MAYBE

Problem:
 active(and(tt(),X)) -> mark(X)
 active(plus(N,0())) -> mark(N)
 active(plus(N,s(M))) -> mark(s(plus(N,M)))
 active(and(X1,X2)) -> and(active(X1),X2)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 and(mark(X1),X2) -> mark(and(X1,X2))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 s(mark(X)) -> mark(s(X))
 proper(and(X1,X2)) -> and(proper(X1),proper(X2))
 proper(tt()) -> ok(tt())
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(s(X)) -> s(proper(X))
 and(ok(X1),ok(X2)) -> ok(and(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(plus(N,s(M))) -> plus#(N,M)
   active#(plus(N,s(M))) -> s#(plus(N,M))
   active#(and(X1,X2)) -> active#(X1)
   active#(and(X1,X2)) -> and#(active(X1),X2)
   active#(plus(X1,X2)) -> active#(X1)
   active#(plus(X1,X2)) -> plus#(active(X1),X2)
   active#(plus(X1,X2)) -> active#(X2)
   active#(plus(X1,X2)) -> plus#(X1,active(X2))
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   and#(mark(X1),X2) -> and#(X1,X2)
   plus#(mark(X1),X2) -> plus#(X1,X2)
   plus#(X1,mark(X2)) -> plus#(X1,X2)
   s#(mark(X)) -> s#(X)
   proper#(and(X1,X2)) -> proper#(X2)
   proper#(and(X1,X2)) -> proper#(X1)
   proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
   proper#(plus(X1,X2)) -> proper#(X2)
   proper#(plus(X1,X2)) -> proper#(X1)
   proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   and#(ok(X1),ok(X2)) -> and#(X1,X2)
   plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
   s#(ok(X)) -> s#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(and(tt(),X)) -> mark(X)
   active(plus(N,0())) -> mark(N)
   active(plus(N,s(M))) -> mark(s(plus(N,M)))
   active(and(X1,X2)) -> and(active(X1),X2)
   active(plus(X1,X2)) -> plus(active(X1),X2)
   active(plus(X1,X2)) -> plus(X1,active(X2))
   active(s(X)) -> s(active(X))
   and(mark(X1),X2) -> mark(and(X1,X2))
   plus(mark(X1),X2) -> mark(plus(X1,X2))
   plus(X1,mark(X2)) -> mark(plus(X1,X2))
   s(mark(X)) -> mark(s(X))
   proper(and(X1,X2)) -> and(proper(X1),proper(X2))
   proper(tt()) -> ok(tt())
   proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(s(X)) -> s(proper(X))
   and(ok(X1),ok(X2)) -> ok(and(X1,X2))
   plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  EDG Processor:
   DPs:
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(and(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    and#(mark(X1),X2) -> and#(X1,X2)
    plus#(mark(X1),X2) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    s#(mark(X)) -> s#(X)
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    s#(ok(X)) -> s#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(and(tt(),X)) -> mark(X)
    active(plus(N,0())) -> mark(N)
    active(plus(N,s(M))) -> mark(s(plus(N,M)))
    active(and(X1,X2)) -> and(active(X1),X2)
    active(plus(X1,X2)) -> plus(active(X1),X2)
    active(plus(X1,X2)) -> plus(X1,active(X2))
    active(s(X)) -> s(active(X))
    and(mark(X1),X2) -> mark(and(X1,X2))
    plus(mark(X1),X2) -> mark(plus(X1,X2))
    plus(X1,mark(X2)) -> mark(plus(X1,X2))
    s(mark(X)) -> mark(s(X))
    proper(and(X1,X2)) -> and(proper(X1),proper(X2))
    proper(tt()) -> ok(tt())
    proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(s(X)) -> s(proper(X))
    and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    s(ok(X)) -> ok(s(X))
    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#(plus(N,s(M))) -> plus#(N,M)
    top#(ok(X)) -> active#(X) -> active#(plus(N,s(M))) -> s#(plus(N,M))
    top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    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#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    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#(and(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    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))
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    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)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    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#(and(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    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#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    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)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    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#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) ->
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    and#(ok(X1),ok(X2)) -> and#(X1,X2) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    and#(ok(X1),ok(X2)) -> and#(X1,X2) ->
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    and#(mark(X1),X2) -> and#(X1,X2) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    and#(mark(X1),X2) -> and#(X1,X2) -> and#(ok(X1),ok(X2)) -> and#(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)
    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#(plus(N,s(M))) -> plus#(N,M)
    active#(s(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    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#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(s(X)) -> s#(active(X))
    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#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    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#(X2) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    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(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(N,s(M))) -> s#(plus(N,M)) ->
    s#(mark(X)) -> s#(X)
    active#(plus(N,s(M))) -> s#(plus(N,M)) ->
    s#(ok(X)) -> s#(X)
    active#(and(X1,X2)) -> and#(active(X1),X2) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    active#(and(X1,X2)) -> and#(active(X1),X2) ->
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(and(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(and(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(and(X1,X2)) -> active#(X1) -> active#(s(X)) -> s#(active(X))
   SCC Processor:
    #sccs: 6
    #rules: 18
    #arcs: 140/841
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     proper#(s(X)) -> proper#(X)
     proper#(plus(X1,X2)) -> proper#(X1)
     proper#(plus(X1,X2)) -> proper#(X2)
     proper#(and(X1,X2)) -> proper#(X1)
     proper#(and(X1,X2)) -> proper#(X2)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     active#(s(X)) -> active#(X)
     active#(plus(X1,X2)) -> active#(X2)
     active#(plus(X1,X2)) -> active#(X1)
     active#(and(X1,X2)) -> active#(X1)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     and#(ok(X1),ok(X2)) -> and#(X1,X2)
     and#(mark(X1),X2) -> and#(X1,X2)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    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(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     s#(ok(X)) -> s#(X)
     s#(mark(X)) -> s#(X)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(plus(X1,X2)) -> plus(active(X1),X2)
     active(plus(X1,X2)) -> plus(X1,active(X2))
     active(s(X)) -> s(active(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     plus(mark(X1),X2) -> mark(plus(X1,X2))
     plus(X1,mark(X2)) -> mark(plus(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(tt()) -> ok(tt())
     proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open