YES

Problem:
 average(s(x),y) -> average(x,s(y))
 average(x,s(s(s(y)))) -> s(average(s(x),y))
 average(0(),0()) -> 0()
 average(0(),s(0())) -> 0()
 average(0(),s(s(0()))) -> s(0())

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                       [1 1 1]     [1 0 0]  
   [average](x0, x1) = [0 1 1]x0 + [0 0 0]x1
                       [0 1 1]     [0 0 0]  ,
   
             [1 0 0]  
   [s](x0) = [0 0 1]x0
             [0 1 0]  ,
   
         [0]
   [0] = [0]
         [1]
  orientation:
                     [1 1 1]    [1 0 0]     [1 1 1]    [1 0 0]                   
   average(s(x),y) = [0 1 1]x + [0 0 0]y >= [0 1 1]x + [0 0 0]y = average(x,s(y))
                     [0 1 1]    [0 0 0]     [0 1 1]    [0 0 0]                   
   
                           [1 1 1]    [1 0 0]     [1 1 1]    [1 0 0]                      
   average(x,s(s(s(y)))) = [0 1 1]x + [0 0 0]y >= [0 1 1]x + [0 0 0]y = s(average(s(x),y))
                           [0 1 1]    [0 0 0]     [0 1 1]    [0 0 0]                      
   
                      [1]    [0]      
   average(0(),0()) = [1] >= [0] = 0()
                      [1]    [1]      
   
                         [1]    [0]      
   average(0(),s(0())) = [1] >= [0] = 0()
                         [1]    [1]      
   
                            [1]    [0]         
   average(0(),s(s(0()))) = [1] >= [1] = s(0())
                            [1]    [0]         
  problem:
   average(s(x),y) -> average(x,s(y))
   average(x,s(s(s(y)))) -> s(average(s(x),y))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [average](x0, x1) = 4x0 + 4x1 + 6,
    
    [s](x0) = x0 + 1
   orientation:
    average(s(x),y) = 4x + 4y + 10 >= 4x + 4y + 10 = average(x,s(y))
    
    average(x,s(s(s(y)))) = 4x + 4y + 18 >= 4x + 4y + 11 = s(average(s(x),y))
   problem:
    average(s(x),y) -> average(x,s(y))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                         [1 1 1]     [1 0 0]  
     [average](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                         [0 0 0]     [0 0 0]  ,
     
               [1 0 0]     [0]
     [s](x0) = [0 1 1]x0 + [0]
               [0 0 0]     [1]
    orientation:
                       [1 1 1]    [1 0 0]    [1]    [1 1 1]    [1 0 0]                   
     average(s(x),y) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y = average(x,s(y))
                       [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]                   
    problem:
     
    Qed