YES

Problem:
 strict:
  f(g(x),y,z) -> f(x,y,g(z))
 weak:
  f(x,y,g(z)) -> f(x,g(y),z)
  f(x,a(),z) -> f(x,g(a()),z)
  f(x,y,z) -> f(x,y,g(z))

Proof:
 Matrix Interpretation Processor: dim=5
  
  interpretation:
                     [1 0 0 0 0]     [1 0 1 1 1]     [1 0 0 1 1]     [0]
                     [1 1 0 0 0]     [1 0 1 0 1]     [0 0 0 0 0]     [0]
   [f](x0, x1, x2) = [0 1 0 0 0]x0 + [1 0 0 0 1]x1 + [0 0 0 0 0]x2 + [1]
                     [0 0 0 0 0]     [0 0 0 1 0]     [0 0 0 0 0]     [0]
                     [0 1 0 0 0]     [0 0 0 0 0]     [0 0 0 0 0]     [0],
   
             [1 0 0 0 1]  
             [0 1 1 0 0]  
   [g](x0) = [0 0 0 0 0]x0
             [0 0 0 1 0]  
             [0 0 0 0 0]  ,
   
         [0]
         [1]
   [a] = [1]
         [1]
         [1]
  orientation:
                 [1 0 0 0 1]    [1 0 1 1 1]    [1 0 0 1 1]    [0]    [1 0 0 0 0]    [1 0 1 1 1]    [1 0 0 1 1]    [0]              
                 [1 1 1 0 1]    [1 0 1 0 1]    [0 0 0 0 0]    [0]    [1 1 0 0 0]    [1 0 1 0 1]    [0 0 0 0 0]    [0]              
   f(g(x),y,z) = [0 1 1 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] >= [0 1 0 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] = f(x,y,g(z))
                 [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]              
                 [0 1 1 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 1 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
   
                 [1 0 0 0 0]    [1 0 1 1 1]    [1 0 0 1 1]    [0]    [1 0 0 0 0]    [1 0 0 1 1]    [1 0 0 1 1]    [0]              
                 [1 1 0 0 0]    [1 0 1 0 1]    [0 0 0 0 0]    [0]    [1 1 0 0 0]    [1 0 0 0 1]    [0 0 0 0 0]    [0]              
   f(x,y,g(z)) = [0 1 0 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] >= [0 1 0 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] = f(x,g(y),z)
                 [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]              
                 [0 1 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 1 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
   
                [1 0 0 0 0]    [1 0 0 1 1]    [3]    [1 0 0 0 0]    [1 0 0 1 1]    [2]                
                [1 1 0 0 0]    [0 0 0 0 0]    [2]    [1 1 0 0 0]    [0 0 0 0 0]    [1]                
   f(x,a(),z) = [0 1 0 0 0]x + [0 0 0 0 0]z + [2] >= [0 1 0 0 0]x + [0 0 0 0 0]z + [2] = f(x,g(a()),z)
                [0 0 0 0 0]    [0 0 0 0 0]    [1]    [0 0 0 0 0]    [0 0 0 0 0]    [1]                
                [0 1 0 0 0]    [0 0 0 0 0]    [0]    [0 1 0 0 0]    [0 0 0 0 0]    [0]                
   
              [1 0 0 0 0]    [1 0 1 1 1]    [1 0 0 1 1]    [0]    [1 0 0 0 0]    [1 0 1 1 1]    [1 0 0 1 1]    [0]              
              [1 1 0 0 0]    [1 0 1 0 1]    [0 0 0 0 0]    [0]    [1 1 0 0 0]    [1 0 1 0 1]    [0 0 0 0 0]    [0]              
   f(x,y,z) = [0 1 0 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] >= [0 1 0 0 0]x + [1 0 0 0 1]y + [0 0 0 0 0]z + [1] = f(x,y,g(z))
              [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 1 0]    [0 0 0 0 0]    [0]              
              [0 1 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 1 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
  problem:
   strict:
    f(g(x),y,z) -> f(x,y,g(z))
   weak:
    f(x,y,g(z)) -> f(x,g(y),z)
    f(x,y,z) -> f(x,y,g(z))
  Matrix Interpretation Processor: dim=5
   
   interpretation:
                      [1 0 1 0 0]     [1 0 0 0 0]     [1 0 0 0 0]     [1]
                      [0 1 0 1 1]     [0 0 0 0 1]     [0 0 0 0 0]     [0]
    [f](x0, x1, x2) = [0 1 0 0 0]x0 + [0 0 0 0 0]x1 + [0 0 0 0 0]x2 + [0]
                      [0 0 1 0 0]     [0 1 0 1 1]     [0 0 0 0 0]     [0]
                      [0 0 0 0 0]     [0 0 0 0 0]     [0 0 0 0 0]     [0],
    
              [1 0 0 0 0]     [0]
              [0 1 0 1 0]     [0]
    [g](x0) = [1 0 1 0 0]x0 + [1]
              [0 0 0 0 1]     [0]
              [0 0 0 0 0]     [0]
   orientation:
                  [2 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [2]    [1 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [1]              
                  [0 1 0 1 1]    [0 0 0 0 1]    [0 0 0 0 0]    [0]    [0 1 0 1 1]    [0 0 0 0 1]    [0 0 0 0 0]    [0]              
    f(g(x),y,z) = [0 1 0 1 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] >= [0 1 0 0 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] = f(x,y,g(z))
                  [1 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [1]    [0 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [0]              
                  [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
    
                  [1 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [1]    [1 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [1]              
                  [0 1 0 1 1]    [0 0 0 0 1]    [0 0 0 0 0]    [0]    [0 1 0 1 1]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
    f(x,y,g(z)) = [0 1 0 0 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] >= [0 1 0 0 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] = f(x,g(y),z)
                  [0 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [0]    [0 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [0]              
                  [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
    
               [1 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [1]    [1 0 1 0 0]    [1 0 0 0 0]    [1 0 0 0 0]    [1]              
               [0 1 0 1 1]    [0 0 0 0 1]    [0 0 0 0 0]    [0]    [0 1 0 1 1]    [0 0 0 0 1]    [0 0 0 0 0]    [0]              
    f(x,y,z) = [0 1 0 0 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] >= [0 1 0 0 0]x + [0 0 0 0 0]y + [0 0 0 0 0]z + [0] = f(x,y,g(z))
               [0 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [0]    [0 0 1 0 0]    [0 1 0 1 1]    [0 0 0 0 0]    [0]              
               [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]              
   problem:
    strict:
     
    weak:
     f(x,y,g(z)) -> f(x,g(y),z)
     f(x,y,z) -> f(x,y,g(z))
   Qed