MAYBE

Problem:
 active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
 active(U12(tt())) -> mark(tt())
 active(U21(tt())) -> mark(tt())
 active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
 active(U32(tt())) -> mark(tt())
 active(U41(tt(),N)) -> mark(N)
 active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
 active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
 active(U61(tt())) -> mark(0())
 active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
 active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
 active(isNat(0())) -> mark(tt())
 active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
 active(isNat(s(V1))) -> mark(U21(isNat(V1)))
 active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
 active(plus(N,0())) -> mark(U41(isNat(N),N))
 active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
 active(x(N,0())) -> mark(U61(isNat(N)))
 active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
 active(U11(X1,X2)) -> U11(active(X1),X2)
 active(U12(X)) -> U12(active(X))
 active(U21(X)) -> U21(active(X))
 active(U31(X1,X2)) -> U31(active(X1),X2)
 active(U32(X)) -> U32(active(X))
 active(U41(X1,X2)) -> U41(active(X1),X2)
 active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
 active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
 active(x(X1,X2)) -> x(active(X1),X2)
 active(x(X1,X2)) -> x(X1,active(X2))
 U11(mark(X1),X2) -> mark(U11(X1,X2))
 U12(mark(X)) -> mark(U12(X))
 U21(mark(X)) -> mark(U21(X))
 U31(mark(X1),X2) -> mark(U31(X1,X2))
 U32(mark(X)) -> mark(U32(X))
 U41(mark(X1),X2) -> mark(U41(X1,X2))
 U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
 U52(mark(X1),X2,X3) -> mark(U52(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))
 U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
 x(mark(X1),X2) -> mark(x(X1,X2))
 x(X1,mark(X2)) -> mark(x(X1,X2))
 proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
 proper(tt()) -> ok(tt())
 proper(U12(X)) -> U12(proper(X))
 proper(isNat(X)) -> isNat(proper(X))
 proper(U21(X)) -> U21(proper(X))
 proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
 proper(U32(X)) -> U32(proper(X))
 proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
 proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
 proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
 proper(x(X1,X2)) -> x(proper(X1),proper(X2))
 U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
 U12(ok(X)) -> ok(U12(X))
 isNat(ok(X)) -> ok(isNat(X))
 U21(ok(X)) -> ok(U21(X))
 U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
 U32(ok(X)) -> ok(U32(X))
 U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
 U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
 U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
 U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
 x(ok(X1),ok(X2)) -> ok(x(X1,X2))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(U11(tt(),V2)) -> isNat#(V2)
   active#(U11(tt(),V2)) -> U12#(isNat(V2))
   active#(U31(tt(),V2)) -> isNat#(V2)
   active#(U31(tt(),V2)) -> U32#(isNat(V2))
   active#(U51(tt(),M,N)) -> isNat#(N)
   active#(U51(tt(),M,N)) -> U52#(isNat(N),M,N)
   active#(U52(tt(),M,N)) -> plus#(N,M)
   active#(U52(tt(),M,N)) -> s#(plus(N,M))
   active#(U71(tt(),M,N)) -> isNat#(N)
   active#(U71(tt(),M,N)) -> U72#(isNat(N),M,N)
   active#(U72(tt(),M,N)) -> x#(N,M)
   active#(U72(tt(),M,N)) -> plus#(x(N,M),N)
   active#(isNat(plus(V1,V2))) -> isNat#(V1)
   active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
   active#(isNat(s(V1))) -> isNat#(V1)
   active#(isNat(s(V1))) -> U21#(isNat(V1))
   active#(isNat(x(V1,V2))) -> isNat#(V1)
   active#(isNat(x(V1,V2))) -> U31#(isNat(V1),V2)
   active#(plus(N,0())) -> isNat#(N)
   active#(plus(N,0())) -> U41#(isNat(N),N)
   active#(plus(N,s(M))) -> isNat#(M)
   active#(plus(N,s(M))) -> U51#(isNat(M),M,N)
   active#(x(N,0())) -> isNat#(N)
   active#(x(N,0())) -> U61#(isNat(N))
   active#(x(N,s(M))) -> isNat#(M)
   active#(x(N,s(M))) -> U71#(isNat(M),M,N)
   active#(U11(X1,X2)) -> active#(X1)
   active#(U11(X1,X2)) -> U11#(active(X1),X2)
   active#(U12(X)) -> active#(X)
   active#(U12(X)) -> U12#(active(X))
   active#(U21(X)) -> active#(X)
   active#(U21(X)) -> U21#(active(X))
   active#(U31(X1,X2)) -> active#(X1)
   active#(U31(X1,X2)) -> U31#(active(X1),X2)
   active#(U32(X)) -> active#(X)
   active#(U32(X)) -> U32#(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#(U52(X1,X2,X3)) -> active#(X1)
   active#(U52(X1,X2,X3)) -> U52#(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#(U72(X1,X2,X3)) -> active#(X1)
   active#(U72(X1,X2,X3)) -> U72#(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))
   U11#(mark(X1),X2) -> U11#(X1,X2)
   U12#(mark(X)) -> U12#(X)
   U21#(mark(X)) -> U21#(X)
   U31#(mark(X1),X2) -> U31#(X1,X2)
   U32#(mark(X)) -> U32#(X)
   U41#(mark(X1),X2) -> U41#(X1,X2)
   U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
   U52#(mark(X1),X2,X3) -> U52#(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)
   U72#(mark(X1),X2,X3) -> U72#(X1,X2,X3)
   x#(mark(X1),X2) -> x#(X1,X2)
   x#(X1,mark(X2)) -> x#(X1,X2)
   proper#(U11(X1,X2)) -> proper#(X2)
   proper#(U11(X1,X2)) -> proper#(X1)
   proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
   proper#(U12(X)) -> proper#(X)
   proper#(U12(X)) -> U12#(proper(X))
   proper#(isNat(X)) -> proper#(X)
   proper#(isNat(X)) -> isNat#(proper(X))
   proper#(U21(X)) -> proper#(X)
   proper#(U21(X)) -> U21#(proper(X))
   proper#(U31(X1,X2)) -> proper#(X2)
   proper#(U31(X1,X2)) -> proper#(X1)
   proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
   proper#(U32(X)) -> proper#(X)
   proper#(U32(X)) -> U32#(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#(U52(X1,X2,X3)) -> proper#(X3)
   proper#(U52(X1,X2,X3)) -> proper#(X2)
   proper#(U52(X1,X2,X3)) -> proper#(X1)
   proper#(U52(X1,X2,X3)) -> U52#(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#(U72(X1,X2,X3)) -> proper#(X3)
   proper#(U72(X1,X2,X3)) -> proper#(X2)
   proper#(U72(X1,X2,X3)) -> proper#(X1)
   proper#(U72(X1,X2,X3)) -> U72#(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))
   U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
   U12#(ok(X)) -> U12#(X)
   isNat#(ok(X)) -> isNat#(X)
   U21#(ok(X)) -> U21#(X)
   U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
   U32#(ok(X)) -> U32#(X)
   U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
   U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
   U52#(ok(X1),ok(X2),ok(X3)) -> U52#(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)
   U72#(ok(X1),ok(X2),ok(X3)) -> U72#(X1,X2,X3)
   x#(ok(X1),ok(X2)) -> x#(X1,X2)
   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(),V2)) -> mark(U12(isNat(V2)))
   active(U12(tt())) -> mark(tt())
   active(U21(tt())) -> mark(tt())
   active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
   active(U32(tt())) -> mark(tt())
   active(U41(tt(),N)) -> mark(N)
   active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
   active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
   active(U61(tt())) -> mark(0())
   active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
   active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
   active(isNat(0())) -> mark(tt())
   active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
   active(plus(N,0())) -> mark(U41(isNat(N),N))
   active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
   active(x(N,0())) -> mark(U61(isNat(N)))
   active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
   active(U11(X1,X2)) -> U11(active(X1),X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X)) -> U21(active(X))
   active(U31(X1,X2)) -> U31(active(X1),X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1,X2)) -> U41(active(X1),X2)
   active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
   active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
   active(x(X1,X2)) -> x(active(X1),X2)
   active(x(X1,X2)) -> x(X1,active(X2))
   U11(mark(X1),X2) -> mark(U11(X1,X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X)) -> mark(U21(X))
   U31(mark(X1),X2) -> mark(U31(X1,X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1),X2) -> mark(U41(X1,X2))
   U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
   U52(mark(X1),X2,X3) -> mark(U52(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))
   U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
   x(mark(X1),X2) -> mark(x(X1,X2))
   x(X1,mark(X2)) -> mark(x(X1,X2))
   proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
   proper(tt()) -> ok(tt())
   proper(U12(X)) -> U12(proper(X))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U21(X)) -> U21(proper(X))
   proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
   proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
   proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
   proper(x(X1,X2)) -> x(proper(X1),proper(X2))
   U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
   U12(ok(X)) -> ok(U12(X))
   isNat(ok(X)) -> ok(isNat(X))
   U21(ok(X)) -> ok(U21(X))
   U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
   U32(ok(X)) -> ok(U32(X))
   U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
   U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
   U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
   U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
   x(ok(X1),ok(X2)) -> ok(x(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U31(tt(),V2)) -> isNat#(V2)
    active#(U31(tt(),V2)) -> U32#(isNat(V2))
    active#(U51(tt(),M,N)) -> isNat#(N)
    active#(U51(tt(),M,N)) -> U52#(isNat(N),M,N)
    active#(U52(tt(),M,N)) -> plus#(N,M)
    active#(U52(tt(),M,N)) -> s#(plus(N,M))
    active#(U71(tt(),M,N)) -> isNat#(N)
    active#(U71(tt(),M,N)) -> U72#(isNat(N),M,N)
    active#(U72(tt(),M,N)) -> x#(N,M)
    active#(U72(tt(),M,N)) -> plus#(x(N,M),N)
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(isNat(x(V1,V2))) -> isNat#(V1)
    active#(isNat(x(V1,V2))) -> U31#(isNat(V1),V2)
    active#(plus(N,0())) -> isNat#(N)
    active#(plus(N,0())) -> U41#(isNat(N),N)
    active#(plus(N,s(M))) -> isNat#(M)
    active#(plus(N,s(M))) -> U51#(isNat(M),M,N)
    active#(x(N,0())) -> isNat#(N)
    active#(x(N,0())) -> U61#(isNat(N))
    active#(x(N,s(M))) -> isNat#(M)
    active#(x(N,s(M))) -> U71#(isNat(M),M,N)
    active#(U11(X1,X2)) -> active#(X1)
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U12(X)) -> active#(X)
    active#(U12(X)) -> U12#(active(X))
    active#(U21(X)) -> active#(X)
    active#(U21(X)) -> U21#(active(X))
    active#(U31(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U32(X)) -> active#(X)
    active#(U32(X)) -> U32#(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#(U52(X1,X2,X3)) -> active#(X1)
    active#(U52(X1,X2,X3)) -> U52#(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#(U72(X1,X2,X3)) -> active#(X1)
    active#(U72(X1,X2,X3)) -> U72#(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))
    U11#(mark(X1),X2) -> U11#(X1,X2)
    U12#(mark(X)) -> U12#(X)
    U21#(mark(X)) -> U21#(X)
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U32#(mark(X)) -> U32#(X)
    U41#(mark(X1),X2) -> U41#(X1,X2)
    U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
    U52#(mark(X1),X2,X3) -> U52#(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)
    U72#(mark(X1),X2,X3) -> U72#(X1,X2,X3)
    x#(mark(X1),X2) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2)
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U12(X)) -> proper#(X)
    proper#(U12(X)) -> U12#(proper(X))
    proper#(isNat(X)) -> proper#(X)
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U21(X)) -> proper#(X)
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U32(X)) -> proper#(X)
    proper#(U32(X)) -> U32#(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#(U52(X1,X2,X3)) -> proper#(X3)
    proper#(U52(X1,X2,X3)) -> proper#(X2)
    proper#(U52(X1,X2,X3)) -> proper#(X1)
    proper#(U52(X1,X2,X3)) -> U52#(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#(U72(X1,X2,X3)) -> proper#(X3)
    proper#(U72(X1,X2,X3)) -> proper#(X2)
    proper#(U72(X1,X2,X3)) -> proper#(X1)
    proper#(U72(X1,X2,X3)) -> U72#(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))
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    U12#(ok(X)) -> U12#(X)
    isNat#(ok(X)) -> isNat#(X)
    U21#(ok(X)) -> U21#(X)
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    U32#(ok(X)) -> U32#(X)
    U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
    U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
    U52#(ok(X1),ok(X2),ok(X3)) -> U52#(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)
    U72#(ok(X1),ok(X2),ok(X3)) -> U72#(X1,X2,X3)
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    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(),V2)) -> mark(U12(isNat(V2)))
    active(U12(tt())) -> mark(tt())
    active(U21(tt())) -> mark(tt())
    active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
    active(U32(tt())) -> mark(tt())
    active(U41(tt(),N)) -> mark(N)
    active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
    active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
    active(U61(tt())) -> mark(0())
    active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
    active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
    active(isNat(0())) -> mark(tt())
    active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
    active(isNat(s(V1))) -> mark(U21(isNat(V1)))
    active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
    active(plus(N,0())) -> mark(U41(isNat(N),N))
    active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
    active(x(N,0())) -> mark(U61(isNat(N)))
    active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
    active(U11(X1,X2)) -> U11(active(X1),X2)
    active(U12(X)) -> U12(active(X))
    active(U21(X)) -> U21(active(X))
    active(U31(X1,X2)) -> U31(active(X1),X2)
    active(U32(X)) -> U32(active(X))
    active(U41(X1,X2)) -> U41(active(X1),X2)
    active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
    active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
    active(x(X1,X2)) -> x(active(X1),X2)
    active(x(X1,X2)) -> x(X1,active(X2))
    U11(mark(X1),X2) -> mark(U11(X1,X2))
    U12(mark(X)) -> mark(U12(X))
    U21(mark(X)) -> mark(U21(X))
    U31(mark(X1),X2) -> mark(U31(X1,X2))
    U32(mark(X)) -> mark(U32(X))
    U41(mark(X1),X2) -> mark(U41(X1,X2))
    U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
    U52(mark(X1),X2,X3) -> mark(U52(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))
    U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
    x(mark(X1),X2) -> mark(x(X1,X2))
    x(X1,mark(X2)) -> mark(x(X1,X2))
    proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
    proper(tt()) -> ok(tt())
    proper(U12(X)) -> U12(proper(X))
    proper(isNat(X)) -> isNat(proper(X))
    proper(U21(X)) -> U21(proper(X))
    proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
    proper(U32(X)) -> U32(proper(X))
    proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
    proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
    proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
    proper(x(X1,X2)) -> x(proper(X1),proper(X2))
    U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
    U12(ok(X)) -> ok(U12(X))
    isNat(ok(X)) -> ok(isNat(X))
    U21(ok(X)) -> ok(U21(X))
    U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
    U32(ok(X)) -> ok(U32(X))
    U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
    U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
    U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
    U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
    x(ok(X1),ok(X2)) -> ok(x(X1,X2))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    ...
   SCC Processor:
    #sccs: 18
    #rules: 77
    #arcs: 2419/18496
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     active#(U11(X1,X2)) -> active#(X1)
     active#(U12(X)) -> active#(X)
     active#(U21(X)) -> active#(X)
     active#(U31(X1,X2)) -> active#(X1)
     active#(U32(X)) -> active#(X)
     active#(U41(X1,X2)) -> active#(X1)
     active#(U51(X1,X2,X3)) -> active#(X1)
     active#(U52(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#(U72(X1,X2,X3)) -> active#(X1)
     active#(x(X1,X2)) -> active#(X1)
     active#(x(X1,X2)) -> active#(X2)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(U11(X1,X2)) -> proper#(X2)
     proper#(U11(X1,X2)) -> proper#(X1)
     proper#(U12(X)) -> proper#(X)
     proper#(isNat(X)) -> proper#(X)
     proper#(U21(X)) -> proper#(X)
     proper#(U31(X1,X2)) -> proper#(X2)
     proper#(U31(X1,X2)) -> proper#(X1)
     proper#(U32(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#(U52(X1,X2,X3)) -> proper#(X3)
     proper#(U52(X1,X2,X3)) -> proper#(X2)
     proper#(U52(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#(U72(X1,X2,X3)) -> proper#(X3)
     proper#(U72(X1,X2,X3)) -> proper#(X2)
     proper#(U72(X1,X2,X3)) -> proper#(X1)
     proper#(x(X1,X2)) -> proper#(X2)
     proper#(x(X1,X2)) -> proper#(X1)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    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(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(x#) = 1
     problem:
      DPs:
       x#(mark(X1),X2) -> x#(X1,X2)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(x#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U72#(mark(X1),X2,X3) -> U72#(X1,X2,X3)
     U72#(ok(X1),ok(X2),ok(X3)) -> U72#(X1,X2,X3)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U72#) = 2
     problem:
      DPs:
       U72#(mark(X1),X2,X3) -> U72#(X1,X2,X3)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U72#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    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(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U71#) = 2
     problem:
      DPs:
       U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U71#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U61#(mark(X)) -> U61#(X)
     U61#(ok(X)) -> U61#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U61#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    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(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 1
     problem:
      DPs:
       plus#(mark(X1),X2) -> plus#(X1,X2)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(plus#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     s#(mark(X)) -> s#(X)
     s#(ok(X)) -> s#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(s#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     U52#(mark(X1),X2,X3) -> U52#(X1,X2,X3)
     U52#(ok(X1),ok(X2),ok(X3)) -> U52#(X1,X2,X3)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U52#) = 2
     problem:
      DPs:
       U52#(mark(X1),X2,X3) -> U52#(X1,X2,X3)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U52#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    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(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U51#) = 2
     problem:
      DPs:
       U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U51#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U41#(mark(X1),X2) -> U41#(X1,X2)
     U41#(ok(X1),ok(X2)) -> U41#(X1,X2)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U41#) = 1
     problem:
      DPs:
       U41#(mark(X1),X2) -> U41#(X1,X2)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U41#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U32#(mark(X)) -> U32#(X)
     U32#(ok(X)) -> U32#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U32#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     U31#(mark(X1),X2) -> U31#(X1,X2)
     U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U31#) = 1
     problem:
      DPs:
       U31#(mark(X1),X2) -> U31#(X1,X2)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U31#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U21#(mark(X)) -> U21#(X)
     U21#(ok(X)) -> U21#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U21#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     isNat#(ok(X)) -> isNat#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(isNat#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     U12#(mark(X)) -> U12#(X)
     U12#(ok(X)) -> U12#(X)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U12#) = 0
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     U11#(mark(X1),X2) -> U11#(X1,X2)
     U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
     active(U32(tt())) -> mark(tt())
     active(U41(tt(),N)) -> mark(N)
     active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
     active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
     active(U61(tt())) -> mark(0())
     active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
     active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
     active(plus(N,0())) -> mark(U41(isNat(N),N))
     active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
     active(x(N,0())) -> mark(U61(isNat(N)))
     active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
     active(U11(X1,X2)) -> U11(active(X1),X2)
     active(U12(X)) -> U12(active(X))
     active(U21(X)) -> U21(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U32(X)) -> U32(active(X))
     active(U41(X1,X2)) -> U41(active(X1),X2)
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     U11(mark(X1),X2) -> mark(U11(X1,X2))
     U12(mark(X)) -> mark(U12(X))
     U21(mark(X)) -> mark(U21(X))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U32(mark(X)) -> mark(U32(X))
     U41(mark(X1),X2) -> mark(U41(X1,X2))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U52(mark(X1),X2,X3) -> mark(U52(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))
     U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(U12(X)) -> U12(proper(X))
     proper(isNat(X)) -> isNat(proper(X))
     proper(U21(X)) -> U21(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U32(X)) -> U32(proper(X))
     proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
     U12(ok(X)) -> ok(U12(X))
     isNat(ok(X)) -> ok(isNat(X))
     U21(ok(X)) -> ok(U21(X))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U32(ok(X)) -> ok(U32(X))
     U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
     U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(U11#) = 1
     problem:
      DPs:
       U11#(mark(X1),X2) -> U11#(X1,X2)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
       active(U32(tt())) -> mark(tt())
       active(U41(tt(),N)) -> mark(N)
       active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
       active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
       active(U61(tt())) -> mark(0())
       active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
       active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
       active(plus(N,0())) -> mark(U41(isNat(N),N))
       active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
       active(x(N,0())) -> mark(U61(isNat(N)))
       active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
       active(U11(X1,X2)) -> U11(active(X1),X2)
       active(U12(X)) -> U12(active(X))
       active(U21(X)) -> U21(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U32(X)) -> U32(active(X))
       active(U41(X1,X2)) -> U41(active(X1),X2)
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       U11(mark(X1),X2) -> mark(U11(X1,X2))
       U12(mark(X)) -> mark(U12(X))
       U21(mark(X)) -> mark(U21(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U32(mark(X)) -> mark(U32(X))
       U41(mark(X1),X2) -> mark(U41(X1,X2))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2,X3) -> mark(U52(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))
       U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
       proper(tt()) -> ok(tt())
       proper(U12(X)) -> U12(proper(X))
       proper(isNat(X)) -> isNat(proper(X))
       proper(U21(X)) -> U21(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U32(X)) -> U32(proper(X))
       proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
       U12(ok(X)) -> ok(U12(X))
       isNat(ok(X)) -> ok(isNat(X))
       U21(ok(X)) -> ok(U21(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U32(ok(X)) -> ok(U32(X))
       U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
       U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(U11#) = 0
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),V2)) -> mark(U32(isNat(V2)))
        active(U32(tt())) -> mark(tt())
        active(U41(tt(),N)) -> mark(N)
        active(U51(tt(),M,N)) -> mark(U52(isNat(N),M,N))
        active(U52(tt(),M,N)) -> mark(s(plus(N,M)))
        active(U61(tt())) -> mark(0())
        active(U71(tt(),M,N)) -> mark(U72(isNat(N),M,N))
        active(U72(tt(),M,N)) -> mark(plus(x(N,M),N))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(isNat(x(V1,V2))) -> mark(U31(isNat(V1),V2))
        active(plus(N,0())) -> mark(U41(isNat(N),N))
        active(plus(N,s(M))) -> mark(U51(isNat(M),M,N))
        active(x(N,0())) -> mark(U61(isNat(N)))
        active(x(N,s(M))) -> mark(U71(isNat(M),M,N))
        active(U11(X1,X2)) -> U11(active(X1),X2)
        active(U12(X)) -> U12(active(X))
        active(U21(X)) -> U21(active(X))
        active(U31(X1,X2)) -> U31(active(X1),X2)
        active(U32(X)) -> U32(active(X))
        active(U41(X1,X2)) -> U41(active(X1),X2)
        active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
        active(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(active(X1),X2,X3)
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        U11(mark(X1),X2) -> mark(U11(X1,X2))
        U12(mark(X)) -> mark(U12(X))
        U21(mark(X)) -> mark(U21(X))
        U31(mark(X1),X2) -> mark(U31(X1,X2))
        U32(mark(X)) -> mark(U32(X))
        U41(mark(X1),X2) -> mark(U41(X1,X2))
        U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
        U52(mark(X1),X2,X3) -> mark(U52(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))
        U72(mark(X1),X2,X3) -> mark(U72(X1,X2,X3))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
        proper(tt()) -> ok(tt())
        proper(U12(X)) -> U12(proper(X))
        proper(isNat(X)) -> isNat(proper(X))
        proper(U21(X)) -> U21(proper(X))
        proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
        proper(U32(X)) -> U32(proper(X))
        proper(U41(X1,X2)) -> U41(proper(X1),proper(X2))
        proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
        proper(U52(X1,X2,X3)) -> U52(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(U72(X1,X2,X3)) -> U72(proper(X1),proper(X2),proper(X3))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
        U12(ok(X)) -> ok(U12(X))
        isNat(ok(X)) -> ok(isNat(X))
        U21(ok(X)) -> ok(U21(X))
        U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
        U32(ok(X)) -> ok(U32(X))
        U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
        U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
        U52(ok(X1),ok(X2),ok(X3)) -> ok(U52(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))
        U72(ok(X1),ok(X2),ok(X3)) -> ok(U72(X1,X2,X3))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed