YES

Problem:
 strict:
  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())
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
                       [2 0]     [2 0]  
   [average](x0, x1) = [2 1]x0 + [0 0]x1,
   
                [2 0]     [2]
   [rand](x0) = [0 2]x0 + [0],
   
               
   [s](x0) = x0,
   
         [0]
   [0] = [2]
  orientation:
                     [2 0]    [2 0]     [2 0]    [2 0]                   
   average(s(x),y) = [2 1]x + [0 0]y >= [2 1]x + [0 0]y = average(x,s(y))
   
                           [2 0]    [2 0]     [2 0]    [2 0]                      
   average(x,s(s(s(y)))) = [2 1]x + [0 0]y >= [2 1]x + [0 0]y = s(average(s(x),y))
   
                      [0]    [0]      
   average(0(),0()) = [2] >= [2] = 0()
   
                         [0]    [0]      
   average(0(),s(0())) = [2] >= [2] = 0()
   
                            [0]    [0]         
   average(0(),s(s(0()))) = [2] >= [2] = s(0())
   
             [2 0]    [2]         
   rand(x) = [0 2]x + [0] >= x = x
   
             [2 0]    [2]    [2 0]    [2]             
   rand(x) = [0 2]x + [0] >= [0 2]x + [0] = rand(s(x))
  problem:
   strict:
    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())
   weak:
    rand(x) -> rand(s(x))
  Matrix Interpretation Processor: dim=2
   
   interpretation:
                        [1 3]     [1 0]  
    [average](x0, x1) = [2 2]x0 + [0 0]x1,
    
                 [2 2]     [2]
    [rand](x0) = [0 0]x0 + [2],
    
                
    [s](x0) = x0,
    
          [0]
    [0] = [1]
   orientation:
                      [1 3]    [1 0]     [1 3]    [1 0]                   
    average(s(x),y) = [2 2]x + [0 0]y >= [2 2]x + [0 0]y = average(x,s(y))
    
                            [1 3]    [1 0]     [1 3]    [1 0]                      
    average(x,s(s(s(y)))) = [2 2]x + [0 0]y >= [2 2]x + [0 0]y = s(average(s(x),y))
    
                       [3]    [0]      
    average(0(),0()) = [2] >= [1] = 0()
    
                          [3]    [0]      
    average(0(),s(0())) = [2] >= [1] = 0()
    
                             [3]    [0]         
    average(0(),s(s(0()))) = [2] >= [1] = s(0())
    
              [2 2]    [2]    [2 2]    [2]             
    rand(x) = [0 0]x + [2] >= [0 0]x + [2] = rand(s(x))
   problem:
    strict:
     average(s(x),y) -> average(x,s(y))
     average(x,s(s(s(y)))) -> s(average(s(x),y))
    weak:
     rand(x) -> rand(s(x))
   Matrix Interpretation Processor: dim=2
    
    interpretation:
                         [2 1]     [2 1]  
     [average](x0, x1) = [0 1]x0 + [0 1]x1,
     
                  [1 0]     [2]
     [rand](x0) = [0 0]x0 + [1],
     
                    [0]
     [s](x0) = x0 + [1]
    orientation:
                       [2 1]    [2 1]    [1]    [2 1]    [2 1]    [1]                  
     average(s(x),y) = [0 1]x + [0 1]y + [1] >= [0 1]x + [0 1]y + [1] = average(x,s(y))
     
                             [2 1]    [2 1]    [3]    [2 1]    [2 1]    [1]                     
     average(x,s(s(s(y)))) = [0 1]x + [0 1]y + [3] >= [0 1]x + [0 1]y + [2] = s(average(s(x),y))
     
               [1 0]    [2]    [1 0]    [2]             
     rand(x) = [0 0]x + [1] >= [0 0]x + [1] = rand(s(x))
    problem:
     strict:
      average(s(x),y) -> average(x,s(y))
     weak:
      rand(x) -> rand(s(x))
    Matrix Interpretation Processor: dim=2
     
     interpretation:
                          [1 2]     [2 0]     [1]
      [average](x0, x1) = [0 3]x0 + [0 0]x1 + [0],
      
                   [1 0]     [1]
      [rand](x0) = [0 0]x0 + [0],
      
                [1 0]     [0]
      [s](x0) = [1 1]x0 + [1]
     orientation:
                        [3 2]    [2 0]    [3]    [1 2]    [2 0]    [1]                  
      average(s(x),y) = [3 3]x + [0 0]y + [3] >= [0 3]x + [0 0]y + [0] = average(x,s(y))
      
                [1 0]    [1]    [1 0]    [1]             
      rand(x) = [0 0]x + [0] >= [0 0]x + [0] = rand(s(x))
     problem:
      strict:
       
      weak:
       rand(x) -> rand(s(x))
     Qed