/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

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

Proof:
 DP Processor:
  DPs:
   active#(plus(N,s(M))) -> plus#(N,M)
   active#(plus(N,s(M))) -> s#(plus(N,M))
   active#(x(N,s(M))) -> x#(N,M)
   active#(x(N,s(M))) -> plus#(x(N,M),N)
   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))
   active#(x(X1,X2)) -> active#(X1)
   active#(x(X1,X2)) -> x#(active(X1),X2)
   active#(x(X1,X2)) -> active#(X2)
   active#(x(X1,X2)) -> x#(X1,active(X2))
   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)
   x#(mark(X1),X2) -> x#(X1,X2)
   x#(X1,mark(X2)) -> x#(X1,X2)
   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))
   proper#(x(X1,X2)) -> proper#(X2)
   proper#(x(X1,X2)) -> proper#(X1)
   proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
   and#(ok(X1),ok(X2)) -> and#(X1,X2)
   plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
   s#(ok(X)) -> s#(X)
   x#(ok(X1),ok(X2)) -> x#(X1,X2)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(and(tt(),X)) -> mark(X)
   active(plus(N,0())) -> mark(N)
   active(plus(N,s(M))) -> mark(s(plus(N,M)))
   active(x(N,0())) -> mark(0())
   active(x(N,s(M))) -> mark(plus(x(N,M),N))
   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))
   active(x(X1,X2)) -> x(active(X1),X2)
   active(x(X1,X2)) -> x(X1,active(X2))
   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))
   x(mark(X1),X2) -> mark(x(X1,X2))
   x(X1,mark(X2)) -> mark(x(X1,X2))
   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))
   proper(x(X1,X2)) -> x(proper(X1),proper(X2))
   and(ok(X1),ok(X2)) -> ok(and(X1,X2))
   plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
   s(ok(X)) -> ok(s(X))
   x(ok(X1),ok(X2)) -> ok(x(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(x(N,s(M))) -> x#(N,M)
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    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))
    active#(x(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> x#(X1,active(X2))
    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)
    x#(mark(X1),X2) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2)
    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))
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    s#(ok(X)) -> s#(X)
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(and(tt(),X)) -> mark(X)
    active(plus(N,0())) -> mark(N)
    active(plus(N,s(M))) -> mark(s(plus(N,M)))
    active(x(N,0())) -> mark(0())
    active(x(N,s(M))) -> mark(plus(x(N,M),N))
    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))
    active(x(X1,X2)) -> x(active(X1),X2)
    active(x(X1,X2)) -> x(X1,active(X2))
    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))
    x(mark(X1),X2) -> mark(x(X1,X2))
    x(X1,mark(X2)) -> mark(x(X1,X2))
    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))
    proper(x(X1,X2)) -> x(proper(X1),proper(X2))
    and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    s(ok(X)) -> ok(s(X))
    x(ok(X1),ok(X2)) -> ok(x(X1,X2))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> active#(X) -> active#(x(X1,X2)) -> x#(X1,active(X2))
    top#(ok(X)) -> active#(X) -> active#(x(X1,X2)) -> active#(X2)
    top#(ok(X)) -> active#(X) -> active#(x(X1,X2)) -> x#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(x(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    top#(ok(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(x(N,s(M))) -> plus#(x(N,M),N)
    top#(ok(X)) -> active#(X) -> active#(x(N,s(M))) -> x#(N,M)
    top#(ok(X)) -> active#(X) -> active#(plus(N,s(M))) -> s#(plus(N,M))
    top#(ok(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(x(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(x(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(x(X1,X2)) -> proper#(X2) -> proper#(s(X)) -> proper#(X)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(x(X1,X2)) -> proper#(X1) -> proper#(s(X)) -> proper#(X)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(x(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2)) ->
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2)) ->
    x#(X1,mark(X2)) -> x#(X1,X2)
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2)) ->
    x#(mark(X1),X2) -> x#(X1,X2)
    proper#(s(X)) -> proper#(X) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(x(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(x(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(plus(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(and(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X)
    proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(plus(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2)) ->
    plus#(ok(X1),ok(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#(mark(X1),X2) -> plus#(X1,X2)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X2) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> x#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(x(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> plus#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(plus(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> proper#(X1) ->
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) ->
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(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#(ok(X1),ok(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)
    and#(mark(X1),X2) -> and#(X1,X2) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    x#(ok(X1),ok(X2)) -> x#(X1,X2) -> x#(ok(X1),ok(X2)) -> x#(X1,X2)
    x#(ok(X1),ok(X2)) -> x#(X1,X2) -> x#(X1,mark(X2)) -> x#(X1,X2)
    x#(ok(X1),ok(X2)) -> x#(X1,X2) -> x#(mark(X1),X2) -> x#(X1,X2)
    x#(mark(X1),X2) -> x#(X1,X2) -> x#(ok(X1),ok(X2)) -> x#(X1,X2)
    x#(mark(X1),X2) -> x#(X1,X2) -> x#(X1,mark(X2)) -> x#(X1,X2)
    x#(mark(X1),X2) -> x#(X1,X2) -> x#(mark(X1),X2) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2) -> x#(ok(X1),ok(X2)) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2) -> x#(X1,mark(X2)) -> x#(X1,X2)
    x#(X1,mark(X2)) -> x#(X1,X2) -> x#(mark(X1),X2) -> x#(X1,X2)
    s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(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#(mark(X1),X2) -> plus#(X1,X2)
    plus#(mark(X1),X2) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(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#(mark(X1),X2) -> plus#(X1,X2)
    plus#(X1,mark(X2)) -> plus#(X1,X2) ->
    plus#(ok(X1),ok(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#(mark(X1),X2) -> plus#(X1,X2)
    active#(x(X1,X2)) -> x#(active(X1),X2) ->
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    active#(x(X1,X2)) -> x#(active(X1),X2) ->
    x#(X1,mark(X2)) -> x#(X1,X2)
    active#(x(X1,X2)) -> x#(active(X1),X2) ->
    x#(mark(X1),X2) -> x#(X1,X2)
    active#(x(X1,X2)) -> x#(X1,active(X2)) ->
    x#(ok(X1),ok(X2)) -> x#(X1,X2)
    active#(x(X1,X2)) -> x#(X1,active(X2)) ->
    x#(X1,mark(X2)) -> x#(X1,X2)
    active#(x(X1,X2)) -> x#(X1,active(X2)) ->
    x#(mark(X1),X2) -> x#(X1,X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(x(X1,X2)) -> active#(X2) -> active#(s(X)) -> active#(X)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(x(N,s(M))) -> x#(N,M)
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(x(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(x(X1,X2)) -> active#(X1) -> active#(s(X)) -> active#(X)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> x#(N,M)
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(x(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(x(N,s(M))) -> x#(N,M) -> x#(ok(X1),ok(X2)) -> x#(X1,X2)
    active#(x(N,s(M))) -> x#(N,M) -> x#(X1,mark(X2)) -> x#(X1,X2)
    active#(x(N,s(M))) -> x#(N,M) ->
    x#(mark(X1),X2) -> x#(X1,X2)
    active#(x(N,s(M))) -> plus#(x(N,M),N) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(x(N,s(M))) -> plus#(x(N,M),N) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    active#(x(N,s(M))) -> plus#(x(N,M),N) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X)
    active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X)
    active#(s(X)) -> active#(X) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(s(X)) -> active#(X) -> active#(x(X1,X2)) -> active#(X2)
    active#(s(X)) -> active#(X) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(x(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X2)
    active#(s(X)) -> active#(X) ->
    active#(plus(X1,X2)) -> plus#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(plus(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(and(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) -> active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(s(X)) -> active#(X) -> active#(x(N,s(M))) -> x#(N,M)
    active#(s(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(s(X)) -> active#(X) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(X1,X2)) -> plus#(active(X1),X2) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(active(X1),X2) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(active(X1),X2) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> plus#(X1,active(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#(mark(X1),X2) -> plus#(X1,X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(X1,X2)) -> active#(X2)
    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#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(x(N,s(M))) -> x#(N,M)
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X2) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    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#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> x#(N,M)
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(plus(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> plus#(N,M)
    active#(plus(N,s(M))) -> s#(plus(N,M)) ->
    s#(ok(X)) -> s#(X)
    active#(plus(N,s(M))) -> s#(plus(N,M)) ->
    s#(mark(X)) -> s#(X)
    active#(plus(N,s(M))) -> plus#(N,M) ->
    plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    active#(plus(N,s(M))) -> plus#(N,M) ->
    plus#(X1,mark(X2)) -> plus#(X1,X2)
    active#(plus(N,s(M))) -> plus#(N,M) ->
    plus#(mark(X1),X2) -> plus#(X1,X2)
    active#(and(X1,X2)) -> and#(active(X1),X2) ->
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    active#(and(X1,X2)) -> and#(active(X1),X2) ->
    and#(mark(X1),X2) -> and#(X1,X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(X1,active(X2))
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> x#(active(X1),X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(and(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> plus#(X1,active(X2))
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(X1,X2)) -> active#(X2)
    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#(X1)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(and(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> plus#(x(N,M),N)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(x(N,s(M))) -> x#(N,M)
    active#(and(X1,X2)) -> active#(X1) ->
    active#(plus(N,s(M))) -> s#(plus(N,M))
    active#(and(X1,X2)) -> active#(X1) -> active#(plus(N,s(M))) -> plus#(N,M)
   SCC Processor:
    #sccs: 7
    #rules: 25
    #arcs: 271/1681
    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(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Usable Rule Processor:
     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(x(N,0())) -> mark(0())
      active(x(N,s(M))) -> mark(plus(x(N,M),N))
      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))
      active(x(X1,X2)) -> x(active(X1),X2)
      active(x(X1,X2)) -> x(X1,active(X2))
      plus(mark(X1),X2) -> mark(plus(X1,X2))
      plus(X1,mark(X2)) -> mark(plus(X1,X2))
      plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
      s(mark(X)) -> mark(s(X))
      s(ok(X)) -> ok(s(X))
      x(mark(X1),X2) -> mark(x(X1,X2))
      x(X1,mark(X2)) -> mark(x(X1,X2))
      x(ok(X1),ok(X2)) -> ok(x(X1,X2))
      and(mark(X1),X2) -> mark(and(X1,X2))
      and(ok(X1),ok(X2)) -> ok(and(X1,X2))
      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))
      proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     LPO Processor:
      argument filtering:
       pi(tt) = []
       pi(and) = [0,1]
       pi(active) = 0
       pi(mark) = [0]
       pi(0) = []
       pi(plus) = [0,1]
       pi(s) = [0]
       pi(x) = [0,1]
       pi(proper) = 0
       pi(ok) = 0
       pi(top#) = 0
      usable rules:
       active(and(tt(),X)) -> mark(X)
       active(plus(N,0())) -> mark(N)
       active(plus(N,s(M))) -> mark(s(plus(N,M)))
       active(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       plus(mark(X1),X2) -> mark(plus(X1,X2))
       plus(X1,mark(X2)) -> mark(plus(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(mark(X)) -> mark(s(X))
       s(ok(X)) -> ok(s(X))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       and(mark(X1),X2) -> mark(and(X1,X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
      precedence:
       x > plus > s ~ and > top# ~ ok ~ proper ~ 0 ~ mark ~ active ~ tt
      problem:
       DPs:
        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(x(N,0())) -> mark(0())
        active(x(N,s(M))) -> mark(plus(x(N,M),N))
        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))
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        plus(mark(X1),X2) -> mark(plus(X1,X2))
        plus(X1,mark(X2)) -> mark(plus(X1,X2))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        s(mark(X)) -> mark(s(X))
        s(ok(X)) -> ok(s(X))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        and(mark(X1),X2) -> mark(and(X1,X2))
        and(ok(X1),ok(X2)) -> ok(and(X1,X2))
        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))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
      Restore Modifier:
       DPs:
        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(x(N,0())) -> mark(0())
        active(x(N,s(M))) -> mark(plus(x(N,M),N))
        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))
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        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))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        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))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        and(ok(X1),ok(X2)) -> ok(and(X1,X2))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        s(ok(X)) -> ok(s(X))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
       Subterm Criterion Processor:
        simple projection:
         pi(and) = 1
         pi(active) = [0,0]
         pi(mark) = 0
         pi(plus) = 0
         pi(s) = [0,0]
         pi(x) = [1,1]
         pi(ok) = [0,0,0]
         pi(top#) = 0
        problem:
         DPs:
          
         TRS:
          active(and(tt(),X)) -> mark(X)
          active(plus(N,0())) -> mark(N)
          active(plus(N,s(M))) -> mark(s(plus(N,M)))
          active(x(N,0())) -> mark(0())
          active(x(N,s(M))) -> mark(plus(x(N,M),N))
          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))
          active(x(X1,X2)) -> x(active(X1),X2)
          active(x(X1,X2)) -> x(X1,active(X2))
          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))
          x(mark(X1),X2) -> mark(x(X1,X2))
          x(X1,mark(X2)) -> mark(x(X1,X2))
          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))
          proper(x(X1,X2)) -> x(proper(X1),proper(X2))
          and(ok(X1),ok(X2)) -> ok(and(X1,X2))
          plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
          s(ok(X)) -> ok(s(X))
          x(ok(X1),ok(X2)) -> ok(x(X1,X2))
          top(mark(X)) -> top(proper(X))
          top(ok(X)) -> top(active(X))
        Qed
    
    DPs:
     active#(and(X1,X2)) -> active#(X1)
     active#(plus(X1,X2)) -> active#(X1)
     active#(plus(X1,X2)) -> active#(X2)
     active#(s(X)) -> active#(X)
     active#(x(X1,X2)) -> active#(X1)
     active#(x(X1,X2)) -> active#(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(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(and(tt(),X)) -> mark(X)
       active(plus(N,0())) -> mark(N)
       active(plus(N,s(M))) -> mark(s(plus(N,M)))
       active(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(and(X1,X2)) -> proper#(X2)
     proper#(and(X1,X2)) -> proper#(X1)
     proper#(plus(X1,X2)) -> proper#(X2)
     proper#(plus(X1,X2)) -> proper#(X1)
     proper#(s(X)) -> proper#(X)
     proper#(x(X1,X2)) -> proper#(X2)
     proper#(x(X1,X2)) -> proper#(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(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(and(tt(),X)) -> mark(X)
       active(plus(N,0())) -> mark(N)
       active(plus(N,s(M))) -> mark(s(plus(N,M)))
       active(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     x#(mark(X1),X2) -> x#(X1,X2)
     x#(X1,mark(X2)) -> x#(X1,X2)
     x#(ok(X1),ok(X2)) -> x#(X1,X2)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(x#) = 0
     problem:
      DPs:
       x#(X1,mark(X2)) -> x#(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(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(x#) = 1
      problem:
       DPs:
        
       TRS:
        active(and(tt(),X)) -> mark(X)
        active(plus(N,0())) -> mark(N)
        active(plus(N,s(M))) -> mark(s(plus(N,M)))
        active(x(N,0())) -> mark(0())
        active(x(N,s(M))) -> mark(plus(x(N,M),N))
        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))
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        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))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        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))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        and(ok(X1),ok(X2)) -> ok(and(X1,X2))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        s(ok(X)) -> ok(s(X))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     s#(mark(X)) -> s#(X)
     s#(ok(X)) -> s#(X)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(s#) = 0
     problem:
      DPs:
       
      TRS:
       active(and(tt(),X)) -> mark(X)
       active(plus(N,0())) -> mark(N)
       active(plus(N,s(M))) -> mark(s(plus(N,M)))
       active(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     plus#(mark(X1),X2) -> plus#(X1,X2)
     plus#(X1,mark(X2)) -> plus#(X1,X2)
     plus#(ok(X1),ok(X2)) -> plus#(X1,X2)
    TRS:
     active(and(tt(),X)) -> mark(X)
     active(plus(N,0())) -> mark(N)
     active(plus(N,s(M))) -> mark(s(plus(N,M)))
     active(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       plus#(X1,mark(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(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(plus#) = 1
      problem:
       DPs:
        
       TRS:
        active(and(tt(),X)) -> mark(X)
        active(plus(N,0())) -> mark(N)
        active(plus(N,s(M))) -> mark(s(plus(N,M)))
        active(x(N,0())) -> mark(0())
        active(x(N,s(M))) -> mark(plus(x(N,M),N))
        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))
        active(x(X1,X2)) -> x(active(X1),X2)
        active(x(X1,X2)) -> x(X1,active(X2))
        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))
        x(mark(X1),X2) -> mark(x(X1,X2))
        x(X1,mark(X2)) -> mark(x(X1,X2))
        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))
        proper(x(X1,X2)) -> x(proper(X1),proper(X2))
        and(ok(X1),ok(X2)) -> ok(and(X1,X2))
        plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
        s(ok(X)) -> ok(s(X))
        x(ok(X1),ok(X2)) -> ok(x(X1,X2))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     and#(mark(X1),X2) -> and#(X1,X2)
     and#(ok(X1),ok(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(x(N,0())) -> mark(0())
     active(x(N,s(M))) -> mark(plus(x(N,M),N))
     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))
     active(x(X1,X2)) -> x(active(X1),X2)
     active(x(X1,X2)) -> x(X1,active(X2))
     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))
     x(mark(X1),X2) -> mark(x(X1,X2))
     x(X1,mark(X2)) -> mark(x(X1,X2))
     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))
     proper(x(X1,X2)) -> x(proper(X1),proper(X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
     s(ok(X)) -> ok(s(X))
     x(ok(X1),ok(X2)) -> ok(x(X1,X2))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(and#) = 0
     problem:
      DPs:
       
      TRS:
       active(and(tt(),X)) -> mark(X)
       active(plus(N,0())) -> mark(N)
       active(plus(N,s(M))) -> mark(s(plus(N,M)))
       active(x(N,0())) -> mark(0())
       active(x(N,s(M))) -> mark(plus(x(N,M),N))
       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))
       active(x(X1,X2)) -> x(active(X1),X2)
       active(x(X1,X2)) -> x(X1,active(X2))
       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))
       x(mark(X1),X2) -> mark(x(X1,X2))
       x(X1,mark(X2)) -> mark(x(X1,X2))
       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))
       proper(x(X1,X2)) -> x(proper(X1),proper(X2))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
       s(ok(X)) -> ok(s(X))
       x(ok(X1),ok(X2)) -> ok(x(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed