active#(X1)
    active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
    active#(U12(X1,X2)) -> active#(X1)
    active#(U12(X1,X2)) -> U12#(active(X1),X2)
    active#(U13(X)) -> active#(X)
    active#(U13(X)) -> U13#(active(X))
    active#(U21(X1,X2)) -> active#(X1)
    active#(U21(X1,X2)) -> U21#(active(X1),X2)
    active#(U22(X)) -> active#(X)
    active#(U22(X)) -> U22#(active(X))
    active#(U31(X1,X2,X3)) -> active#(X1)
    active#(U31(X1,X2,X3)) -> U31#(active(X1),X2,X3)
    active#(U32(X1,X2)) -> active#(X1)
    active#(U32(X1,X2)) -> U32#(active(X1),X2)
    active#(U33(X)) -> active#(X)
    active#(U33(X)) -> U33#(active(X))
    active#(U41(X1,X2)) -> active#(X1)
    active#(U41(X1,X2)) -> U41#(active(X1),X2)
    active#(U51(X1,X2,X3)) -> active#(X1)
    active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(U61(X)) -> active#(X)
    active#(U61(X)) -> U61#(active(X))
    active#(U71(X1,X2,X3)) -> active#(X1)
    active#(U71(X1,X2,X3)) -> U71#(active(X1),X2,X3)
    active#(x(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(and(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> and#(active(X1),X2)
    U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
    U12#(mark(X1),X2) -> U12#(X1,X2)
    U13#(mark(X)) -> U13#(X)
    U21#(mark(X1),X2) -> U21#(X1,X2)
    U22#(mark(X)) -> U22#(X)
    U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3)
    U32#(mark(X1),X2) -> U32#(X1,X2)
    U33#(mark(X)) -> U33#(X)
    U41#(mark(X1),X2) -> U41#(X1,X2)
    U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
    s#(mark(X)) -> s#(X)
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    U61#(mark(X)) -> U61#(X)
    U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
    x#(mark(X1),X2) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2)
    and#(mark(X1),X2) -> and#(X1,X2)
    proper#(U11(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
    proper#(U12(X1,X2)) -> proper#(X2)
    proper#(U12(X1,X2)) -> proper#(X1)
    proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2))
    proper#(isNat(X)) -> proper#(X)
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U13(X)) -> proper#(X)
    proper#(U13(X)) -> U13#(proper(X))
    proper#(U21(X1,X2)) -> proper#(X2)
    proper#(U21(X1,X2)) -> proper#(X1)
    proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
    proper#(U22(X)) -> proper#(X)
    proper#(U22(X)) -> U22#(proper(X))
    proper#(U31(X1,X2,X3)) -> proper#(X3)
    proper#(U31(X1,X2,X3)) -> proper#(X2)
    proper#(U31(X1,X2,X3)) -> proper#(X1)
    proper#(U31(X1,X2,X3)) -> U31#(proper(X1),proper(X2),proper(X3))
    proper#(U32(X1,X2)) -> proper#(X2)
    proper#(U32(X1,X2)) -> proper#(X1)
    proper#(U32(X1,X2)) -> U32#(proper(X1),proper(X2))
    proper#(U33(X)) -> proper#(X)
    proper#(U33(X)) -> U33#(proper(X))
    proper#(U41(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2))
    proper#(U51(X1,X2,X3)) -> proper#(X3)
    proper#(U51(X1,X2,X3)) -> proper#(X2)
    proper#(U51(X1,X2,X3)) -> proper#(X1)
    proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U61(X)) -> proper#(X)
    proper#(U61(X)) -> U61#(proper(X))
    proper#(U71(X1,X2,X3)) -> proper#(X3)
    proper#(U71(X1,X2,X3)) -> proper#(X2)
    proper#(U71(X1,X2,X3)) -> proper#(X1)
    proper#(U71(X1,X2,X3)) -> U71#(proper(X1),proper(X2),proper(X3))
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(isNatKind(X)) -> proper#(X)
    proper#(isNatKind(X)) -> isNatKind#(proper(X))
    U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
    U12#(ok(X1),ok(X2)) -> U12#(X1,X2)
    isNat#(ok(X)) -> isNat#(X)
    U13#(ok(X)) -> U13#(X)
    U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
    U22#(ok(X)) -> U22#(X)
    U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3)
    U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
    U33#(ok(X)) -> U33#(X)
    U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
    U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
    s#(ok(X)) -> s#(X)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    U61#(ok(X)) -> U61#(X)
    U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    isNatKind#(ok(X)) -> isNatKind#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
    active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
    active(U13(tt())) -> mark(tt())
    active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
    active(U22(tt())) -> mark(tt())
    active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
    active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
    active(U33(tt())) -> mark(tt())
    active(U41(tt(),N)) -> mark(N)
    active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
    active(U61(tt())) -> mark(0())
    active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
    active(and(tt(),X)) -> mark(X)
    active(isNat(0())) -> mark(tt())
    active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
    active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
    active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
    active(isNatKind(0())) -> mark(tt())
    active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
    active(isNatKind(s(V1))) -> mark(isNatKind(V1))
    active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
    active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
    active(plus(N,s(M))) ->
    mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
    active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
    active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
    active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
    active(U12(X1,X2)) -> U12(active(X1),X2)
    active(U13(X)) -> U13(active(X))
    active(U21(X1,X2)) -> U21(active(X1),X2)
    active(U22(X)) -> U22(active(X))
    active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
    active(U32(X1,X2)) -> U32(active(X1),X2)
    active(U33(X)) -> U33(active(X))
    active(U41(X1,X2)) -> U41(active(X1),X2)
    active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
    active(s(X)) -> s(active(X))
    active(plus(X1,X2)) -> plus(active(X1),X2)
    active(plus(X1,X2)) -> plus(X1,active(X2))
    active(U61(X)) -> U61(active(X))
    active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
    active(x(X1,X2)) -> x(active(X1),X2)
    active(x(X1,X2)) -> x(X1,active(X2))
    active(and(X1,X2)) -> and(active(X1),X2)
    U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
    U12(mark(X1),X2) -> mark(U12(X1,X2))
    U13(mark(X)) -> mark(U13(X))
    U21(mark(X1),X2) -> mark(U21(X1,X2))
    U22(mark(X)) -> mark(U22(X))
    U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
    U32(mark(X1),X2) -> mark(U32(X1,X2))
    U33(mark(X)) -> mark(U33(X))
    U41(mark(X1),X2) -> mark(U41(X1,X2))
    U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
    s(mark(X)) -> mark(s(X))
    plus(mark(X1),X2) -> mark(plus(X1,X2))
    plus(X1,mark(X2)) -> mark(plus(X1,X2))
    U61(mark(X)) -> mark(U61(X))
    U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
    x(mark(X1),X2) -> mark(x(X1,X2))
    x(X1,mark(X2)) -> mark(x(X1,X2))
    and(mark(X1),X2) -> mark(and(X1,X2))
    proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
    proper(tt()) -> ok(tt())
    proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
    proper(isNat(X)) -> isNat(proper(X))
    proper(U13(X)) -> U13(proper(X))
    proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
    proper(U22(X)) -> U22(proper(X))
    proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
    proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
    proper(U33(X)) -> U33(proper(X))
    proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
    proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
    proper(s(X)) -> s(proper(X))
    proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
    proper(U61(X)) -> U61(proper(X))
    proper(0()) -> ok(0())
    proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
    proper(x(X1,X2)) -> x(proper(X1),proper(X2))
    proper(and(X1,X2)) -> and(proper(X1),proper(X2))
    proper(isNatKind(X)) -> isNatKind(proper(X))
    U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
    U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
    isNat(ok(X)) -> ok(isNat(X))
    U13(ok(X)) -> ok(U13(X))
    U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
    U22(ok(X)) -> ok(U22(X))
    U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
    U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
    U33(ok(X)) -> ok(U33(X))
    U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
    U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
    s(ok(X)) -> ok(s(X))
    plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    U61(ok(X)) -> ok(U61(X))
    U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
    x(ok(X1),ok(X2)) -> ok(x(X1,X2))
    and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    isNatKind(ok(X)) -> ok(isNatKind(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    ...
   SCC Processor:
    #sccs: 21
    #rules: 89
    #arcs: 3709/33124
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     active#(U11(X1,X2,X3)) -> active#(X1)
     active#(U12(X1,X2)) -> active#(X1)
     active#(U13(X)) -> active#(X)
     active#(U21(X1,X2)) -> active#(X1)
     active#(U22(X)) -> active#(X)
     active#(U31(X1,X2,X3)) -> active#(X1)
     active#(U32(X1,X2)) -> active#(X1)
     active#(U33(X)) -> active#(X)
     active#(U41(X1,X2)) -> active#(X1)
     active#(U51(X1,X2,X3)) -> active#(X1)
     active#(s(X)) -> active#(X)
     active#(plus(X1,X2)) -> active#(X1)
     active#(plus(X1,X2)) -> active#(X2)
     active#(U61(X)) -> active#(X)
     active#(U71(X1,X2,X3)) -> active#(X1)
     active#(x(X1,X2)) -> active#(X1)
     active#(x(X1,X2)) -> active#(X2)
     active#(and(X1,X2)) -> active#(X1)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     proper#(U11(X1,X2,X3)) -> proper#(X3)
     proper#(U11(X1,X2,X3)) -> proper#(X2)
     proper#(U11(X1,X2,X3)) -> proper#(X1)
     proper#(U12(X1,X2)) -> proper#(X2)
     proper#(U12(X1,X2)) -> proper#(X1)
     proper#(isNat(X)) -> proper#(X)
     proper#(U13(X)) -> proper#(X)
     proper#(U21(X1,X2)) -> proper#(X2)
     proper#(U21(X1,X2)) -> proper#(X1)
     proper#(U22(X)) -> proper#(X)
     proper#(U31(X1,X2,X3)) -> proper#(X3)
     proper#(U31(X1,X2,X3)) -> proper#(X2)
     proper#(U31(X1,X2,X3)) -> proper#(X1)
     proper#(U32(X1,X2)) -> proper#(X2)
     proper#(U32(X1,X2)) -> proper#(X1)
     proper#(U33(X)) -> proper#(X)
     proper#(U41(X1,X2)) -> proper#(X2)
     proper#(U41(X1,X2)) -> proper#(X1)
     proper#(U51(X1,X2,X3)) -> proper#(X3)
     proper#(U51(X1,X2,X3)) -> proper#(X2)
     proper#(U51(X1,X2,X3)) -> proper#(X1)
     proper#(s(X)) -> proper#(X)
     proper#(plus(X1,X2)) -> proper#(X2)
     proper#(plus(X1,X2)) -> proper#(X1)
     proper#(U61(X)) -> proper#(X)
     proper#(U71(X1,X2,X3)) -> proper#(X3)
     proper#(U71(X1,X2,X3)) -> proper#(X2)
     proper#(U71(X1,X2,X3)) -> proper#(X1)
     proper#(x(X1,X2)) -> proper#(X2)
     proper#(x(X1,X2)) -> proper#(X1)
     proper#(and(X1,X2)) -> proper#(X2)
     proper#(and(X1,X2)) -> proper#(X1)
     proper#(isNatKind(X)) -> proper#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     isNatKind#(ok(X)) -> isNatKind#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     and#(mark(X1),X2) -> and#(X1,X2)
     and#(ok(X1),ok(X2)) -> and#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     x#(mark(X1),X2) -> x#(X1,X2)
     x#(X1,mark(X2)) -> x#(X1,X2)
     x#(ok(X1),ok(X2)) -> x#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
     U71#(ok(X1),ok(X2),ok(X3)) -> U71#(X1,X2,X3)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U61#(mark(X)) -> U61#(X)
     U61#(ok(X)) -> U61#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     plus#(mark(X1),X2) -> plus#(X1,X2)
     plus#(X1,mark(X2)) -> plus#(X1,X2)
     plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     s#(mark(X)) -> s#(X)
     s#(ok(X)) -> s#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
     U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U41#(mark(X1),X2) -> U41#(X1,X2)
     U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U33#(mark(X)) -> U33#(X)
     U33#(ok(X)) -> U33#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U32#(mark(X1),X2) -> U32#(X1,X2)
     U32#(ok(X1),ok(X2)) -> U32#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U31#(mark(X1),X2,X3) -> U31#(X1,X2,X3)
     U31#(ok(X1),ok(X2),ok(X3)) -> U31#(X1,X2,X3)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U22#(mark(X)) -> U22#(X)
     U22#(ok(X)) -> U22#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U21#(mark(X1),X2) -> U21#(X1,X2)
     U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U13#(mark(X)) -> U13#(X)
     U13#(ok(X)) -> U13#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     isNat#(ok(X)) -> isNat#(X)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U12#(mark(X1),X2) -> U12#(X1,X2)
     U12#(ok(X1),ok(X2)) -> U12#(X1,X2)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
     U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
    TRS:
     active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
     active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
     active(U13(tt())) -> mark(tt())
     active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
     active(U22(tt())) -> mark(tt())
     active(U31(tt(),V1,V2)) -> mark(U32(isNat(V1),V2))
     active(U32(tt(),V2)) -> mark(U33(isNat(V2)))
     active(U33(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(and(tt(),X)) -> mark(X)
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
     active(isNat(x(V1,V2))) -> mark(U31(and(isNatKind(V1),isNatKind(V2)),V1,V2))
     active(isNatKind(0())) -> mark(tt())
     active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(isNatKind(s(V1))) -> mark(isNatKind(V1))
     active(isNatKind(x(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
     active(plus(N,0())) -> mark(U41(and(isNat(N),isNatKind(N)),N))
     active(plus(N,s(M))) ->
     mark(U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(x(N,0())) -> mark(U61(and(isNat(N),isNatKind(N))))
     active(x(N,s(M))) -> mark(U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(U12(X1,X2)) -> U12(active(X1),X2)
     active(U13(X)) -> U13(active(X))
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U22(X)) -> U22(active(X))
     active(U31(X1,X2,X3)) -> U31(active(X1),X2,X3)
     active(U32(X1,X2)) -> U32(active(X1),X2)
     active(U33(X)) -> U33(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(s(X)) -> s(active(X))
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(U61(X)) -> U61(active(X))
     active(U71(X1,X2,X3)) -> U71(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     active(and(X1,X2)) -> and(active(X1),X2)
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U12(mark(X1),X2) -> mark(U12(X1,X2))
     U13(mark(X)) -> mark(U13(X))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U22(mark(X)) -> mark(U22(X))
     U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
     U32(mark(X1),X2) -> mark(U32(X1,X2))
     U33(mark(X)) -> mark(U33(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     s(mark(X)) -> mark(s(X))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     U61(mark(X)) -> mark(U61(X))
     U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U13(X)) -> U13(proper(X))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U22(X)) -> U22(proper(X))
     proper(U31(X1,X2,X3)) -> U31(proper(X1),proper(X2),proper(X3))
     proper(U32(X1,X2)) -> U32(proper(X1),proper(X2))
     proper(U33(X)) -> U33(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(U61(X)) -> U61(proper(X))
     proper(0()) -> ok(0())
     proper(U71(X1,X2,X3)) -> U71(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNatKind(X)) -> isNatKind(proper(X))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
     isNat(ok(X)) -> ok(isNat(X))
     U13(ok(X)) -> ok(U13(X))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U22(ok(X)) -> ok(U22(X))
     U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
     U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
     U33(ok(X)) -> ok(U33(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     U61(ok(X)) -> ok(U61(X))
     U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isNatKind(ok(X)) -> ok(isNatKind(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open