YES

Problem:
 f(n__f(n__a())) -> f(n__g(n__f(n__a())))
 f(X) -> n__f(X)
 a() -> n__a()
 g(X) -> n__g(X)
 activate(n__f(X)) -> f(X)
 activate(n__a()) -> a()
 activate(n__g(X)) -> g(activate(X))
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   f#(n__f(n__a())) -> f#(n__g(n__f(n__a())))
   activate#(n__f(X)) -> f#(X)
   activate#(n__a()) -> a#()
   activate#(n__g(X)) -> activate#(X)
   activate#(n__g(X)) -> g#(activate(X))
  TRS:
   f(n__f(n__a())) -> f(n__g(n__f(n__a())))
   f(X) -> n__f(X)
   a() -> n__a()
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a()) -> a()
   activate(n__g(X)) -> g(activate(X))
   activate(X) -> X
  Matrix Interpretation Processor: dim=3
   
   usable rules:
    
   interpretation:
    [activate#](x0) = [0 0 1]x0 + [1],
    
    [g#](x0) = [0],
    
    [a#] = 0,
    
    [f#](x0) = [0 1 0]x0 + [1],
    
                     [1 0 1]     [1]
    [activate](x0) = [0 1 0]x0 + [0]
                     [1 0 0]     [0],
    
              [0 0 0]  
    [g](x0) = [0 0 1]x0
              [0 0 0]  ,
    
          [0]
    [a] = [0]
          [0],
    
                 [0 0 0]     [0]
    [n__g](x0) = [0 0 0]x0 + [0]
                 [0 1 1]     [1],
    
              [0 1 0]  
    [f](x0) = [0 0 0]x0
              [0 1 1]  ,
    
                 [0 0 1]     [0]
    [n__f](x0) = [1 1 0]x0 + [1]
                 [0 1 1]     [1],
    
             [1]
    [n__a] = [1]
             [0]
   orientation:
    f#(n__f(n__a())) = 4 >= 1 = f#(n__g(n__f(n__a())))
    
    activate#(n__f(X)) = [0 1 1]X + [2] >= [0 1 0]X + [1] = f#(X)
    
    activate#(n__a()) = 1 >= 0 = a#()
    
    activate#(n__g(X)) = [0 1 1]X + [2] >= [0 0 1]X + [1] = activate#(X)
    
    activate#(n__g(X)) = [0 1 1]X + [2] >= [0] = g#(activate(X))
    
                      [3]    [0]                        
    f(n__f(n__a())) = [0] >= [0] = f(n__g(n__f(n__a())))
                      [5]    [6]                        
    
           [0 1 0]     [0 0 1]    [0]          
    f(X) = [0 0 0]X >= [1 1 0]X + [1] = n__f(X)
           [0 1 1]     [0 1 1]    [1]          
    
          [0]    [1]         
    a() = [0] >= [1] = n__a()
          [0]    [0]         
    
           [0 0 0]     [0 0 0]    [0]          
    g(X) = [0 0 1]X >= [0 0 0]X + [0] = n__g(X)
           [0 0 0]     [0 1 1]    [1]          
    
                        [0 1 2]    [2]    [0 1 0]        
    activate(n__f(X)) = [1 1 0]X + [1] >= [0 0 0]X = f(X)
                        [0 0 1]    [0]    [0 1 1]        
    
                       [2]    [0]      
    activate(n__a()) = [1] >= [0] = a()
                       [1]    [0]      
    
                        [0 1 1]    [2]    [0 0 0]                  
    activate(n__g(X)) = [0 0 0]X + [0] >= [1 0 0]X = g(activate(X))
                        [0 0 0]    [0]    [0 0 0]                  
    
                  [1 0 1]    [1]         
    activate(X) = [0 1 0]X + [0] >= X = X
                  [1 0 0]    [0]         
   problem:
    DPs:
     
    TRS:
     f(n__f(n__a())) -> f(n__g(n__f(n__a())))
     f(X) -> n__f(X)
     a() -> n__a()
     g(X) -> n__g(X)
     activate(n__f(X)) -> f(X)
     activate(n__a()) -> a()
     activate(n__g(X)) -> g(activate(X))
     activate(X) -> X
   Qed