YES

Problem:
 f(a(),x) -> f(b(),f(c(),x))
 f(a(),f(b(),x)) -> f(b(),f(a(),x))
 f(d(),f(c(),x)) -> f(d(),f(a(),x))
 f(a(),f(c(),x)) -> f(c(),f(a(),x))

Proof:
 Extended Uncurrying Processor:
  application symbol: f
  symbol table:
   d ==> d0/0 d1/1
   c ==> c0/0 c1/1
   b ==> b0/0 b1/1
   a ==> a0/0 a1/1
   
  uncurry-rules:
   f(a0(),x1) -> a1(x1)
   f(b0(),x3) -> b1(x3)
   f(c0(),x5) -> c1(x5)
   f(d0(),x7) -> d1(x7)
  eta-rules:
   
  problem:
   a1(x) -> b1(c1(x))
   a1(b1(x)) -> b1(a1(x))
   d1(c1(x)) -> d1(a1(x))
   a1(c1(x)) -> c1(a1(x))
   f(a0(),x1) -> a1(x1)
   f(b0(),x3) -> b1(x3)
   f(c0(),x5) -> c1(x5)
   f(d0(),x7) -> d1(x7)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
               [1 0 0]  
    [b1](x0) = [0 0 0]x0
               [0 0 0]  ,
    
                  [1 0 1]     [1 1 1]     [0]
    [f](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [0]
                  [0 0 0]     [1 0 1]     [1],
    
           [0]
    [a0] = [0]
           [0],
    
               [1 0 0]  
    [c1](x0) = [0 0 0]x0
               [0 0 0]  ,
    
           [1]
    [b0] = [0]
           [1],
    
               [1 0 0]     [0]
    [d1](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [1],
    
           [1]
    [d0] = [0]
           [1],
    
           [1]
    [c0] = [0]
           [0],
    
               [1 0 0]     [0]
    [a1](x0) = [0 0 0]x0 + [0]
               [1 0 0]     [1]
   orientation:
            [1 0 0]    [0]    [1 0 0]             
    a1(x) = [0 0 0]x + [0] >= [0 0 0]x = b1(c1(x))
            [1 0 0]    [1]    [0 0 0]             
    
                [1 0 0]    [0]    [1 0 0]             
    a1(b1(x)) = [0 0 0]x + [0] >= [0 0 0]x = b1(a1(x))
                [1 0 0]    [1]    [0 0 0]             
    
                [1 0 0]    [0]    [1 0 0]    [0]            
    d1(c1(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = d1(a1(x))
                [0 0 0]    [1]    [0 0 0]    [1]            
    
                [1 0 0]    [0]    [1 0 0]             
    a1(c1(x)) = [0 0 0]x + [0] >= [0 0 0]x = c1(a1(x))
                [1 0 0]    [1]    [0 0 0]             
    
                 [1 1 1]     [0]    [1 0 0]     [0]         
    f(a0(),x1) = [0 0 1]x1 + [0] >= [0 0 0]x1 + [0] = a1(x1)
                 [1 0 1]     [1]    [1 0 0]     [1]         
    
                 [1 1 1]     [2]    [1 0 0]           
    f(b0(),x3) = [0 0 1]x3 + [0] >= [0 0 0]x3 = b1(x3)
                 [1 0 1]     [1]    [0 0 0]           
    
                 [1 1 1]     [1]    [1 0 0]           
    f(c0(),x5) = [0 0 1]x5 + [0] >= [0 0 0]x5 = c1(x5)
                 [1 0 1]     [1]    [0 0 0]           
    
                 [1 1 1]     [2]    [1 0 0]     [0]         
    f(d0(),x7) = [0 0 1]x7 + [0] >= [0 0 0]x7 + [0] = d1(x7)
                 [1 0 1]     [1]    [0 0 0]     [1]         
   problem:
    a1(x) -> b1(c1(x))
    a1(b1(x)) -> b1(a1(x))
    d1(c1(x)) -> d1(a1(x))
    a1(c1(x)) -> c1(a1(x))
    f(a0(),x1) -> a1(x1)
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                [1 0 0]  
     [b1](x0) = [0 0 0]x0
                [0 0 0]  ,
     
                   [1 0 0]     [1 0 1]     [1]
     [f](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [0]
                   [0 0 0]     [1 1 1]     [0],
     
            [0]
     [a0] = [0]
            [0],
     
                [1 0 0]     [0]
     [c1](x0) = [0 1 0]x0 + [1]
                [0 0 0]     [0],
     
                [1 1 0]  
     [d1](x0) = [0 1 0]x0
                [0 0 0]  ,
     
                [1 0 0]     [1]
     [a1](x0) = [0 1 0]x0 + [0]
                [1 1 1]     [0]
    orientation:
             [1 0 0]    [1]    [1 0 0]             
     a1(x) = [0 1 0]x + [0] >= [0 0 0]x = b1(c1(x))
             [1 1 1]    [0]    [0 0 0]             
     
                 [1 0 0]    [1]    [1 0 0]    [1]            
     a1(b1(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = b1(a1(x))
                 [1 0 0]    [0]    [0 0 0]    [0]            
     
                 [1 1 0]    [1]    [1 1 0]    [1]            
     d1(c1(x)) = [0 1 0]x + [1] >= [0 1 0]x + [0] = d1(a1(x))
                 [0 0 0]    [0]    [0 0 0]    [0]            
     
                 [1 0 0]    [1]    [1 0 0]    [1]            
     a1(c1(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = c1(a1(x))
                 [1 1 0]    [1]    [0 0 0]    [0]            
     
                  [1 0 1]     [1]    [1 0 0]     [1]         
     f(a0(),x1) = [0 1 0]x1 + [0] >= [0 1 0]x1 + [0] = a1(x1)
                  [1 1 1]     [0]    [1 1 1]     [0]         
    problem:
     a1(b1(x)) -> b1(a1(x))
     d1(c1(x)) -> d1(a1(x))
     a1(c1(x)) -> c1(a1(x))
     f(a0(),x1) -> a1(x1)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [b1](x0) = x0,
      
      [f](x0, x1) = 5x0 + 4x1 + 3,
      
      [a0] = 4,
      
      [c1](x0) = x0 + 6,
      
      [d1](x0) = 2x0 + 1,
      
      [a1](x0) = x0 + 6
     orientation:
      a1(b1(x)) = x + 6 >= x + 6 = b1(a1(x))
      
      d1(c1(x)) = 2x + 13 >= 2x + 13 = d1(a1(x))
      
      a1(c1(x)) = x + 12 >= x + 12 = c1(a1(x))
      
      f(a0(),x1) = 4x1 + 23 >= x1 + 6 = a1(x1)
     problem:
      a1(b1(x)) -> b1(a1(x))
      d1(c1(x)) -> d1(a1(x))
      a1(c1(x)) -> c1(a1(x))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [b1](x0) = 2x0 + 4,
       
       [c1](x0) = 4x0 + 1,
       
       [d1](x0) = 2x0 + 1,
       
       [a1](x0) = x0
      orientation:
       a1(b1(x)) = 2x + 4 >= 2x + 4 = b1(a1(x))
       
       d1(c1(x)) = 8x + 3 >= 2x + 1 = d1(a1(x))
       
       a1(c1(x)) = 4x + 1 >= 4x + 1 = c1(a1(x))
      problem:
       a1(b1(x)) -> b1(a1(x))
       a1(c1(x)) -> c1(a1(x))
      Matrix Interpretation Processor: dim=3
       
       interpretation:
                   [1 0 1]     [0]
        [b1](x0) = [0 1 0]x0 + [1]
                   [0 1 0]     [1],
        
                   [1 0 1]     [1]
        [c1](x0) = [0 1 0]x0 + [0]
                   [0 1 0]     [0],
        
                   [1 0 1]  
        [a1](x0) = [0 1 0]x0
                   [0 1 0]  
       orientation:
                    [1 1 1]    [1]    [1 1 1]    [0]            
        a1(b1(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = b1(a1(x))
                    [0 1 0]    [1]    [0 1 0]    [1]            
        
                    [1 1 1]    [1]    [1 1 1]    [1]            
        a1(c1(x)) = [0 1 0]x + [0] >= [0 1 0]x + [0] = c1(a1(x))
                    [0 1 0]    [0]    [0 1 0]    [0]            
       problem:
        a1(c1(x)) -> c1(a1(x))
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                    [1 0 0]     [0]
         [c1](x0) = [0 1 0]x0 + [1]
                    [0 0 0]     [0],
         
                    [1 1 0]  
         [a1](x0) = [0 1 0]x0
                    [0 0 0]  
        orientation:
                     [1 1 0]    [1]    [1 1 0]    [0]            
         a1(c1(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = c1(a1(x))
                     [0 0 0]    [0]    [0 0 0]    [0]            
        problem:
         
        Qed