YES

Problem:
 active(f(a(),X,X)) -> mark(f(X,b(),b()))
 active(b()) -> mark(a())
 active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
 f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
 proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
 proper(a()) -> ok(a())
 proper(b()) -> ok(b())
 f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(a(),X,X)) -> f#(X,b(),b())
   active#(f(X1,X2,X3)) -> active#(X2)
   active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3)
   f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
   proper#(f(X1,X2,X3)) -> proper#(X3)
   proper#(f(X1,X2,X3)) -> proper#(X2)
   proper#(f(X1,X2,X3)) -> proper#(X1)
   proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
   f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
   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(a(),X,X)) -> mark(f(X,b(),b()))
   active(b()) -> mark(a())
   active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
   f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
   proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
   proper(a()) -> ok(a())
   proper(b()) -> ok(b())
   f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(f(a(),X,X)) -> f#(X,b(),b())
    active#(f(X1,X2,X3)) -> active#(X2)
    active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3)
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    proper#(f(X1,X2,X3)) -> proper#(X3)
    proper#(f(X1,X2,X3)) -> proper#(X2)
    proper#(f(X1,X2,X3)) -> proper#(X1)
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    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(a(),X,X)) -> mark(f(X,b(),b()))
    active(b()) -> mark(a())
    active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
    f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
    proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
    proper(a()) -> ok(a())
    proper(b()) -> ok(b())
    f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
    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#(f(X1,X2,X3)) -> f#(X1,active(X2),X3)
    top#(ok(X)) -> active#(X) -> active#(f(X1,X2,X3)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(f(a(),X,X)) -> f#(X,b(),b())
    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#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(f(X1,X2,X3)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) ->
    proper#(f(X1,X2,X3)) -> proper#(X3)
    proper#(f(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
    proper#(f(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X1,X2,X3)) -> proper#(X1)
    proper#(f(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X1,X2,X3)) -> proper#(X2)
    proper#(f(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X1,X2,X3)) -> proper#(X3)
    proper#(f(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
    proper#(f(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X1,X2,X3)) -> proper#(X1)
    proper#(f(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X1,X2,X3)) -> proper#(X2)
    proper#(f(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X1,X2,X3)) -> proper#(X3)
    proper#(f(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3))
    proper#(f(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X1,X2,X3)) -> proper#(X1)
    proper#(f(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X1,X2,X3)) -> proper#(X2)
    proper#(f(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X1,X2,X3)) -> proper#(X3)
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) ->
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    proper#(f(X1,X2,X3)) -> f#(proper(X1),proper(X2),proper(X3)) ->
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) ->
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3) ->
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3) ->
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3) ->
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    active#(f(a(),X,X)) -> f#(X,b(),b()) ->
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    active#(f(a(),X,X)) -> f#(X,b(),b()) ->
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) ->
    f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3) ->
    f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
    active#(f(X1,X2,X3)) -> active#(X2) ->
    active#(f(X1,X2,X3)) -> f#(X1,active(X2),X3)
    active#(f(X1,X2,X3)) -> active#(X2) ->
    active#(f(X1,X2,X3)) -> active#(X2)
    active#(f(X1,X2,X3)) -> active#(X2) -> active#(f(a(),X,X)) -> f#(X,b(),b())
   SCC Processor:
    #sccs: 4
    #rules: 8
    #arcs: 40/169
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(f(a(),X,X)) -> mark(f(X,b(),b()))
     active(b()) -> mark(a())
     active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
     f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
     proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
     proper(a()) -> ok(a())
     proper(b()) -> ok(b())
     f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor: dim=2
     
     interpretation:
      [top#](x0) = [2 0]x0,
      
                  [2]
      [top](x0) = [0],
      
                   
      [ok](x0) = x0,
      
                       
      [proper](x0) = x0,
      
                   [1 0]     [1]
      [mark](x0) = [0 0]x0 + [0],
      
            [2]
      [b] = [0],
      
                     [1 0]  
      [active](x0) = [0 2]x0,
      
                        [0 2]     [1 0]     [0 3]     [0]
      [f](x0, x1, x2) = [0 0]x0 + [0 0]x1 + [0 0]x2 + [2],
      
            [0]
      [a] = [2]
     orientation:
      top#(ok(X)) = [2 0]X >= [2 0]X = top#(active(X))
      
      top#(mark(X)) = [2 0]X + [2] >= [2 0]X = top#(proper(X))
      
                           [1 3]    [4]    [0 2]    [3]                     
      active(f(a(),X,X)) = [0 0]X + [4] >= [0 0]X + [0] = mark(f(X,b(),b()))
      
                    [2]    [1]            
      active(b()) = [0] >= [0] = mark(a())
      
                            [0 2]     [1 0]     [0 3]     [0]    [0 2]     [1 0]     [0 3]     [0]                      
      active(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [4] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] = f(X1,active(X2),X3)
      
                          [0 2]     [1 0]     [0 3]     [1]    [0 2]     [1 0]     [0 3]     [1]                    
      f(X1,mark(X2),X3) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [0] = mark(f(X1,X2,X3))
      
                            [0 2]     [1 0]     [0 3]     [0]    [0 2]     [1 0]     [0 3]     [0]                                      
      proper(f(X1,X2,X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] = f(proper(X1),proper(X2),proper(X3))
      
                    [0]    [0]          
      proper(a()) = [2] >= [2] = ok(a())
      
                    [2]    [2]          
      proper(b()) = [0] >= [0] = ok(b())
      
                                [0 2]     [1 0]     [0 3]     [0]    [0 2]     [1 0]     [0 3]     [0]                  
      f(ok(X1),ok(X2),ok(X3)) = [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] >= [0 0]X1 + [0 0]X2 + [0 0]X3 + [2] = ok(f(X1,X2,X3))
      
                     [2]    [2]                 
      top(mark(X)) = [0] >= [0] = top(proper(X))
      
                   [2]    [2]                 
      top(ok(X)) = [0] >= [0] = top(active(X))
     problem:
      DPs:
       top#(ok(X)) -> top#(active(X))
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
       f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
       proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
       proper(a()) -> ok(a())
       proper(b()) -> ok(b())
       f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [top#](x0) = x0 + 0,
       
       [top](x0) = x0 + 8,
       
       [ok](x0) = x0 + 5,
       
       [proper](x0) = x0 + 5,
       
       [mark](x0) = x0 + 0,
       
       [b] = 0,
       
       [active](x0) = 0,
       
       [f](x0, x1, x2) = x1,
       
       [a] = 0
      orientation:
       top#(ok(X)) = X + 5 >= 0 = top#(active(X))
       
       active(f(a(),X,X)) = 0 >= 0 = mark(f(X,b(),b()))
       
       active(b()) = 0 >= 0 = mark(a())
       
       active(f(X1,X2,X3)) = 0 >= 0 = f(X1,active(X2),X3)
       
       f(X1,mark(X2),X3) = X2 + 0 >= X2 + 0 = mark(f(X1,X2,X3))
       
       proper(f(X1,X2,X3)) = X2 + 5 >= X2 + 5 = f(proper(X1),proper(X2),proper(X3))
       
       proper(a()) = 5 >= 5 = ok(a())
       
       proper(b()) = 5 >= 5 = ok(b())
       
       f(ok(X1),ok(X2),ok(X3)) = X2 + 5 >= X2 + 5 = ok(f(X1,X2,X3))
       
       top(mark(X)) = X + 8 >= X + 8 = top(proper(X))
       
       top(ok(X)) = X + 8 >= 8 = top(active(X))
      problem:
       DPs:
        
       TRS:
        active(f(a(),X,X)) -> mark(f(X,b(),b()))
        active(b()) -> mark(a())
        active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
        f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
        proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
        proper(a()) -> ok(a())
        proper(b()) -> ok(b())
        f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     active#(f(X1,X2,X3)) -> active#(X2)
    TRS:
     active(f(a(),X,X)) -> mark(f(X,b(),b()))
     active(b()) -> mark(a())
     active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
     f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
     proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
     proper(a()) -> ok(a())
     proper(b()) -> ok(b())
     f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
       f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
       proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
       proper(a()) -> ok(a())
       proper(b()) -> ok(b())
       f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(f(X1,X2,X3)) -> proper#(X3)
     proper#(f(X1,X2,X3)) -> proper#(X2)
     proper#(f(X1,X2,X3)) -> proper#(X1)
    TRS:
     active(f(a(),X,X)) -> mark(f(X,b(),b()))
     active(b()) -> mark(a())
     active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
     f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
     proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
     proper(a()) -> ok(a())
     proper(b()) -> ok(b())
     f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
       f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
       proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
       proper(a()) -> ok(a())
       proper(b()) -> ok(b())
       f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
     f#(ok(X1),ok(X2),ok(X3)) -> f#(X1,X2,X3)
    TRS:
     active(f(a(),X,X)) -> mark(f(X,b(),b()))
     active(b()) -> mark(a())
     active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
     f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
     proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
     proper(a()) -> ok(a())
     proper(b()) -> ok(b())
     f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(f#) = 2
     problem:
      DPs:
       f#(X1,mark(X2),X3) -> f#(X1,X2,X3)
      TRS:
       active(f(a(),X,X)) -> mark(f(X,b(),b()))
       active(b()) -> mark(a())
       active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
       f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
       proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
       proper(a()) -> ok(a())
       proper(b()) -> ok(b())
       f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(f#) = 1
      problem:
       DPs:
        
       TRS:
        active(f(a(),X,X)) -> mark(f(X,b(),b()))
        active(b()) -> mark(a())
        active(f(X1,X2,X3)) -> f(X1,active(X2),X3)
        f(X1,mark(X2),X3) -> mark(f(X1,X2,X3))
        proper(f(X1,X2,X3)) -> f(proper(X1),proper(X2),proper(X3))
        proper(a()) -> ok(a())
        proper(b()) -> ok(b())
        f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed