MAYBE

Problem:
 active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
 active(U12(tt())) -> mark(tt())
 active(U21(tt())) -> mark(tt())
 active(U31(tt(),N)) -> mark(N)
 active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
 active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
 active(isNat(0())) -> mark(tt())
 active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
 active(isNat(s(V1))) -> mark(U21(isNat(V1)))
 active(plus(N,0())) -> mark(U31(isNat(N),N))
 active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
 active(U42(X1,X2,X3)) -> U42(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))
 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))
 U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
 U42(mark(X1),X2,X3) -> mark(U42(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))
 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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
 proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
 proper(s(X)) -> s(proper(X))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 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))
 U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
 U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
 s(ok(X)) -> ok(s(X))
 plus(ok(X1),ok(X2)) -> ok(plus(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#(U41(tt(),M,N)) -> isNat#(N)
   active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
   active#(U42(tt(),M,N)) -> plus#(N,M)
   active#(U42(tt(),M,N)) -> s#(plus(N,M))
   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#(plus(N,0())) -> isNat#(N)
   active#(plus(N,0())) -> U31#(isNat(N),N)
   active#(plus(N,s(M))) -> isNat#(M)
   active#(plus(N,s(M))) -> U41#(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#(U41(X1,X2,X3)) -> active#(X1)
   active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
   active#(U42(X1,X2,X3)) -> active#(X1)
   active#(U42(X1,X2,X3)) -> U42#(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))
   U11#(mark(X1),X2) -> U11#(X1,X2)
   U12#(mark(X)) -> U12#(X)
   U21#(mark(X)) -> U21#(X)
   U31#(mark(X1),X2) -> U31#(X1,X2)
   U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
   U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
   s#(mark(X)) -> s#(X)
   plus#(mark(X1),X2) -> plus#(X1,X2)
   plus#(X1,mark(X2)) -> plus#(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#(U41(X1,X2,X3)) -> proper#(X3)
   proper#(U41(X1,X2,X3)) -> proper#(X2)
   proper#(U41(X1,X2,X3)) -> proper#(X1)
   proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
   proper#(U42(X1,X2,X3)) -> proper#(X3)
   proper#(U42(X1,X2,X3)) -> proper#(X2)
   proper#(U42(X1,X2,X3)) -> proper#(X1)
   proper#(U42(X1,X2,X3)) -> U42#(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))
   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)
   U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
   U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
   s#(ok(X)) -> s#(X)
   plus#(ok(X1),ok(X2)) -> plus#(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(),N)) -> mark(N)
   active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
   active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
   active(isNat(0())) -> mark(tt())
   active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(plus(N,0())) -> mark(U31(isNat(N),N))
   active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
   active(U42(X1,X2,X3)) -> U42(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))
   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))
   U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
   U42(mark(X1),X2,X3) -> mark(U42(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))
   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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
   proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   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))
   U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
   U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  EDG Processor:
   DPs:
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    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#(plus(N,0())) -> isNat#(N)
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(plus(N,s(M))) -> isNat#(M)
    active#(plus(N,s(M))) -> U41#(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#(U41(X1,X2,X3)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> U42#(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))
    U11#(mark(X1),X2) -> U11#(X1,X2)
    U12#(mark(X)) -> U12#(X)
    U21#(mark(X)) -> U21#(X)
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    s#(mark(X)) -> s#(X)
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(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#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> U42#(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))
    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)
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    s#(ok(X)) -> s#(X)
    plus#(ok(X1),ok(X2)) -> plus#(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(),N)) -> mark(N)
    active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
    active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
    active(isNat(0())) -> mark(tt())
    active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
    active(isNat(s(V1))) -> mark(U21(isNat(V1)))
    active(plus(N,0())) -> mark(U31(isNat(N),N))
    active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
    active(U42(X1,X2,X3)) -> U42(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))
    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))
    U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
    U42(mark(X1),X2,X3) -> mark(U42(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))
    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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
    proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
    proper(s(X)) -> s(proper(X))
    proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    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))
    U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
    U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
    s(ok(X)) -> ok(s(X))
    plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> active#(X) -> active#(U11(tt(),V2)) -> isNat#(V2)
    top#(ok(X)) -> active#(X) -> active#(U11(tt(),V2)) -> U12#(isNat(V2))
    top#(ok(X)) -> active#(X) -> active#(U41(tt(),M,N)) -> isNat#(N)
    top#(ok(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    top#(ok(X)) -> active#(X) -> active#(U42(tt(),M,N)) -> plus#(N,M)
    top#(ok(X)) -> active#(X) -> active#(U42(tt(),M,N)) -> s#(plus(N,M))
    top#(ok(X)) -> active#(X) -> active#(isNat(plus(V1,V2))) -> isNat#(V1)
    top#(ok(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    top#(ok(X)) -> active#(X) -> active#(isNat(s(V1))) -> isNat#(V1)
    top#(ok(X)) -> active#(X) -> active#(isNat(s(V1))) -> U21#(isNat(V1))
    top#(ok(X)) -> active#(X) -> active#(plus(N,0())) -> isNat#(N)
    top#(ok(X)) -> active#(X) -> active#(plus(N,0())) -> U31#(isNat(N),N)
    top#(ok(X)) -> active#(X) -> active#(plus(N,s(M))) -> isNat#(M)
    top#(ok(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    top#(ok(X)) -> active#(X) -> active#(U11(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(U12(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(U12(X)) -> U12#(active(X))
    top#(ok(X)) -> active#(X) -> active#(U21(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(U21(X)) -> U21#(active(X))
    top#(ok(X)) -> active#(X) -> active#(U31(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(U41(X1,X2,X3)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    top#(ok(X)) -> active#(X) -> active#(U42(X1,X2,X3)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(U12(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(U12(X)) -> U12#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(isNat(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(U21(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(U21(X)) -> U21#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(U31(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(U31(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X3)
    top#(mark(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    top#(mark(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X3)
    top#(mark(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(U11(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(U12(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(U12(X)) -> U12#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(isNat(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(U21(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(U21(X)) -> U21#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(U31(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(U31(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(s(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(s(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(s(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X)
    proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(s(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3)) ->
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3)) ->
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(s(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X3) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) ->
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) ->
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) ->
    U31#(mark(X1),X2) -> U31#(X1,X2)
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) ->
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U21(X)) -> proper#(X) -> proper#(U12(X)) -> proper#(X)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U21(X)) -> proper#(X) -> proper#(isNat(X)) -> proper#(X)
    proper#(U21(X)) -> proper#(X) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U21(X)) -> proper#(X) -> proper#(U21(X)) -> proper#(X)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U21(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U21(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U21(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U21(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U21(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(U21(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(U21(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U21(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U21(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U21(X)) -> U21#(proper(X)) ->
    U21#(mark(X)) -> U21#(X)
    proper#(U21(X)) -> U21#(proper(X)) -> U21#(ok(X)) -> U21#(X)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U12(X)) -> proper#(X) -> proper#(U12(X)) -> proper#(X)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U12(X)) -> proper#(X) -> proper#(isNat(X)) -> proper#(X)
    proper#(U12(X)) -> proper#(X) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U12(X)) -> proper#(X) -> proper#(U21(X)) -> proper#(X)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U12(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U12(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U12(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U12(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U12(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(U12(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(U12(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U12(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U12(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U12(X)) -> U12#(proper(X)) ->
    U12#(mark(X)) -> U12#(X)
    proper#(U12(X)) -> U12#(proper(X)) -> U12#(ok(X)) -> U12#(X)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(isNat(X)) -> proper#(X) -> proper#(U12(X)) -> proper#(X)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(isNat(X)) -> proper#(X) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(isNat(X)) -> proper#(X) -> proper#(U21(X)) -> proper#(X)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(isNat(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(isNat(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(isNat(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(isNat(X)) -> isNat#(proper(X)) ->
    isNat#(ok(X)) -> isNat#(X)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U12(X)) -> U12#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(isNat(X)) -> isNat#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U21(X)) -> U21#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(U42(X1,X2,X3)) -> U42#(proper(X1),proper(X2),proper(X3))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(U11(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) ->
    U11#(mark(X1),X2) -> U11#(X1,X2)
    proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) ->
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) ->
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) ->
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) ->
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) ->
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2) ->
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2) ->
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    U31#(mark(X1),X2) -> U31#(X1,X2) ->
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U31#(mark(X1),X2) -> U31#(X1,X2) -> U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    U21#(ok(X)) -> U21#(X) -> U21#(mark(X)) -> U21#(X)
    U21#(ok(X)) -> U21#(X) -> U21#(ok(X)) -> U21#(X)
    U21#(mark(X)) -> U21#(X) -> U21#(mark(X)) -> U21#(X)
    U21#(mark(X)) -> U21#(X) -> U21#(ok(X)) -> U21#(X)
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2) ->
    U11#(mark(X1),X2) -> U11#(X1,X2)
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2) ->
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    U11#(mark(X1),X2) -> U11#(X1,X2) ->
    U11#(mark(X1),X2) -> U11#(X1,X2)
    U11#(mark(X1),X2) -> U11#(X1,X2) -> U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    plus#(mark(X1),X2) -> plus#(X1,X2) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(mark(X1),X2) -> plus#(X1,X2) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    plus#(mark(X1),X2) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3) ->
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3) ->
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) ->
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3) ->
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    U12#(ok(X)) -> U12#(X) -> U12#(mark(X)) -> U12#(X)
    U12#(ok(X)) -> U12#(X) -> U12#(ok(X)) -> U12#(X)
    U12#(mark(X)) -> U12#(X) -> U12#(mark(X)) -> U12#(X)
    U12#(mark(X)) -> U12#(X) -> U12#(ok(X)) -> U12#(X)
    isNat#(ok(X)) -> isNat#(X) -> isNat#(ok(X)) -> isNat#(X)
    active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X)
    active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X)
    active#(s(X)) -> active#(X) -> active#(U11(tt(),V2)) -> isNat#(V2)
    active#(s(X)) -> active#(X) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(s(X)) -> active#(X) -> active#(U41(tt(),M,N)) -> isNat#(N)
    active#(s(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(s(X)) -> active#(X) -> active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(s(X)) -> active#(X) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(s(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(s(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(s(X)) -> active#(X) -> active#(isNat(s(V1))) -> isNat#(V1)
    active#(s(X)) -> active#(X) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(s(X)) -> active#(X) -> active#(plus(N,0())) -> isNat#(N)
    active#(s(X)) -> active#(X) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(s(X)) -> active#(X) -> active#(plus(N,s(M))) -> isNat#(M)
    active#(s(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(s(X)) -> active#(X) -> active#(U11(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(U12(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(U12(X)) -> U12#(active(X))
    active#(s(X)) -> active#(X) -> active#(U21(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(U21(X)) -> U21#(active(X))
    active#(s(X)) -> active#(X) -> active#(U31(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(U41(X1,X2,X3)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(s(X)) -> active#(X) -> active#(U42(X1,X2,X3)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2)
    active#(s(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(plus(X1,X2)) -> plus#(active(X1),X2) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(active(X1),X2) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(X1,active(X2)) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(X1,active(X2)) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U12(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U12(X)) -> U12#(active(X))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U21(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U21(X)) -> U21#(active(X))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> U12#(active(X))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> U21#(active(X))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(plus(N,0())) -> U31#(isNat(N),N) ->
    U31#(mark(X1),X2) -> U31#(X1,X2)
    active#(plus(N,0())) -> U31#(isNat(N),N) ->
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    active#(plus(N,0())) -> isNat#(N) ->
    isNat#(ok(X)) -> isNat#(X)
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N) ->
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    active#(U42(tt(),M,N)) -> s#(plus(N,M)) ->
    s#(mark(X)) -> s#(X)
    active#(U42(tt(),M,N)) -> s#(plus(N,M)) ->
    s#(ok(X)) -> s#(X)
    active#(U42(tt(),M,N)) -> plus#(N,M) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    active#(U42(tt(),M,N)) -> plus#(N,M) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    active#(U42(tt(),M,N)) -> plus#(N,M) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3) ->
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3) ->
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U12(X)) -> active#(X)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U21(X)) -> active#(X)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U42(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N) ->
    U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N) ->
    U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
    active#(U41(tt(),M,N)) -> isNat#(N) ->
    isNat#(ok(X)) -> isNat#(X)
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) ->
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) ->
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U12(X)) -> active#(X)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U21(X)) -> active#(X)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U41(X1,X2,X3)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(U31(X1,X2)) -> U31#(active(X1),X2) ->
    U31#(mark(X1),X2) -> U31#(X1,X2)
    active#(U31(X1,X2)) -> U31#(active(X1),X2) ->
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> active#(X)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> active#(X)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U31(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(U21(X)) -> U21#(active(X)) ->
    U21#(mark(X)) -> U21#(X)
    active#(U21(X)) -> U21#(active(X)) -> U21#(ok(X)) -> U21#(X)
    active#(U21(X)) -> active#(X) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U21(X)) -> active#(X) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U21(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U21(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U21(X)) -> active#(X) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U21(X)) -> active#(X) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U21(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U21(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U21(X)) -> active#(X) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U21(X)) -> active#(X) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U21(X)) -> active#(X) -> active#(plus(N,0())) -> isNat#(N)
    active#(U21(X)) -> active#(X) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U21(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U21(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U21(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U21(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U21(X)) -> active#(X) -> active#(U12(X)) -> active#(X)
    active#(U21(X)) -> active#(X) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U21(X)) -> active#(X) -> active#(U21(X)) -> active#(X)
    active#(U21(X)) -> active#(X) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U21(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U21(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U21(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U21(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U21(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U21(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U21(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(U21(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(U21(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U21(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U21(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U21(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(U12(X)) -> U12#(active(X)) ->
    U12#(mark(X)) -> U12#(X)
    active#(U12(X)) -> U12#(active(X)) -> U12#(ok(X)) -> U12#(X)
    active#(U12(X)) -> active#(X) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U12(X)) -> active#(X) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U12(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U12(X)) -> active#(X) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U12(X)) -> active#(X) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U12(X)) -> active#(X) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U12(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U12(X)) -> active#(X) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U12(X)) -> active#(X) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U12(X)) -> active#(X) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U12(X)) -> active#(X) -> active#(plus(N,0())) -> isNat#(N)
    active#(U12(X)) -> active#(X) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U12(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U12(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U12(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U12(X)) -> active#(X) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U12(X)) -> active#(X) -> active#(U12(X)) -> active#(X)
    active#(U12(X)) -> active#(X) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U12(X)) -> active#(X) -> active#(U21(X)) -> active#(X)
    active#(U12(X)) -> active#(X) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U12(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U12(X)) -> active#(X) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U12(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U12(X)) -> active#(X) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U12(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U12(X)) -> active#(X) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U12(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(U12(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(U12(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U12(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U12(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U12(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(isNat(s(V1))) -> U21#(isNat(V1)) ->
    U21#(mark(X)) -> U21#(X)
    active#(isNat(s(V1))) -> U21#(isNat(V1)) ->
    U21#(ok(X)) -> U21#(X)
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2) ->
    U11#(mark(X1),X2) -> U11#(X1,X2)
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2) ->
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    active#(isNat(plus(V1,V2))) -> isNat#(V1) ->
    isNat#(ok(X)) -> isNat#(X)
    active#(U11(tt(),V2)) -> U12#(isNat(V2)) ->
    U12#(mark(X)) -> U12#(X)
    active#(U11(tt(),V2)) -> U12#(isNat(V2)) ->
    U12#(ok(X)) -> U12#(X)
    active#(U11(tt(),V2)) -> isNat#(V2) ->
    isNat#(ok(X)) -> isNat#(X)
    active#(U11(X1,X2)) -> U11#(active(X1),X2) ->
    U11#(mark(X1),X2) -> U11#(X1,X2)
    active#(U11(X1,X2)) -> U11#(active(X1),X2) ->
    U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> isNat#(V2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U11(tt(),V2)) -> U12#(isNat(V2))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> isNat#(N)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U41(tt(),M,N)) -> U42#(isNat(N),M,N)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> plus#(N,M)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U42(tt(),M,N)) -> s#(plus(N,M))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> isNat#(V1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(isNat(plus(V1,V2))) -> U11#(isNat(V1),V2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> isNat#(V1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(isNat(s(V1))) -> U21#(isNat(V1))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> isNat#(N)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(N,0())) -> U31#(isNat(N),N)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> isNat#(M)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> U41#(isNat(M),M,N)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> active#(X1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U11(X1,X2)) -> U11#(active(X1),X2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> active#(X)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U12(X)) -> U12#(active(X))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> active#(X)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U21(X)) -> U21#(active(X))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> active#(X1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> active#(X1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(U42(X1,X2,X3)) -> U42#(active(X1),X2,X3)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(U11(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(U11(X1,X2)) -> active#(X1) -> active#(plus(X1,X2)) -> plus#(X1,active(X2))
   SCC Processor:
    #sccs: 12
    #rules: 45
    #arcs: 847/6241
    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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     proper#(plus(X1,X2)) -> proper#(X1)
     proper#(plus(X1,X2)) -> proper#(X2)
     proper#(s(X)) -> proper#(X)
     proper#(U42(X1,X2,X3)) -> proper#(X1)
     proper#(U42(X1,X2,X3)) -> proper#(X2)
     proper#(U42(X1,X2,X3)) -> proper#(X3)
     proper#(U41(X1,X2,X3)) -> proper#(X1)
     proper#(U41(X1,X2,X3)) -> proper#(X2)
     proper#(U41(X1,X2,X3)) -> proper#(X3)
     proper#(U31(X1,X2)) -> proper#(X1)
     proper#(U31(X1,X2)) -> proper#(X2)
     proper#(U21(X)) -> proper#(X)
     proper#(isNat(X)) -> proper#(X)
     proper#(U12(X)) -> proper#(X)
     proper#(U11(X1,X2)) -> proper#(X1)
     proper#(U11(X1,X2)) -> proper#(X2)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [proper#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [0] = 0,
      
      [s](x0) = x0 + 1,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [U42](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U41](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U31](x0, x1) = x0 + x1 + 1,
      
      [U21](x0) = x0 + 1,
      
      [mark](x0) = 1,
      
      [U12](x0) = x0 + 1,
      
      [isNat](x0) = x0 + 1,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x0 + x1 + 1,
      
      [tt] = 0
     orientation:
      proper#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1)
      
      proper#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2)
      
      proper#(s(X)) = X + 2 >= X + 1 = proper#(X)
      
      proper#(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1)
      
      proper#(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2)
      
      proper#(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3)
      
      proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1)
      
      proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2)
      
      proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3)
      
      proper#(U31(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1)
      
      proper#(U31(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2)
      
      proper#(U21(X)) = X + 2 >= X + 1 = proper#(X)
      
      proper#(isNat(X)) = X + 2 >= X + 1 = proper#(X)
      
      proper#(U12(X)) = X + 2 >= X + 1 = proper#(X)
      
      proper#(U11(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1)
      
      proper#(U11(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2)
      
      active(U11(tt(),V2)) = V2 + 1 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = N + 1 >= 1 = mark(N)
      
      active(U41(tt(),M,N)) = M + N + 1 >= 1 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M + N + 1 >= 1 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V1 + V2 + 2 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 + 2 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 1 >= 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 2 >= 1 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
      
      active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X2 + 2 >= 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = 2 >= 1 = mark(U12(X))
      
      U21(mark(X)) = 2 >= 1 = mark(U21(X))
      
      U31(mark(X1),X2) = X2 + 2 >= 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 2 >= 1 = mark(s(X))
      
      plus(mark(X1),X2) = X2 + 2 >= 1 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + 2 >= 1 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 0 >= 0 = ok(tt())
      
      proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X))
      
      proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 0 >= 0 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X))
      
      isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     active#(plus(X1,X2)) -> active#(X2)
     active#(plus(X1,X2)) -> active#(X1)
     active#(s(X)) -> active#(X)
     active#(U42(X1,X2,X3)) -> active#(X1)
     active#(U41(X1,X2,X3)) -> active#(X1)
     active#(U31(X1,X2)) -> active#(X1)
     active#(U21(X)) -> active#(X)
     active#(U12(X)) -> active#(X)
     active#(U11(X1,X2)) -> active#(X1)
    TRS:
     active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
     active(U12(tt())) -> mark(tt())
     active(U21(tt())) -> mark(tt())
     active(U31(tt(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [active#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = 1,
      
      [proper](x0) = x0,
      
      [0] = 1,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [U42](x0, x1, x2) = x0 + x2,
      
      [U41](x0, x1, x2) = x0,
      
      [U31](x0, x1) = x0,
      
      [U21](x0) = x0,
      
      [mark](x0) = 1,
      
      [U12](x0) = x0,
      
      [isNat](x0) = 1,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x0 + 1,
      
      [tt] = 1
     orientation:
      active#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2)
      
      active#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1)
      
      active#(s(X)) = X + 1 >= X + 1 = active#(X)
      
      active#(U42(X1,X2,X3)) = X1 + X3 + 1 >= X1 + 1 = active#(X1)
      
      active#(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(X1)
      
      active#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1)
      
      active#(U21(X)) = X + 1 >= X + 1 = active#(X)
      
      active#(U12(X)) = X + 1 >= X + 1 = active#(X)
      
      active#(U11(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1)
      
      active(U11(tt(),V2)) = 2 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = 1 >= 1 = mark(N)
      
      active(U41(tt(),M,N)) = 1 >= 1 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = N + 1 >= 1 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 2 >= 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 1 >= 1 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X >= X = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X3 >= X1 + X3 = U42(active(X1),X2,X3)
      
      active(s(X)) = X >= X = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = 2 >= 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = 1 >= 1 = mark(U12(X))
      
      U21(mark(X)) = 1 >= 1 = mark(U21(X))
      
      U31(mark(X1),X2) = 1 >= 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = 1 >= 1 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X3 + 1 >= 1 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 1 >= 1 = mark(s(X))
      
      plus(mark(X1),X2) = X2 + 2 >= 1 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + 2 >= 1 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 1 >= 1 = ok(tt())
      
      proper(U12(X)) = X >= X = U12(proper(X))
      
      proper(isNat(X)) = 1 >= 1 = isNat(proper(X))
      
      proper(U21(X)) = X >= X = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 >= X1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 >= X1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X1 + X3 >= X1 + X3 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X >= X = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 1 >= 1 = ok(0())
      
      U11(ok(X1),ok(X2)) = 2 >= 1 = ok(U11(X1,X2))
      
      U12(ok(X)) = 1 >= 1 = ok(U12(X))
      
      isNat(ok(X)) = 1 >= 1 = ok(isNat(X))
      
      U21(ok(X)) = 1 >= 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = 1 >= 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = 1 >= 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = 2 >= 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = 1 >= 1 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = 3 >= 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       active#(s(X)) -> active#(X)
       active#(U42(X1,X2,X3)) -> active#(X1)
       active#(U41(X1,X2,X3)) -> active#(X1)
       active#(U31(X1,X2)) -> active#(X1)
       active#(U21(X)) -> active#(X)
       active#(U12(X)) -> active#(X)
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [active#](x0) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = x0,
       
       [proper](x0) = x0,
       
       [0] = 1,
       
       [s](x0) = x0 + 1,
       
       [plus](x0, x1) = x1 + 1,
       
       [U42](x0, x1, x2) = x0,
       
       [U41](x0, x1, x2) = x0 + 1,
       
       [U31](x0, x1) = x0 + 1,
       
       [U21](x0) = x0,
       
       [mark](x0) = 0,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 1,
       
       [active](x0) = x0,
       
       [U11](x0, x1) = 0,
       
       [tt] = 0
      orientation:
       active#(s(X)) = X + 2 >= X + 1 = active#(X)
       
       active#(U42(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(X1)
       
       active#(U41(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1)
       
       active#(U31(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1)
       
       active#(U21(X)) = X + 1 >= X + 1 = active#(X)
       
       active#(U12(X)) = X + 1 >= X + 1 = active#(X)
       
       active(U11(tt(),V2)) = 0 >= 0 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 0 >= 0 = mark(tt())
       
       active(U21(tt())) = 0 >= 0 = mark(tt())
       
       active(U31(tt(),N)) = 1 >= 0 = mark(N)
       
       active(U41(tt(),M,N)) = 1 >= 0 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = 0 >= 0 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 0 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 0 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 0 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = 2 >= 0 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + 2 >= 0 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2)
       
       active(U12(X)) = X >= X = U12(active(X))
       
       active(U21(X)) = X >= X = U21(active(X))
       
       active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 >= X1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2))
       
       U12(mark(X)) = 0 >= 0 = mark(U12(X))
       
       U21(mark(X)) = 0 >= 0 = mark(U21(X))
       
       U31(mark(X1),X2) = 1 >= 0 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = 1 >= 0 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = 0 >= 0 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = 1 >= 0 = mark(s(X))
       
       plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = 1 >= 0 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = X >= X = U12(proper(X))
       
       proper(isNat(X)) = 1 >= 1 = isNat(proper(X))
       
       proper(U21(X)) = X >= X = U21(proper(X))
       
       proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = X1 >= X1 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
       
       proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2))
       
       proper(0()) = 1 >= 1 = ok(0())
       
       U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2))
       
       U12(ok(X)) = X >= X = ok(U12(X))
       
       isNat(ok(X)) = 1 >= 1 = ok(isNat(X))
       
       U21(ok(X)) = X >= X = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        active#(U42(X1,X2,X3)) -> active#(X1)
        active#(U21(X)) -> active#(X)
        active#(U12(X)) -> active#(X)
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [active#](x0) = x0 + 1,
        
        [top](x0) = 0,
        
        [ok](x0) = 0,
        
        [proper](x0) = x0,
        
        [0] = 1,
        
        [s](x0) = 1,
        
        [plus](x0, x1) = x0 + x1,
        
        [U42](x0, x1, x2) = x0 + 1,
        
        [U41](x0, x1, x2) = x0,
        
        [U31](x0, x1) = x0,
        
        [U21](x0) = x0 + 1,
        
        [mark](x0) = 1,
        
        [U12](x0) = x0,
        
        [isNat](x0) = x0 + 1,
        
        [active](x0) = x0,
        
        [U11](x0, x1) = 1,
        
        [tt] = 1
       orientation:
        active#(U42(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1)
        
        active#(U21(X)) = X + 2 >= X + 1 = active#(X)
        
        active#(U12(X)) = X + 1 >= X + 1 = active#(X)
        
        active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
        
        active(U12(tt())) = 1 >= 1 = mark(tt())
        
        active(U21(tt())) = 2 >= 1 = mark(tt())
        
        active(U31(tt(),N)) = 1 >= 1 = mark(N)
        
        active(U41(tt(),M,N)) = 1 >= 1 = mark(U42(isNat(N),M,N))
        
        active(U42(tt(),M,N)) = 2 >= 1 = mark(s(plus(N,M)))
        
        active(isNat(0())) = 2 >= 1 = mark(tt())
        
        active(isNat(plus(V1,V2))) = V1 + V2 + 1 >= 1 = mark(U11(isNat(V1),V2))
        
        active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNat(V1)))
        
        active(plus(N,0())) = N + 1 >= 1 = mark(U31(isNat(N),N))
        
        active(plus(N,s(M))) = N + 1 >= 1 = mark(U41(isNat(M),M,N))
        
        active(U11(X1,X2)) = 1 >= 1 = U11(active(X1),X2)
        
        active(U12(X)) = X >= X = U12(active(X))
        
        active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
        
        active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2)
        
        active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3)
        
        active(U42(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2,X3)
        
        active(s(X)) = 1 >= 1 = s(active(X))
        
        active(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(active(X1),X2)
        
        active(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(X1,active(X2))
        
        U11(mark(X1),X2) = 1 >= 1 = mark(U11(X1,X2))
        
        U12(mark(X)) = 1 >= 1 = mark(U12(X))
        
        U21(mark(X)) = 2 >= 1 = mark(U21(X))
        
        U31(mark(X1),X2) = 1 >= 1 = mark(U31(X1,X2))
        
        U41(mark(X1),X2,X3) = 1 >= 1 = mark(U41(X1,X2,X3))
        
        U42(mark(X1),X2,X3) = 2 >= 1 = mark(U42(X1,X2,X3))
        
        s(mark(X)) = 1 >= 1 = mark(s(X))
        
        plus(mark(X1),X2) = X2 + 1 >= 1 = mark(plus(X1,X2))
        
        plus(X1,mark(X2)) = X1 + 1 >= 1 = mark(plus(X1,X2))
        
        proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2))
        
        proper(tt()) = 1 >= 0 = ok(tt())
        
        proper(U12(X)) = X >= X = U12(proper(X))
        
        proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
        
        proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
        
        proper(U31(X1,X2)) = X1 >= X1 = U31(proper(X1),proper(X2))
        
        proper(U41(X1,X2,X3)) = X1 >= X1 = U41(proper(X1),proper(X2),proper(X3))
        
        proper(U42(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2),proper(X3))
        
        proper(s(X)) = 1 >= 1 = s(proper(X))
        
        proper(plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(proper(X1),proper(X2))
        
        proper(0()) = 1 >= 0 = ok(0())
        
        U11(ok(X1),ok(X2)) = 1 >= 0 = ok(U11(X1,X2))
        
        U12(ok(X)) = 0 >= 0 = ok(U12(X))
        
        isNat(ok(X)) = 1 >= 0 = ok(isNat(X))
        
        U21(ok(X)) = 1 >= 0 = ok(U21(X))
        
        U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2))
        
        U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3))
        
        U42(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U42(X1,X2,X3))
        
        s(ok(X)) = 1 >= 0 = ok(s(X))
        
        plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2))
        
        top(mark(X)) = 0 >= 0 = top(proper(X))
        
        top(ok(X)) = 0 >= 0 = top(active(X))
       problem:
        DPs:
         active#(U12(X)) -> active#(X)
        TRS:
         active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
         active(U12(tt())) -> mark(tt())
         active(U21(tt())) -> mark(tt())
         active(U31(tt(),N)) -> mark(N)
         active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
         active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
         active(isNat(0())) -> mark(tt())
         active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
         active(isNat(s(V1))) -> mark(U21(isNat(V1)))
         active(plus(N,0())) -> mark(U31(isNat(N),N))
         active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
         active(U42(X1,X2,X3)) -> U42(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))
         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))
         U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
         U42(mark(X1),X2,X3) -> mark(U42(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))
         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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
         proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
         proper(s(X)) -> s(proper(X))
         proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
         proper(0()) -> ok(0())
         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))
         U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
         U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
         s(ok(X)) -> ok(s(X))
         plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
         top(mark(X)) -> top(proper(X))
         top(ok(X)) -> top(active(X))
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [active#](x0) = x0 + 1,
         
         [top](x0) = 0,
         
         [ok](x0) = x0,
         
         [proper](x0) = x0,
         
         [0] = 0,
         
         [s](x0) = 1,
         
         [plus](x0, x1) = 1,
         
         [U42](x0, x1, x2) = 1,
         
         [U41](x0, x1, x2) = 1,
         
         [U31](x0, x1) = 1,
         
         [U21](x0) = 1,
         
         [mark](x0) = 1,
         
         [U12](x0) = x0 + 1,
         
         [isNat](x0) = 1,
         
         [active](x0) = x0,
         
         [U11](x0, x1) = 1,
         
         [tt] = 0
        orientation:
         active#(U12(X)) = X + 2 >= X + 1 = active#(X)
         
         active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
         
         active(U12(tt())) = 1 >= 1 = mark(tt())
         
         active(U21(tt())) = 1 >= 1 = mark(tt())
         
         active(U31(tt(),N)) = 1 >= 1 = mark(N)
         
         active(U41(tt(),M,N)) = 1 >= 1 = mark(U42(isNat(N),M,N))
         
         active(U42(tt(),M,N)) = 1 >= 1 = mark(s(plus(N,M)))
         
         active(isNat(0())) = 1 >= 1 = mark(tt())
         
         active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
         
         active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
         
         active(plus(N,0())) = 1 >= 1 = mark(U31(isNat(N),N))
         
         active(plus(N,s(M))) = 1 >= 1 = mark(U41(isNat(M),M,N))
         
         active(U11(X1,X2)) = 1 >= 1 = U11(active(X1),X2)
         
         active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
         
         active(U21(X)) = 1 >= 1 = U21(active(X))
         
         active(U31(X1,X2)) = 1 >= 1 = U31(active(X1),X2)
         
         active(U41(X1,X2,X3)) = 1 >= 1 = U41(active(X1),X2,X3)
         
         active(U42(X1,X2,X3)) = 1 >= 1 = U42(active(X1),X2,X3)
         
         active(s(X)) = 1 >= 1 = s(active(X))
         
         active(plus(X1,X2)) = 1 >= 1 = plus(active(X1),X2)
         
         active(plus(X1,X2)) = 1 >= 1 = plus(X1,active(X2))
         
         U11(mark(X1),X2) = 1 >= 1 = mark(U11(X1,X2))
         
         U12(mark(X)) = 2 >= 1 = mark(U12(X))
         
         U21(mark(X)) = 1 >= 1 = mark(U21(X))
         
         U31(mark(X1),X2) = 1 >= 1 = mark(U31(X1,X2))
         
         U41(mark(X1),X2,X3) = 1 >= 1 = mark(U41(X1,X2,X3))
         
         U42(mark(X1),X2,X3) = 1 >= 1 = mark(U42(X1,X2,X3))
         
         s(mark(X)) = 1 >= 1 = mark(s(X))
         
         plus(mark(X1),X2) = 1 >= 1 = mark(plus(X1,X2))
         
         plus(X1,mark(X2)) = 1 >= 1 = mark(plus(X1,X2))
         
         proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2))
         
         proper(tt()) = 0 >= 0 = ok(tt())
         
         proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X))
         
         proper(isNat(X)) = 1 >= 1 = isNat(proper(X))
         
         proper(U21(X)) = 1 >= 1 = U21(proper(X))
         
         proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2))
         
         proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3))
         
         proper(U42(X1,X2,X3)) = 1 >= 1 = U42(proper(X1),proper(X2),proper(X3))
         
         proper(s(X)) = 1 >= 1 = s(proper(X))
         
         proper(plus(X1,X2)) = 1 >= 1 = plus(proper(X1),proper(X2))
         
         proper(0()) = 0 >= 0 = ok(0())
         
         U11(ok(X1),ok(X2)) = 1 >= 1 = ok(U11(X1,X2))
         
         U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X))
         
         isNat(ok(X)) = 1 >= 1 = ok(isNat(X))
         
         U21(ok(X)) = 1 >= 1 = ok(U21(X))
         
         U31(ok(X1),ok(X2)) = 1 >= 1 = ok(U31(X1,X2))
         
         U41(ok(X1),ok(X2),ok(X3)) = 1 >= 1 = ok(U41(X1,X2,X3))
         
         U42(ok(X1),ok(X2),ok(X3)) = 1 >= 1 = ok(U42(X1,X2,X3))
         
         s(ok(X)) = 1 >= 1 = ok(s(X))
         
         plus(ok(X1),ok(X2)) = 1 >= 1 = ok(plus(X1,X2))
         
         top(mark(X)) = 0 >= 0 = top(proper(X))
         
         top(ok(X)) = 0 >= 0 = top(active(X))
        problem:
         DPs:
          
         TRS:
          active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
          active(U12(tt())) -> mark(tt())
          active(U21(tt())) -> mark(tt())
          active(U31(tt(),N)) -> mark(N)
          active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
          active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
          active(isNat(0())) -> mark(tt())
          active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
          active(isNat(s(V1))) -> mark(U21(isNat(V1)))
          active(plus(N,0())) -> mark(U31(isNat(N),N))
          active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
          active(U42(X1,X2,X3)) -> U42(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))
          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))
          U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
          U42(mark(X1),X2,X3) -> mark(U42(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))
          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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
          proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
          proper(s(X)) -> s(proper(X))
          proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
          proper(0()) -> ok(0())
          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))
          U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
          U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
          s(ok(X)) -> ok(s(X))
          plus(ok(X1),ok(X2)) -> ok(plus(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [isNat#](x0) = x0,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0 + 1,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x1,
      
      [U42](x0, x1, x2) = x1,
      
      [U41](x0, x1, x2) = x2 + 1,
      
      [U31](x0, x1) = x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = 0,
      
      [U12](x0) = x0 + 1,
      
      [isNat](x0) = x0,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x1,
      
      [tt] = 0
     orientation:
      isNat#(ok(X)) = X + 1 >= X = isNat#(X)
      
      active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 0 = mark(tt())
      
      active(U21(tt())) = 0 >= 0 = mark(tt())
      
      active(U31(tt(),N)) = N >= 0 = mark(N)
      
      active(U41(tt(),M,N)) = N + 1 >= 0 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M >= 0 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 0 >= 0 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = 0 >= 0 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M >= 0 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X2 >= X2 = U42(active(X1),X2,X3)
      
      active(s(X)) = X >= X = s(active(X))
      
      active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
      
      U12(mark(X)) = 1 >= 0 = mark(U12(X))
      
      U21(mark(X)) = 0 >= 0 = mark(U21(X))
      
      U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X2 >= 0 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 0 >= 0 = mark(s(X))
      
      plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 1 >= 1 = ok(tt())
      
      proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
      
      proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
      
      proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 1 >= 1 = ok(0())
      
      U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
      
      isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       
      TRS:
       active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
       active(U12(tt())) -> mark(tt())
       active(U21(tt())) -> mark(tt())
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     U11#(ok(X1),ok(X2)) -> U11#(X1,X2)
     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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U11#](x0, x1) = x1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0 + 1,
      
      [0] = 0,
      
      [s](x0) = x0 + 1,
      
      [plus](x0, x1) = x0,
      
      [U42](x0, x1, x2) = x2,
      
      [U41](x0, x1, x2) = x0,
      
      [U31](x0, x1) = x0,
      
      [U21](x0) = x0,
      
      [mark](x0) = 0,
      
      [U12](x0) = x0 + 1,
      
      [isNat](x0) = x0,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x0 + 1,
      
      [tt] = 1
     orientation:
      U11#(ok(X1),ok(X2)) = X2 + 1 >= X2 = U11#(X1,X2)
      
      U11#(mark(X1),X2) = X2 >= X2 = U11#(X1,X2)
      
      active(U11(tt(),V2)) = 2 >= 0 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 2 >= 0 = mark(tt())
      
      active(U21(tt())) = 1 >= 0 = mark(tt())
      
      active(U31(tt(),N)) = 1 >= 0 = mark(N)
      
      active(U41(tt(),M,N)) = 1 >= 0 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = N >= 0 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 0 >= 0 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V1 >= 0 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N >= 0 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = N >= 0 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X3 >= X3 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 >= X1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 >= X1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = 1 >= 0 = mark(U11(X1,X2))
      
      U12(mark(X)) = 1 >= 0 = mark(U12(X))
      
      U21(mark(X)) = 0 >= 0 = mark(U21(X))
      
      U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X3 >= 0 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 1 >= 0 = mark(s(X))
      
      plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 >= 0 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 + 2 >= X1 + 2 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 2 >= 2 = ok(tt())
      
      proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
      
      proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 2 >= X + 2 = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 1 >= 1 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
      
      isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 2 >= X + 2 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U11#](x0, x1) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = 0,
       
       [proper](x0) = 0,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x0 + x1,
       
       [U42](x0, x1, x2) = x0 + x1 + x2,
       
       [U41](x0, x1, x2) = x0 + x1 + x2,
       
       [U31](x0, x1) = x0 + x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = x0 + 1,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 0,
       
       [active](x0) = x0 + 1,
       
       [U11](x0, x1) = x0,
       
       [tt] = 0
      orientation:
       U11#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U11#(X1,X2)
       
       active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 1 = mark(tt())
       
       active(U21(tt())) = 1 >= 1 = mark(tt())
       
       active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
       
       active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 1 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
       
       active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
       
       U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
       
       U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
       
       U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
       
       plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = 0 >= 0 = U12(proper(X))
       
       proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
       
       proper(U21(X)) = 0 >= 0 = U21(proper(X))
       
       proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 0 >= 0 = s(proper(X))
       
       proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
       
       proper(0()) = 0 >= 0 = ok(0())
       
       U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2))
       
       U12(ok(X)) = 0 >= 0 = ok(U12(X))
       
       isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
       
       U21(ok(X)) = 0 >= 0 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = 0 >= 0 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U12#(ok(X)) -> U12#(X)
     U12#(mark(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U12#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [U42](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U41](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U31](x0, x1) = x0 + x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = x0 + 1,
      
      [U12](x0) = x0,
      
      [isNat](x0) = 0,
      
      [active](x0) = x0 + 1,
      
      [U11](x0, x1) = x0,
      
      [tt] = 0
     orientation:
      U12#(ok(X)) = X + 1 >= X + 1 = U12#(X)
      
      U12#(mark(X)) = X + 2 >= X + 1 = U12#(X)
      
      active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
      
      active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 2 >= N + 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
      
      active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
      
      U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
      
      U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
      
      plus(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 >= X1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 0 >= 0 = ok(tt())
      
      proper(U12(X)) = X >= X = U12(proper(X))
      
      proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
      
      proper(U21(X)) = X >= X = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X >= X = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 0 >= 0 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X >= X = ok(U12(X))
      
      isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
      
      U21(ok(X)) = X >= X = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X >= X = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U12#](x0) = x0,
       
       [top](x0) = 0,
       
       [ok](x0) = x0 + 1,
       
       [proper](x0) = x0 + 1,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x1,
       
       [U42](x0, x1, x2) = x1,
       
       [U41](x0, x1, x2) = x2 + 1,
       
       [U31](x0, x1) = x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = 0,
       
       [U12](x0) = x0 + 1,
       
       [isNat](x0) = x0,
       
       [active](x0) = x0,
       
       [U11](x0, x1) = x1,
       
       [tt] = 0
      orientation:
       U12#(ok(X)) = X + 1 >= X = U12#(X)
       
       active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 0 = mark(tt())
       
       active(U21(tt())) = 0 >= 0 = mark(tt())
       
       active(U31(tt(),N)) = N >= 0 = mark(N)
       
       active(U41(tt(),M,N)) = N + 1 >= 0 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M >= 0 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 0 >= 0 = mark(tt())
       
       active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = 0 >= 0 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M >= 0 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X >= X = U21(active(X))
       
       active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X2 >= X2 = U42(active(X1),X2,X3)
       
       active(s(X)) = X >= X = s(active(X))
       
       active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
       
       U12(mark(X)) = 1 >= 0 = mark(U12(X))
       
       U21(mark(X)) = 0 >= 0 = mark(U21(X))
       
       U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X2 >= 0 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = 0 >= 0 = mark(s(X))
       
       plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 1 >= 1 = ok(tt())
       
       proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
       
       proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
       
       proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
       
       proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
       
       proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2))
       
       proper(0()) = 1 >= 1 = ok(0())
       
       U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
       
       U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
       
       isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
       
       U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U21#(ok(X)) -> U21#(X)
     U21#(mark(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U21#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [U42](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U41](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U31](x0, x1) = x0 + x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = x0 + 1,
      
      [U12](x0) = x0,
      
      [isNat](x0) = 0,
      
      [active](x0) = x0 + 1,
      
      [U11](x0, x1) = x0,
      
      [tt] = 0
     orientation:
      U21#(ok(X)) = X + 1 >= X + 1 = U21#(X)
      
      U21#(mark(X)) = X + 2 >= X + 1 = U21#(X)
      
      active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
      
      active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 2 >= N + 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
      
      active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
      
      U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
      
      U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
      
      plus(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 >= X1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 0 >= 0 = ok(tt())
      
      proper(U12(X)) = X >= X = U12(proper(X))
      
      proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
      
      proper(U21(X)) = X >= X = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X >= X = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 0 >= 0 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X >= X = ok(U12(X))
      
      isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
      
      U21(ok(X)) = X >= X = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X >= X = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U21#](x0) = x0,
       
       [top](x0) = 0,
       
       [ok](x0) = x0 + 1,
       
       [proper](x0) = x0 + 1,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x1,
       
       [U42](x0, x1, x2) = x1,
       
       [U41](x0, x1, x2) = x2 + 1,
       
       [U31](x0, x1) = x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = 0,
       
       [U12](x0) = x0 + 1,
       
       [isNat](x0) = x0,
       
       [active](x0) = x0,
       
       [U11](x0, x1) = x1,
       
       [tt] = 0
      orientation:
       U21#(ok(X)) = X + 1 >= X = U21#(X)
       
       active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 0 = mark(tt())
       
       active(U21(tt())) = 0 >= 0 = mark(tt())
       
       active(U31(tt(),N)) = N >= 0 = mark(N)
       
       active(U41(tt(),M,N)) = N + 1 >= 0 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M >= 0 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 0 >= 0 = mark(tt())
       
       active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = 0 >= 0 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M >= 0 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X >= X = U21(active(X))
       
       active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X2 >= X2 = U42(active(X1),X2,X3)
       
       active(s(X)) = X >= X = s(active(X))
       
       active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
       
       U12(mark(X)) = 1 >= 0 = mark(U12(X))
       
       U21(mark(X)) = 0 >= 0 = mark(U21(X))
       
       U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X2 >= 0 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = 0 >= 0 = mark(s(X))
       
       plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 1 >= 1 = ok(tt())
       
       proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
       
       proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
       
       proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
       
       proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
       
       proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2))
       
       proper(0()) = 1 >= 1 = ok(0())
       
       U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
       
       U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
       
       isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
       
       U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
     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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U31#](x0, x1) = x1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0 + 1,
      
      [0] = 0,
      
      [s](x0) = x0 + 1,
      
      [plus](x0, x1) = x0,
      
      [U42](x0, x1, x2) = x2,
      
      [U41](x0, x1, x2) = x0,
      
      [U31](x0, x1) = x0,
      
      [U21](x0) = x0,
      
      [mark](x0) = 0,
      
      [U12](x0) = x0 + 1,
      
      [isNat](x0) = x0,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x0 + 1,
      
      [tt] = 1
     orientation:
      U31#(ok(X1),ok(X2)) = X2 + 1 >= X2 = U31#(X1,X2)
      
      U31#(mark(X1),X2) = X2 >= X2 = U31#(X1,X2)
      
      active(U11(tt(),V2)) = 2 >= 0 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 2 >= 0 = mark(tt())
      
      active(U21(tt())) = 1 >= 0 = mark(tt())
      
      active(U31(tt(),N)) = 1 >= 0 = mark(N)
      
      active(U41(tt(),M,N)) = 1 >= 0 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = N >= 0 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 0 >= 0 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V1 >= 0 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N >= 0 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = N >= 0 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X3 >= X3 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 >= X1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 >= X1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = 1 >= 0 = mark(U11(X1,X2))
      
      U12(mark(X)) = 1 >= 0 = mark(U12(X))
      
      U21(mark(X)) = 0 >= 0 = mark(U21(X))
      
      U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X3 >= 0 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 1 >= 0 = mark(s(X))
      
      plus(mark(X1),X2) = 0 >= 0 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 >= 0 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 + 2 >= X1 + 2 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 2 >= 2 = ok(tt())
      
      proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
      
      proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 2 >= X + 2 = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + 1 >= X1 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 1 >= 1 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
      
      isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 2 >= X + 2 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U31#](x0, x1) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = 0,
       
       [proper](x0) = 0,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x0 + x1,
       
       [U42](x0, x1, x2) = x0 + x1 + x2,
       
       [U41](x0, x1, x2) = x0 + x1 + x2,
       
       [U31](x0, x1) = x0 + x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = x0 + 1,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 0,
       
       [active](x0) = x0 + 1,
       
       [U11](x0, x1) = x0,
       
       [tt] = 0
      orientation:
       U31#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U31#(X1,X2)
       
       active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 1 = mark(tt())
       
       active(U21(tt())) = 1 >= 1 = mark(tt())
       
       active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
       
       active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 1 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
       
       active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
       
       U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
       
       U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
       
       U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
       
       plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = 0 >= 0 = U12(proper(X))
       
       proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
       
       proper(U21(X)) = 0 >= 0 = U21(proper(X))
       
       proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 0 >= 0 = s(proper(X))
       
       proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
       
       proper(0()) = 0 >= 0 = ok(0())
       
       U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2))
       
       U12(ok(X)) = 0 >= 0 = ok(U12(X))
       
       isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
       
       U21(ok(X)) = 0 >= 0 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = 0 >= 0 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
     U41#(mark(X1),X2,X3) -> U41#(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U41#](x0, x1, x2) = x1 + x2,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0 + 1,
      
      [0] = 1,
      
      [s](x0) = x0 + 1,
      
      [plus](x0, x1) = x1 + 1,
      
      [U42](x0, x1, x2) = x2 + 1,
      
      [U41](x0, x1, x2) = x0 + 1,
      
      [U31](x0, x1) = x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = 0,
      
      [U12](x0) = x0,
      
      [isNat](x0) = x0 + 1,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x1,
      
      [tt] = 0
     orientation:
      U41#(ok(X1),ok(X2),ok(X3)) = X2 + X3 + 2 >= X2 + X3 = U41#(X1,X2,X3)
      
      U41#(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41#(X1,X2,X3)
      
      active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 0 >= 0 = mark(tt())
      
      active(U21(tt())) = 0 >= 0 = mark(tt())
      
      active(U31(tt(),N)) = N >= 0 = mark(N)
      
      active(U41(tt(),M,N)) = 1 >= 0 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = N + 1 >= 0 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 2 >= 0 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V2 + 2 >= 0 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 + 2 >= 0 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = 2 >= 0 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + 2 >= 0 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
      
      active(U12(X)) = X >= X = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
      
      U12(mark(X)) = 0 >= 0 = mark(U12(X))
      
      U21(mark(X)) = 0 >= 0 = mark(U21(X))
      
      U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = 1 >= 0 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 1 >= 0 = mark(s(X))
      
      plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = 1 >= 0 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 1 >= 1 = ok(tt())
      
      proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X))
      
      proper(isNat(X)) = X + 2 >= X + 2 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 2 >= X + 2 = s(proper(X))
      
      proper(plus(X1,X2)) = X2 + 2 >= X2 + 2 = plus(proper(X1),proper(X2))
      
      proper(0()) = 2 >= 2 = ok(0())
      
      U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X))
      
      isNat(ok(X)) = X + 2 >= X + 2 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 2 >= X + 2 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       U41#(mark(X1),X2,X3) -> U41#(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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U41#](x0, x1, x2) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = 0,
       
       [proper](x0) = 0,
       
       [0] = 1,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x0 + x1,
       
       [U42](x0, x1, x2) = x0 + x1 + x2,
       
       [U41](x0, x1, x2) = x0 + x1 + x2,
       
       [U31](x0, x1) = x0 + x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = x0 + 1,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 0,
       
       [active](x0) = x0 + 1,
       
       [U11](x0, x1) = x0,
       
       [tt] = 0
      orientation:
       U41#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U41#(X1,X2,X3)
       
       active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 1 = mark(tt())
       
       active(U21(tt())) = 1 >= 1 = mark(tt())
       
       active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
       
       active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 1 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = N + 2 >= N + 1 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
       
       active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
       
       U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
       
       U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
       
       U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
       
       plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = 0 >= 0 = U12(proper(X))
       
       proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
       
       proper(U21(X)) = 0 >= 0 = U21(proper(X))
       
       proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 0 >= 0 = s(proper(X))
       
       proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
       
       proper(0()) = 0 >= 0 = ok(0())
       
       U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2))
       
       U12(ok(X)) = 0 >= 0 = ok(U12(X))
       
       isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
       
       U21(ok(X)) = 0 >= 0 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = 0 >= 0 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     U42#(ok(X1),ok(X2),ok(X3)) -> U42#(X1,X2,X3)
     U42#(mark(X1),X2,X3) -> U42#(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [U42#](x0, x1, x2) = x1 + x2,
      
      [top](x0) = 0,
      
      [ok](x0) = x0 + 1,
      
      [proper](x0) = x0 + 1,
      
      [0] = 1,
      
      [s](x0) = x0 + 1,
      
      [plus](x0, x1) = x1 + 1,
      
      [U42](x0, x1, x2) = x2 + 1,
      
      [U41](x0, x1, x2) = x0 + 1,
      
      [U31](x0, x1) = x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = 0,
      
      [U12](x0) = x0,
      
      [isNat](x0) = x0 + 1,
      
      [active](x0) = x0,
      
      [U11](x0, x1) = x1,
      
      [tt] = 0
     orientation:
      U42#(ok(X1),ok(X2),ok(X3)) = X2 + X3 + 2 >= X2 + X3 = U42#(X1,X2,X3)
      
      U42#(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U42#(X1,X2,X3)
      
      active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 0 >= 0 = mark(tt())
      
      active(U21(tt())) = 0 >= 0 = mark(tt())
      
      active(U31(tt(),N)) = N >= 0 = mark(N)
      
      active(U41(tt(),M,N)) = 1 >= 0 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = N + 1 >= 0 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 2 >= 0 = mark(tt())
      
      active(isNat(plus(V1,V2))) = V2 + 2 >= 0 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = V1 + 2 >= 0 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = 2 >= 0 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + 2 >= 0 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
      
      active(U12(X)) = X >= X = U12(active(X))
      
      active(U21(X)) = X >= X = U21(active(X))
      
      active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
      
      U12(mark(X)) = 0 >= 0 = mark(U12(X))
      
      U21(mark(X)) = 0 >= 0 = mark(U21(X))
      
      U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = 1 >= 0 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = 1 >= 0 = mark(s(X))
      
      plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = 1 >= 0 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 1 >= 1 = ok(tt())
      
      proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X))
      
      proper(isNat(X)) = X + 2 >= X + 2 = isNat(proper(X))
      
      proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
      
      proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X + 2 >= X + 2 = s(proper(X))
      
      proper(plus(X1,X2)) = X2 + 2 >= X2 + 2 = plus(proper(X1),proper(X2))
      
      proper(0()) = 2 >= 2 = ok(0())
      
      U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X))
      
      isNat(ok(X)) = X + 2 >= X + 2 = ok(isNat(X))
      
      U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X + 2 >= X + 2 = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       U42#(mark(X1),X2,X3) -> U42#(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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [U42#](x0, x1, x2) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = 0,
       
       [proper](x0) = 0,
       
       [0] = 1,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x0 + x1,
       
       [U42](x0, x1, x2) = x0 + x1 + x2,
       
       [U41](x0, x1, x2) = x0 + x1 + x2,
       
       [U31](x0, x1) = x0 + x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = x0 + 1,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 0,
       
       [active](x0) = x0 + 1,
       
       [U11](x0, x1) = x0,
       
       [tt] = 0
      orientation:
       U42#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U42#(X1,X2,X3)
       
       active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 1 = mark(tt())
       
       active(U21(tt())) = 1 >= 1 = mark(tt())
       
       active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
       
       active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 1 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = N + 2 >= N + 1 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
       
       active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
       
       U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
       
       U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
       
       U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
       
       plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = 0 >= 0 = U12(proper(X))
       
       proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
       
       proper(U21(X)) = 0 >= 0 = U21(proper(X))
       
       proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 0 >= 0 = s(proper(X))
       
       proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
       
       proper(0()) = 0 >= 0 = ok(0())
       
       U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2))
       
       U12(ok(X)) = 0 >= 0 = ok(U12(X))
       
       isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
       
       U21(ok(X)) = 0 >= 0 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = 0 >= 0 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = 0 >= 0 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     s#(ok(X)) -> s#(X)
     s#(mark(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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [s#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + x1 + 1,
      
      [U42](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U41](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [U31](x0, x1) = x0 + x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = x0 + 1,
      
      [U12](x0) = x0,
      
      [isNat](x0) = 0,
      
      [active](x0) = x0 + 1,
      
      [U11](x0, x1) = x0,
      
      [tt] = 0
     orientation:
      s#(ok(X)) = X + 1 >= X + 1 = s#(X)
      
      s#(mark(X)) = X + 2 >= X + 1 = s#(X)
      
      active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
      
      active(U41(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M + N + 2 >= M + N + 2 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 2 >= N + 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 2 >= M + N + 2 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
      
      active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
      
      U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
      
      U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
      
      plus(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = X1 >= X1 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 0 >= 0 = ok(tt())
      
      proper(U12(X)) = X >= X = U12(proper(X))
      
      proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
      
      proper(U21(X)) = X >= X = U21(proper(X))
      
      proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = X >= X = s(proper(X))
      
      proper(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(proper(X1),proper(X2))
      
      proper(0()) = 0 >= 0 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X >= X = ok(U12(X))
      
      isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
      
      U21(ok(X)) = X >= X = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X >= X = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [s#](x0) = x0,
       
       [top](x0) = 0,
       
       [ok](x0) = x0 + 1,
       
       [proper](x0) = x0 + 1,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x1,
       
       [U42](x0, x1, x2) = x1,
       
       [U41](x0, x1, x2) = x2 + 1,
       
       [U31](x0, x1) = x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = 0,
       
       [U12](x0) = x0 + 1,
       
       [isNat](x0) = x0,
       
       [active](x0) = x0,
       
       [U11](x0, x1) = x1,
       
       [tt] = 0
      orientation:
       s#(ok(X)) = X + 1 >= X = s#(X)
       
       active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 0 = mark(tt())
       
       active(U21(tt())) = 0 >= 0 = mark(tt())
       
       active(U31(tt(),N)) = N >= 0 = mark(N)
       
       active(U41(tt(),M,N)) = N + 1 >= 0 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M >= 0 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 0 >= 0 = mark(tt())
       
       active(isNat(plus(V1,V2))) = V2 >= 0 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = 0 >= 0 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M >= 0 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X >= X = U21(active(X))
       
       active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X2 >= X2 = U42(active(X1),X2,X3)
       
       active(s(X)) = X >= X = s(active(X))
       
       active(plus(X1,X2)) = X2 >= X2 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X2 >= X2 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
       
       U12(mark(X)) = 1 >= 0 = mark(U12(X))
       
       U21(mark(X)) = 0 >= 0 = mark(U21(X))
       
       U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X2 >= 0 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = 0 >= 0 = mark(s(X))
       
       plus(mark(X1),X2) = X2 >= 0 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = 0 >= 0 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 1 >= 1 = ok(tt())
       
       proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X))
       
       proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
       
       proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X))
       
       proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
       
       proper(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(proper(X1),proper(X2))
       
       proper(0()) = 1 >= 1 = ok(0())
       
       U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
       
       U12(ok(X)) = X + 2 >= X + 2 = ok(U12(X))
       
       isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
       
       U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
       
       plus(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
        active(U12(tt())) -> mark(tt())
        active(U21(tt())) -> mark(tt())
        active(U31(tt(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
     plus#(X1,mark(X2)) -> plus#(X1,X2)
     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(),N)) -> mark(N)
     active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
     active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
     active(isNat(0())) -> mark(tt())
     active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
     active(isNat(s(V1))) -> mark(U21(isNat(V1)))
     active(plus(N,0())) -> mark(U31(isNat(N),N))
     active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2,X3)) -> U42(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))
     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))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U42(mark(X1),X2,X3) -> mark(U42(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))
     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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
     proper(s(X)) -> s(proper(X))
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     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))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
     s(ok(X)) -> ok(s(X))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [plus#](x0, x1) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = 0,
      
      [0] = 0,
      
      [s](x0) = x0,
      
      [plus](x0, x1) = x0 + x1,
      
      [U42](x0, x1, x2) = x0 + x1 + x2,
      
      [U41](x0, x1, x2) = x0 + x1 + x2,
      
      [U31](x0, x1) = x0 + x1,
      
      [U21](x0) = x0,
      
      [mark](x0) = x0 + 1,
      
      [U12](x0) = x0,
      
      [isNat](x0) = 0,
      
      [active](x0) = x0 + 1,
      
      [U11](x0, x1) = x0,
      
      [tt] = 0
     orientation:
      plus#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = plus#(X1,X2)
      
      plus#(X1,mark(X2)) = X1 + 1 >= X1 + 1 = plus#(X1,X2)
      
      plus#(mark(X1),X2) = X1 + 2 >= X1 + 1 = plus#(X1,X2)
      
      active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
      
      active(U12(tt())) = 1 >= 1 = mark(tt())
      
      active(U21(tt())) = 1 >= 1 = mark(tt())
      
      active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
      
      active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
      
      active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
      
      active(isNat(0())) = 1 >= 1 = mark(tt())
      
      active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
      
      active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
      
      active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(isNat(N),N))
      
      active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
      
      active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
      
      active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
      
      active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
      
      active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
      
      active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
      
      active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
      
      active(s(X)) = X + 1 >= X + 1 = s(active(X))
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
      
      active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
      
      U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
      
      U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
      
      U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
      
      U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
      
      U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
      
      U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
      
      s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
      
      plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
      
      plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
      
      proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
      
      proper(tt()) = 0 >= 0 = ok(tt())
      
      proper(U12(X)) = 0 >= 0 = U12(proper(X))
      
      proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
      
      proper(U21(X)) = 0 >= 0 = U21(proper(X))
      
      proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
      
      proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
      
      proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
      
      proper(s(X)) = 0 >= 0 = s(proper(X))
      
      proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
      
      proper(0()) = 0 >= 0 = ok(0())
      
      U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2))
      
      U12(ok(X)) = X >= X = ok(U12(X))
      
      isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
      
      U21(ok(X)) = X >= X = ok(U21(X))
      
      U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2))
      
      U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3))
      
      U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U42(X1,X2,X3))
      
      s(ok(X)) = X >= X = ok(s(X))
      
      plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
       plus#(X1,mark(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(),N)) -> mark(N)
       active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
       active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
       active(isNat(0())) -> mark(tt())
       active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
       active(isNat(s(V1))) -> mark(U21(isNat(V1)))
       active(plus(N,0())) -> mark(U31(isNat(N),N))
       active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2,X3)) -> U42(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))
       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))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2,X3) -> mark(U42(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))
       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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
       proper(s(X)) -> s(proper(X))
       proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       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))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
       s(ok(X)) -> ok(s(X))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [plus#](x0, x1) = x1 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = x0,
       
       [proper](x0) = 0,
       
       [0] = 0,
       
       [s](x0) = x0,
       
       [plus](x0, x1) = x0 + x1,
       
       [U42](x0, x1, x2) = x0 + x1 + x2,
       
       [U41](x0, x1, x2) = x0 + x1 + x2,
       
       [U31](x0, x1) = x0 + x1,
       
       [U21](x0) = x0,
       
       [mark](x0) = x0 + 1,
       
       [U12](x0) = x0,
       
       [isNat](x0) = 0,
       
       [active](x0) = x0 + 1,
       
       [U11](x0, x1) = x0,
       
       [tt] = 0
      orientation:
       plus#(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = plus#(X1,X2)
       
       plus#(X1,mark(X2)) = X2 + 2 >= X2 + 1 = plus#(X1,X2)
       
       active(U11(tt(),V2)) = 1 >= 1 = mark(U12(isNat(V2)))
       
       active(U12(tt())) = 1 >= 1 = mark(tt())
       
       active(U21(tt())) = 1 >= 1 = mark(tt())
       
       active(U31(tt(),N)) = N + 1 >= N + 1 = mark(N)
       
       active(U41(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(U42(isNat(N),M,N))
       
       active(U42(tt(),M,N)) = M + N + 1 >= M + N + 1 = mark(s(plus(N,M)))
       
       active(isNat(0())) = 1 >= 1 = mark(tt())
       
       active(isNat(plus(V1,V2))) = 1 >= 1 = mark(U11(isNat(V1),V2))
       
       active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1)))
       
       active(plus(N,0())) = N + 1 >= N + 1 = mark(U31(isNat(N),N))
       
       active(plus(N,s(M))) = M + N + 1 >= M + N + 1 = mark(U41(isNat(M),M,N))
       
       active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2)
       
       active(U12(X)) = X + 1 >= X + 1 = U12(active(X))
       
       active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
       
       active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2)
       
       active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3)
       
       active(U42(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U42(active(X1),X2,X3)
       
       active(s(X)) = X + 1 >= X + 1 = s(active(X))
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(active(X1),X2)
       
       active(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,active(X2))
       
       U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2))
       
       U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X))
       
       U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X))
       
       U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U31(X1,X2))
       
       U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U41(X1,X2,X3))
       
       U42(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U42(X1,X2,X3))
       
       s(mark(X)) = X + 1 >= X + 1 = mark(s(X))
       
       plus(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       plus(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(plus(X1,X2))
       
       proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2))
       
       proper(tt()) = 0 >= 0 = ok(tt())
       
       proper(U12(X)) = 0 >= 0 = U12(proper(X))
       
       proper(isNat(X)) = 0 >= 0 = isNat(proper(X))
       
       proper(U21(X)) = 0 >= 0 = U21(proper(X))
       
       proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2))
       
       proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3))
       
       proper(U42(X1,X2,X3)) = 0 >= 0 = U42(proper(X1),proper(X2),proper(X3))
       
       proper(s(X)) = 0 >= 0 = s(proper(X))
       
       proper(plus(X1,X2)) = 0 >= 0 = plus(proper(X1),proper(X2))
       
       proper(0()) = 0 >= 0 = ok(0())
       
       U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2))
       
       U12(ok(X)) = X >= X = ok(U12(X))
       
       isNat(ok(X)) = 0 >= 0 = ok(isNat(X))
       
       U21(ok(X)) = X >= X = ok(U21(X))
       
       U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2))
       
       U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3))
       
       U42(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U42(X1,X2,X3))
       
       s(ok(X)) = X >= X = ok(s(X))
       
       plus(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(plus(X1,X2))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      problem:
       DPs:
        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(),N)) -> mark(N)
        active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
        active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
        active(isNat(0())) -> mark(tt())
        active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
        active(isNat(s(V1))) -> mark(U21(isNat(V1)))
        active(plus(N,0())) -> mark(U31(isNat(N),N))
        active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
        active(U42(X1,X2,X3)) -> U42(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))
        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))
        U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
        U42(mark(X1),X2,X3) -> mark(U42(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))
        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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
        proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
        proper(s(X)) -> s(proper(X))
        proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
        proper(0()) -> ok(0())
        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))
        U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
        U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
        s(ok(X)) -> ok(s(X))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [plus#](x0, x1) = x1 + 1,
        
        [top](x0) = 0,
        
        [ok](x0) = x0 + 1,
        
        [proper](x0) = x0 + 1,
        
        [0] = 0,
        
        [s](x0) = x0,
        
        [plus](x0, x1) = x1 + 1,
        
        [U42](x0, x1, x2) = x2,
        
        [U41](x0, x1, x2) = x1,
        
        [U31](x0, x1) = x0,
        
        [U21](x0) = x0 + 1,
        
        [mark](x0) = 0,
        
        [U12](x0) = x0,
        
        [isNat](x0) = x0,
        
        [active](x0) = x0,
        
        [U11](x0, x1) = x1,
        
        [tt] = 0
       orientation:
        plus#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = plus#(X1,X2)
        
        active(U11(tt(),V2)) = V2 >= 0 = mark(U12(isNat(V2)))
        
        active(U12(tt())) = 0 >= 0 = mark(tt())
        
        active(U21(tt())) = 1 >= 0 = mark(tt())
        
        active(U31(tt(),N)) = 0 >= 0 = mark(N)
        
        active(U41(tt(),M,N)) = M >= 0 = mark(U42(isNat(N),M,N))
        
        active(U42(tt(),M,N)) = N >= 0 = mark(s(plus(N,M)))
        
        active(isNat(0())) = 0 >= 0 = mark(tt())
        
        active(isNat(plus(V1,V2))) = V2 + 1 >= 0 = mark(U11(isNat(V1),V2))
        
        active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1)))
        
        active(plus(N,0())) = 1 >= 0 = mark(U31(isNat(N),N))
        
        active(plus(N,s(M))) = M + 1 >= 0 = mark(U41(isNat(M),M,N))
        
        active(U11(X1,X2)) = X2 >= X2 = U11(active(X1),X2)
        
        active(U12(X)) = X >= X = U12(active(X))
        
        active(U21(X)) = X + 1 >= X + 1 = U21(active(X))
        
        active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2)
        
        active(U41(X1,X2,X3)) = X2 >= X2 = U41(active(X1),X2,X3)
        
        active(U42(X1,X2,X3)) = X3 >= X3 = U42(active(X1),X2,X3)
        
        active(s(X)) = X >= X = s(active(X))
        
        active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(active(X1),X2)
        
        active(plus(X1,X2)) = X2 + 1 >= X2 + 1 = plus(X1,active(X2))
        
        U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2))
        
        U12(mark(X)) = 0 >= 0 = mark(U12(X))
        
        U21(mark(X)) = 1 >= 0 = mark(U21(X))
        
        U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2))
        
        U41(mark(X1),X2,X3) = X2 >= 0 = mark(U41(X1,X2,X3))
        
        U42(mark(X1),X2,X3) = X3 >= 0 = mark(U42(X1,X2,X3))
        
        s(mark(X)) = 0 >= 0 = mark(s(X))
        
        plus(mark(X1),X2) = X2 + 1 >= 0 = mark(plus(X1,X2))
        
        plus(X1,mark(X2)) = 1 >= 0 = mark(plus(X1,X2))
        
        proper(U11(X1,X2)) = X2 + 1 >= X2 + 1 = U11(proper(X1),proper(X2))
        
        proper(tt()) = 1 >= 1 = ok(tt())
        
        proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X))
        
        proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X))
        
        proper(U21(X)) = X + 2 >= X + 2 = U21(proper(X))
        
        proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2))
        
        proper(U41(X1,X2,X3)) = X2 + 1 >= X2 + 1 = U41(proper(X1),proper(X2),proper(X3))
        
        proper(U42(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U42(proper(X1),proper(X2),proper(X3))
        
        proper(s(X)) = X + 1 >= X + 1 = s(proper(X))
        
        proper(plus(X1,X2)) = X2 + 2 >= X2 + 2 = plus(proper(X1),proper(X2))
        
        proper(0()) = 1 >= 1 = ok(0())
        
        U11(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2))
        
        U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X))
        
        isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X))
        
        U21(ok(X)) = X + 2 >= X + 2 = ok(U21(X))
        
        U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2))
        
        U41(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2,X3))
        
        U42(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U42(X1,X2,X3))
        
        s(ok(X)) = X + 1 >= X + 1 = ok(s(X))
        
        plus(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(plus(X1,X2))
        
        top(mark(X)) = 0 >= 0 = top(proper(X))
        
        top(ok(X)) = 0 >= 0 = top(active(X))
       problem:
        DPs:
         
        TRS:
         active(U11(tt(),V2)) -> mark(U12(isNat(V2)))
         active(U12(tt())) -> mark(tt())
         active(U21(tt())) -> mark(tt())
         active(U31(tt(),N)) -> mark(N)
         active(U41(tt(),M,N)) -> mark(U42(isNat(N),M,N))
         active(U42(tt(),M,N)) -> mark(s(plus(N,M)))
         active(isNat(0())) -> mark(tt())
         active(isNat(plus(V1,V2))) -> mark(U11(isNat(V1),V2))
         active(isNat(s(V1))) -> mark(U21(isNat(V1)))
         active(plus(N,0())) -> mark(U31(isNat(N),N))
         active(plus(N,s(M))) -> mark(U41(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(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
         active(U42(X1,X2,X3)) -> U42(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))
         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))
         U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
         U42(mark(X1),X2,X3) -> mark(U42(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))
         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(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
         proper(U42(X1,X2,X3)) -> U42(proper(X1),proper(X2),proper(X3))
         proper(s(X)) -> s(proper(X))
         proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
         proper(0()) -> ok(0())
         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))
         U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
         U42(ok(X1),ok(X2),ok(X3)) -> ok(U42(X1,X2,X3))
         s(ok(X)) -> ok(s(X))
         plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
         top(mark(X)) -> top(proper(X))
         top(ok(X)) -> top(active(X))
       Qed