8.02/2.73	YES
8.02/2.74	
8.02/2.74	Problem:
8.02/2.74	 strict:
8.02/2.74	  average(s(x),y) -> average(x,s(y))
8.02/2.74	  average(x,s(s(s(y)))) -> s(average(s(x),y))
8.02/2.74	  average(0(),0()) -> 0()
8.02/2.74	  average(0(),s(0())) -> 0()
8.02/2.74	  average(0(),s(s(0()))) -> s(0())
8.02/2.74	 weak:
8.02/2.74	  rand(x) -> x
8.02/2.74	  rand(x) -> rand(s(x))
8.02/2.74	
8.02/2.74	Proof:
8.02/2.74	 Matrix Interpretation Processor: dim=2
8.02/2.74	  
8.02/2.74	  interpretation:
8.02/2.74	                [1 2]     [2]
8.02/2.74	   [rand](x0) = [0 1]x0 + [0],
8.02/2.74	   
8.02/2.74	         [0]
8.02/2.74	   [0] = [0],
8.02/2.74	   
8.02/2.74	                       [1 2]     [1 2]     [2]
8.02/2.74	   [average](x0, x1) = [0 0]x0 + [0 0]x1 + [0],
8.02/2.74	   
8.02/2.74	             [1 2]  
8.02/2.74	   [s](x0) = [0 0]x0
8.02/2.74	  orientation:
8.02/2.74	                     [1 2]    [1 2]    [2]    [1 2]    [1 2]    [2]                  
8.02/2.74	   average(s(x),y) = [0 0]x + [0 0]y + [0] >= [0 0]x + [0 0]y + [0] = average(x,s(y))
8.02/2.74	   
8.02/2.74	                           [1 2]    [1 2]    [2]    [1 2]    [1 2]    [2]                     
8.02/2.74	   average(x,s(s(s(y)))) = [0 0]x + [0 0]y + [0] >= [0 0]x + [0 0]y + [0] = s(average(s(x),y))
8.02/2.74	   
8.02/2.74	                      [2]    [0]      
8.02/2.74	   average(0(),0()) = [0] >= [0] = 0()
8.02/2.74	   
8.02/2.74	                         [2]    [0]      
8.02/2.74	   average(0(),s(0())) = [0] >= [0] = 0()
8.02/2.74	   
8.02/2.74	                            [2]    [0]         
8.02/2.74	   average(0(),s(s(0()))) = [0] >= [0] = s(0())
8.02/2.74	   
8.02/2.74	             [1 2]    [2]         
8.02/2.74	   rand(x) = [0 1]x + [0] >= x = x
8.02/2.74	   
8.02/2.74	             [1 2]    [2]    [1 2]    [2]             
8.02/2.74	   rand(x) = [0 1]x + [0] >= [0 0]x + [0] = rand(s(x))
8.02/2.74	  problem:
8.02/2.74	   strict:
8.02/2.74	    average(s(x),y) -> average(x,s(y))
8.02/2.74	    average(x,s(s(s(y)))) -> s(average(s(x),y))
8.02/2.74	   weak:
8.02/2.74	    rand(x) -> rand(s(x))
8.02/2.74	  Matrix Interpretation Processor: dim=5
8.02/2.74	   
8.02/2.74	   interpretation:
8.02/2.74	                 [1 1 1 0 0]  
8.02/2.74	                 [0 1 0 0 0]  
8.02/2.74	    [rand](x0) = [0 1 0 0 0]x0
8.02/2.74	                 [0 0 0 0 0]  
8.02/2.74	                 [1 0 0 0 0]  ,
8.02/2.74	    
8.02/2.74	                        [1 0 0 0 1]     [1 0 0 0 1]  
8.02/2.74	                        [0 0 0 0 0]     [0 0 0 0 0]  
8.02/2.74	    [average](x0, x1) = [0 0 0 0 0]x0 + [1 0 1 0 0]x1
8.02/2.74	                        [0 0 0 0 0]     [0 0 0 0 0]  
8.02/2.74	                        [0 0 0 0 1]     [0 0 0 0 1]  ,
8.02/2.74	    
8.02/2.74	              [1 0 0 0 0]     [0]
8.02/2.74	              [0 0 0 0 0]     [0]
8.02/2.74	    [s](x0) = [0 0 0 0 0]x0 + [0]
8.02/2.74	              [0 0 0 0 0]     [0]
8.02/2.74	              [0 0 0 0 1]     [1]
8.02/2.74	   orientation:
8.02/2.74	                      [1 0 0 0 1]    [1 0 0 0 1]    [1]    [1 0 0 0 1]    [1 0 0 0 1]    [1]                  
8.02/2.74	                      [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                  
8.02/2.74	    average(s(x),y) = [0 0 0 0 0]x + [1 0 1 0 0]y + [0] >= [0 0 0 0 0]x + [1 0 0 0 0]y + [0] = average(x,s(y))
8.02/2.74	                      [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                  
8.02/2.74	                      [0 0 0 0 1]    [0 0 0 0 1]    [1]    [0 0 0 0 1]    [0 0 0 0 1]    [1]                  
8.02/2.74	    
8.02/2.74	                            [1 0 0 0 1]    [1 0 0 0 1]    [3]    [1 0 0 0 1]    [1 0 0 0 1]    [1]                     
8.02/2.74	                            [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                     
8.02/2.74	    average(x,s(s(s(y)))) = [0 0 0 0 0]x + [1 0 0 0 0]y + [0] >= [0 0 0 0 0]x + [0 0 0 0 0]y + [0] = s(average(s(x),y))
8.02/2.74	                            [0 0 0 0 0]    [0 0 0 0 0]    [0]    [0 0 0 0 0]    [0 0 0 0 0]    [0]                     
8.02/2.74	                            [0 0 0 0 1]    [0 0 0 0 1]    [3]    [0 0 0 0 1]    [0 0 0 0 1]    [2]                     
8.02/2.74	    
8.02/2.74	              [1 1 1 0 0]     [1 0 0 0 0]              
8.02/2.74	              [0 1 0 0 0]     [0 0 0 0 0]              
8.02/2.74	    rand(x) = [0 1 0 0 0]x >= [0 0 0 0 0]x = rand(s(x))
8.02/2.74	              [0 0 0 0 0]     [0 0 0 0 0]              
8.02/2.74	              [1 0 0 0 0]     [1 0 0 0 0]              
8.02/2.74	   problem:
8.02/2.74	    strict:
8.02/2.74	     average(s(x),y) -> average(x,s(y))
8.02/2.74	    weak:
8.02/2.74	     rand(x) -> rand(s(x))
8.02/2.74	   Matrix Interpretation Processor: dim=2
8.02/2.74	    
8.02/2.74	    interpretation:
8.02/2.74	                  [2 0]     [2]
8.02/2.74	     [rand](x0) = [0 0]x0 + [2],
8.02/2.74	     
8.02/2.74	                         [2 1]     [2 0]  
8.02/2.74	     [average](x0, x1) = [2 1]x0 + [0 1]x1,
8.02/2.74	     
8.02/2.74	                    [0]
8.02/2.74	     [s](x0) = x0 + [2]
8.02/2.74	    orientation:
8.02/2.74	                       [2 1]    [2 0]    [2]    [2 1]    [2 0]    [0]                  
8.02/2.74	     average(s(x),y) = [2 1]x + [0 1]y + [2] >= [2 1]x + [0 1]y + [2] = average(x,s(y))
8.02/2.74	     
8.02/2.74	               [2 0]    [2]    [2 0]    [2]             
8.02/2.74	     rand(x) = [0 0]x + [2] >= [0 0]x + [2] = rand(s(x))
8.02/2.74	    problem:
8.02/2.74	     strict:
8.02/2.74	      
8.02/2.74	     weak:
8.02/2.74	      rand(x) -> rand(s(x))
8.02/2.74	    Qed
8.02/2.74	EOF