YES

Problem:
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 if(0(),y,z) -> y
 if(s(x),y,z) -> z
 half(double(x)) -> x

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                      [1 0 0]            
   [if](x0, x1, x2) = [0 0 0]x0 + x1 + x2
                      [0 0 0]            ,
   
                      [1 0 0]  
   [-](x0, x1) = x0 + [0 0 0]x1
                      [0 0 0]  ,
   
                [1 0 1]     [0]
   [half](x0) = [0 0 1]x0 + [0]
                [1 0 0]     [1],
   
               
   [s](x0) = x0
               ,
   
                  [1 0 1]     [0]
   [double](x0) = [0 0 0]x0 + [1]
                  [0 1 0]     [0],
   
         [0]
   [0] = [1]
         [1]
  orientation:
                 [1]    [0]      
   double(0()) = [1] >= [1] = 0()
                 [1]    [1]      
   
                  [1 0 1]    [0]    [1 0 1]    [0]                  
   double(s(x)) = [0 0 0]x + [1] >= [0 0 0]x + [1] = s(s(double(x)))
                  [0 1 0]    [0]    [0 1 0]    [0]                  
   
               [1]    [0]      
   half(0()) = [1] >= [1] = 0()
               [1]    [1]      
   
                  [1]    [0]      
   half(s(0())) = [1] >= [1] = 0()
                  [1]    [1]      
   
                   [1 0 1]    [0]    [1 0 1]    [0]             
   half(s(s(x))) = [0 0 1]x + [0] >= [0 0 1]x + [0] = s(half(x))
                   [1 0 0]    [1]    [1 0 0]    [1]             
   
                        
   -(x,0()) = x >= x = x
                        
   
                      [1 0 0]         [1 0 0]          
   -(s(x),s(y)) = x + [0 0 0]y >= x + [0 0 0]y = -(x,y)
                      [0 0 0]         [0 0 0]          
   
                               
   if(0(),y,z) = y + z >= y = y
                               
   
                  [1 0 0]                  
   if(s(x),y,z) = [0 0 0]x + y + z >= z = z
                  [0 0 0]                  
   
                     [1 1 1]    [0]         
   half(double(x)) = [0 1 0]x + [0] >= x = x
                     [1 0 1]    [1]         
  problem:
   double(s(x)) -> s(s(double(x)))
   half(s(s(x))) -> s(half(x))
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   if(0(),y,z) -> y
   if(s(x),y,z) -> z
   half(double(x)) -> x
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                       [1 1 0]            
    [if](x0, x1, x2) = [0 0 0]x0 + x1 + x2
                       [0 0 0]            ,
    
                  [1 0 0]     [1 0 1]  
    [-](x0, x1) = [1 1 0]x0 + [0 0 0]x1
                  [0 0 1]     [0 0 0]  ,
    
                 [1 0 0]     [1]
    [half](x0) = [0 1 0]x0 + [0]
                 [1 0 1]     [0],
    
              [1 1 0]  
    [s](x0) = [0 0 0]x0
              [0 0 1]  ,
    
                        [0]
    [double](x0) = x0 + [0]
                        [1],
    
          [0]
    [0] = [1]
          [0]
   orientation:
                   [1 1 0]    [0]    [1 1 0]    [0]                  
    double(s(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = s(s(double(x)))
                   [0 0 1]    [1]    [0 0 1]    [1]                  
    
                    [1 1 0]    [1]    [1 1 0]    [1]             
    half(s(s(x))) = [0 0 0]x + [0] >= [0 0 0]x + [0] = s(half(x))
                    [1 1 1]    [0]    [1 0 1]    [0]             
    
               [1 0 0]          
    -(x,0()) = [1 1 0]x >= x = x
               [0 0 1]          
    
                   [1 1 0]    [1 1 1]     [1 0 0]    [1 0 1]          
    -(s(x),s(y)) = [1 1 0]x + [0 0 0]y >= [1 1 0]x + [0 0 0]y = -(x,y)
                   [0 0 1]    [0 0 0]     [0 0 1]    [0 0 0]          
    
                          [1]         
    if(0(),y,z) = y + z + [0] >= y = y
                          [0]         
    
                   [1 1 0]                  
    if(s(x),y,z) = [0 0 0]x + y + z >= z = z
                   [0 0 0]                  
    
                      [1 0 0]    [1]         
    half(double(x)) = [0 1 0]x + [0] >= x = x
                      [1 0 1]    [1]         
   problem:
    double(s(x)) -> s(s(double(x)))
    half(s(s(x))) -> s(half(x))
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    if(s(x),y,z) -> z
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                        [1 0 0]     [1 0 0]       
     [if](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + x2
                        [0 0 0]     [0 0 0]       ,
     
                   [1 0 0]     [1 1 1]  
     [-](x0, x1) = [0 1 1]x0 + [0 0 0]x1
                   [0 1 1]     [1 0 0]  ,
     
                  [1 1 0]  
     [half](x0) = [1 1 0]x0
                  [1 1 0]  ,
     
               [1 0 0]     [0]
     [s](x0) = [0 0 1]x0 + [0]
               [0 1 0]     [1],
     
                    [1 1 1]  
     [double](x0) = [0 1 1]x0
                    [0 1 1]  ,
     
           [1]
     [0] = [0]
           [0]
    orientation:
                    [1 1 1]    [1]    [1 1 1]    [0]                  
     double(s(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = s(s(double(x)))
                    [0 1 1]    [1]    [0 1 1]    [1]                  
     
                     [1 1 0]    [1]    [1 1 0]    [0]             
     half(s(s(x))) = [1 1 0]x + [1] >= [1 1 0]x + [0] = s(half(x))
                     [1 1 0]    [1]    [1 1 0]    [1]             
     
                [1 0 0]    [1]         
     -(x,0()) = [0 1 1]x + [0] >= x = x
                [0 1 1]    [1]         
     
                    [1 0 0]    [1 1 1]    [1]    [1 0 0]    [1 1 1]          
     -(s(x),s(y)) = [0 1 1]x + [0 0 0]y + [1] >= [0 1 1]x + [0 0 0]y = -(x,y)
                    [0 1 1]    [1 0 0]    [1]    [0 1 1]    [1 0 0]          
     
                    [1 0 0]    [1 0 0]              
     if(s(x),y,z) = [0 0 1]x + [0 0 0]y + z >= z = z
                    [0 0 0]    [0 0 0]              
    problem:
     if(s(x),y,z) -> z
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                         [1 0 0]     [1 0 0]       
      [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + x2
                         [0 1 0]     [0 0 0]       ,
      
                [1 0 0]     [1]
      [s](x0) = [0 0 1]x0 + [0]
                [0 0 0]     [0]
     orientation:
                     [1 0 0]    [1 0 0]        [1]         
      if(s(x),y,z) = [0 0 0]x + [0 0 0]y + z + [0] >= z = z
                     [0 0 1]    [0 0 0]        [0]         
     problem:
      
     Qed