MAYBE

Problem:
 active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
 active(f(X1,X2)) -> f(active(X1),X2)
 active(g(X)) -> g(active(X))
 f(mark(X1),X2) -> mark(f(X1,X2))
 g(mark(X)) -> mark(g(X))
 proper(f(X1,X2)) -> f(proper(X1),proper(X2))
 proper(g(X)) -> g(proper(X))
 f(ok(X1),ok(X2)) -> ok(f(X1,X2))
 g(ok(X)) -> ok(g(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
   active#(f(X1,X2)) -> active#(X1)
   active#(f(X1,X2)) -> f#(active(X1),X2)
   active#(g(X)) -> active#(X)
   active#(g(X)) -> g#(active(X))
   f#(mark(X1),X2) -> f#(X1,X2)
   g#(mark(X)) -> g#(X)
   proper#(f(X1,X2)) -> proper#(X2)
   proper#(f(X1,X2)) -> proper#(X1)
   proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
   proper#(g(X)) -> proper#(X)
   proper#(g(X)) -> g#(proper(X))
   f#(ok(X1),ok(X2)) -> f#(X1,X2)
   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(g(X),Y)) -> mark(f(X,f(g(X),Y)))
   active(f(X1,X2)) -> f(active(X1),X2)
   active(g(X)) -> g(active(X))
   f(mark(X1),X2) -> mark(f(X1,X2))
   g(mark(X)) -> mark(g(X))
   proper(f(X1,X2)) -> f(proper(X1),proper(X2))
   proper(g(X)) -> g(proper(X))
   f(ok(X1),ok(X2)) -> ok(f(X1,X2))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Usable Rule Processor:
   DPs:
    active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
    active#(f(X1,X2)) -> active#(X1)
    active#(f(X1,X2)) -> f#(active(X1),X2)
    active#(g(X)) -> active#(X)
    active#(g(X)) -> g#(active(X))
    f#(mark(X1),X2) -> f#(X1,X2)
    g#(mark(X)) -> g#(X)
    proper#(f(X1,X2)) -> proper#(X2)
    proper#(f(X1,X2)) -> proper#(X1)
    proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
    proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> g#(proper(X))
    f#(ok(X1),ok(X2)) -> f#(X1,X2)
    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:
    f12(x,y) -> x
    f12(x,y) -> y
    g(mark(X)) -> mark(g(X))
    g(ok(X)) -> ok(g(X))
    f(mark(X1),X2) -> mark(f(X1,X2))
    f(ok(X1),ok(X2)) -> ok(f(X1,X2))
    active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
    active(f(X1,X2)) -> f(active(X1),X2)
    active(g(X)) -> g(active(X))
    proper(f(X1,X2)) -> f(proper(X1),proper(X2))
    proper(g(X)) -> g(proper(X))
   EDG Processor:
    DPs:
     active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
     active#(f(X1,X2)) -> active#(X1)
     active#(f(X1,X2)) -> f#(active(X1),X2)
     active#(g(X)) -> active#(X)
     active#(g(X)) -> g#(active(X))
     f#(mark(X1),X2) -> f#(X1,X2)
     g#(mark(X)) -> g#(X)
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(g(X)) -> proper#(X)
     proper#(g(X)) -> g#(proper(X))
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     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:
     f12(x,y) -> x
     f12(x,y) -> y
     g(mark(X)) -> mark(g(X))
     g(ok(X)) -> ok(g(X))
     f(mark(X1),X2) -> mark(f(X1,X2))
     f(ok(X1),ok(X2)) -> ok(f(X1,X2))
     active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
     active(f(X1,X2)) -> f(active(X1),X2)
     active(g(X)) -> g(active(X))
     proper(f(X1,X2)) -> f(proper(X1),proper(X2))
     proper(g(X)) -> g(proper(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#(f(g(X),Y)) -> f#(X,f(g(X),Y))
     top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
     top#(ok(X)) -> active#(X) -> active#(f(X1,X2)) -> f#(active(X1),X2)
     top#(ok(X)) -> active#(X) -> active#(g(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(g(X)) -> g#(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#(f(X1,X2)) -> proper#(X2)
     top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
     top#(mark(X)) -> proper#(X) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
     top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(g(X)) -> proper#(X)
     proper#(f(X1,X2)) -> proper#(X2) ->
     proper#(g(X)) -> g#(proper(X))
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> proper#(X2)
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> proper#(X1)
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(g(X)) -> proper#(X)
     proper#(f(X1,X2)) -> proper#(X1) ->
     proper#(g(X)) -> g#(proper(X))
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2)) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X2)
     proper#(g(X)) -> proper#(X) -> proper#(f(X1,X2)) -> proper#(X1)
     proper#(g(X)) -> proper#(X) ->
     proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
     proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
     proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
     proper#(g(X)) -> g#(proper(X)) -> g#(mark(X)) -> g#(X)
     proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X)
     g#(ok(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
     g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
     g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
     g#(mark(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
     f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
     f#(ok(X1),ok(X2)) -> f#(X1,X2) -> f#(ok(X1),ok(X2)) -> f#(X1,X2)
     f#(mark(X1),X2) -> f#(X1,X2) -> f#(mark(X1),X2) -> f#(X1,X2)
     f#(mark(X1),X2) -> f#(X1,X2) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     active#(f(g(X),Y)) -> f#(X,f(g(X),Y)) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     active#(f(g(X),Y)) -> f#(X,f(g(X),Y)) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     active#(f(X1,X2)) -> f#(active(X1),X2) ->
     f#(mark(X1),X2) -> f#(X1,X2)
     active#(f(X1,X2)) -> f#(active(X1),X2) ->
     f#(ok(X1),ok(X2)) -> f#(X1,X2)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(X1,X2)) -> active#(X1)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(f(X1,X2)) -> f#(active(X1),X2)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(g(X)) -> active#(X)
     active#(f(X1,X2)) -> active#(X1) ->
     active#(g(X)) -> g#(active(X))
     active#(g(X)) -> g#(active(X)) -> g#(mark(X)) -> g#(X)
     active#(g(X)) -> g#(active(X)) -> g#(ok(X)) -> g#(X)
     active#(g(X)) -> active#(X) ->
     active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
     active#(g(X)) -> active#(X) -> active#(f(X1,X2)) -> active#(X1)
     active#(g(X)) -> active#(X) ->
     active#(f(X1,X2)) -> f#(active(X1),X2)
     active#(g(X)) -> active#(X) -> active#(g(X)) -> active#(X)
     active#(g(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
    Restore Modifier:
     DPs:
      active#(f(g(X),Y)) -> f#(X,f(g(X),Y))
      active#(f(X1,X2)) -> active#(X1)
      active#(f(X1,X2)) -> f#(active(X1),X2)
      active#(g(X)) -> active#(X)
      active#(g(X)) -> g#(active(X))
      f#(mark(X1),X2) -> f#(X1,X2)
      g#(mark(X)) -> g#(X)
      proper#(f(X1,X2)) -> proper#(X2)
      proper#(f(X1,X2)) -> proper#(X1)
      proper#(f(X1,X2)) -> f#(proper(X1),proper(X2))
      proper#(g(X)) -> proper#(X)
      proper#(g(X)) -> g#(proper(X))
      f#(ok(X1),ok(X2)) -> f#(X1,X2)
      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(g(X),Y)) -> mark(f(X,f(g(X),Y)))
      active(f(X1,X2)) -> f(active(X1),X2)
      active(g(X)) -> g(active(X))
      f(mark(X1),X2) -> mark(f(X1,X2))
      g(mark(X)) -> mark(g(X))
      proper(f(X1,X2)) -> f(proper(X1),proper(X2))
      proper(g(X)) -> g(proper(X))
      f(ok(X1),ok(X2)) -> ok(f(X1,X2))
      g(ok(X)) -> ok(g(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     SCC Processor:
      #sccs: 5
      #rules: 11
      #arcs: 61/324
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       proper#(g(X)) -> proper#(X)
       proper#(f(X1,X2)) -> proper#(X1)
       proper#(f(X1,X2)) -> proper#(X2)
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       active#(g(X)) -> active#(X)
       active#(f(X1,X2)) -> active#(X1)
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       f#(ok(X1),ok(X2)) -> f#(X1,X2)
       f#(mark(X1),X2) -> f#(X1,X2)
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       g#(ok(X)) -> g#(X)
       g#(mark(X)) -> g#(X)
      TRS:
       active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
       active(f(X1,X2)) -> f(active(X1),X2)
       active(g(X)) -> g(active(X))
       f(mark(X1),X2) -> mark(f(X1,X2))
       g(mark(X)) -> mark(g(X))
       proper(f(X1,X2)) -> f(proper(X1),proper(X2))
       proper(g(X)) -> g(proper(X))
       f(ok(X1),ok(X2)) -> ok(f(X1,X2))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open