YES

Problem:
 active(f(f(a()))) -> mark(f(g(f(a()))))
 active(f(X)) -> f(active(X))
 f(mark(X)) -> mark(f(X))
 proper(f(X)) -> f(proper(X))
 proper(a()) -> ok(a())
 proper(g(X)) -> g(proper(X))
 f(ok(X)) -> ok(f(X))
 g(ok(X)) -> ok(g(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(f(a()))) -> g#(f(a()))
   active#(f(f(a()))) -> f#(g(f(a())))
   active#(f(X)) -> active#(X)
   active#(f(X)) -> f#(active(X))
   f#(mark(X)) -> f#(X)
   proper#(f(X)) -> proper#(X)
   proper#(f(X)) -> f#(proper(X))
   proper#(g(X)) -> proper#(X)
   proper#(g(X)) -> g#(proper(X))
   f#(ok(X)) -> f#(X)
   g#(ok(X)) -> g#(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(f(f(a()))) -> mark(f(g(f(a()))))
   active(f(X)) -> f(active(X))
   f(mark(X)) -> mark(f(X))
   proper(f(X)) -> f(proper(X))
   proper(a()) -> ok(a())
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  CDG Processor:
   DPs:
    active#(f(f(a()))) -> g#(f(a()))
    active#(f(f(a()))) -> f#(g(f(a())))
    active#(f(X)) -> active#(X)
    active#(f(X)) -> f#(active(X))
    f#(mark(X)) -> f#(X)
    proper#(f(X)) -> proper#(X)
    proper#(f(X)) -> f#(proper(X))
    proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> g#(proper(X))
    f#(ok(X)) -> f#(X)
    g#(ok(X)) -> g#(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(f(f(a()))) -> mark(f(g(f(a()))))
    active(f(X)) -> f(active(X))
    f(mark(X)) -> mark(f(X))
    proper(f(X)) -> f(proper(X))
    proper(a()) -> ok(a())
    proper(g(X)) -> g(proper(X))
    f(ok(X)) -> ok(f(X))
    g(ok(X)) -> ok(g(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(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#(ok(X)) -> top#(active(X))
   SCC Processor:
    #sccs: 1
    #rules: 1
    #arcs: 4/225
    DPs:
     top#(ok(X)) -> top#(active(X))
    TRS:
     active(f(f(a()))) -> mark(f(g(f(a()))))
     active(f(X)) -> f(active(X))
     f(mark(X)) -> mark(f(X))
     proper(f(X)) -> f(proper(X))
     proper(a()) -> ok(a())
     proper(g(X)) -> g(proper(X))
     f(ok(X)) -> ok(f(X))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [top#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = 1,
      
      [proper](x0) = x0 + 1,
      
      [mark](x0) = 0,
      
      [g](x0) = x0,
      
      [active](x0) = 0,
      
      [f](x0) = x0,
      
      [a] = 1
     orientation:
      top#(ok(X)) = 2 >= 1 = top#(active(X))
      
      active(f(f(a()))) = 0 >= 0 = mark(f(g(f(a()))))
      
      active(f(X)) = 0 >= 0 = f(active(X))
      
      f(mark(X)) = 0 >= 0 = mark(f(X))
      
      proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
      
      proper(a()) = 2 >= 1 = ok(a())
      
      proper(g(X)) = X + 1 >= X + 1 = g(proper(X))
      
      f(ok(X)) = 1 >= 1 = ok(f(X))
      
      g(ok(X)) = 1 >= 1 = ok(g(X))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       
      TRS:
       active(f(f(a()))) -> mark(f(g(f(a()))))
       active(f(X)) -> f(active(X))
       f(mark(X)) -> mark(f(X))
       proper(f(X)) -> f(proper(X))
       proper(a()) -> ok(a())
       proper(g(X)) -> g(proper(X))
       f(ok(X)) -> ok(f(X))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed