YES

Problem:
 sum(0()) -> 0()
 sum(s(x)) -> +(sum(x),s(x))
 sum1(0()) -> 0()
 sum1(s(x)) -> s(+(sum1(x),+(x,x)))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [1 1 1]     [0]
   [sum1](x0) = [1 1 1]x0 + [1]
                [1 1 1]     [1],
   
               [1 1 0]     [0]
   [sum](x0) = [0 0 0]x0 + [0]
               [1 1 0]     [1],
   
                 [1 0 0]     [1 0 0]     [0]
   [+](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                 [0 0 0]     [0 0 0]     [1],
   
         [0]
   [0] = [0]
         [1],
   
             [1 0 0]  
   [s](x0) = [1 1 1]x0
             [1 0 1]  
  orientation:
              [0]    [0]      
   sum(0()) = [0] >= [0] = 0()
              [1]    [1]      
   
               [2 1 1]    [0]    [2 1 0]    [0]                 
   sum(s(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = +(sum(x),s(x))
               [2 1 1]    [1]    [0 0 0]    [1]                 
   
               [1]    [0]      
   sum1(0()) = [2] >= [0] = 0()
               [2]    [1]      
   
                [3 1 2]    [0]    [3 1 1]    [0]                       
   sum1(s(x)) = [3 1 2]x + [1] >= [3 1 1]x + [1] = s(+(sum1(x),+(x,x)))
                [3 1 2]    [1]    [3 1 1]    [1]                       
  problem:
   sum(0()) -> 0()
   sum(s(x)) -> +(sum(x),s(x))
   sum1(s(x)) -> s(+(sum1(x),+(x,x)))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                 [1 1 1]     [0]
    [sum1](x0) = [1 1 1]x0 + [1]
                 [1 1 1]     [1],
    
                [1 1 1]     [1]
    [sum](x0) = [1 0 1]x0 + [1]
                [1 1 0]     [0],
    
                  [1 0 0]     [1 0 0]     [0]
    [+](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [1]
                  [0 0 0]     [0 0 0]     [0],
    
          [0]
    [0] = [0]
          [0],
    
              [1 0 0]  
    [s](x0) = [1 0 1]x0
              [1 1 0]  
   orientation:
               [1]    [0]      
    sum(0()) = [1] >= [0] = 0()
               [0]    [0]      
    
                [3 1 1]    [1]    [2 1 1]    [1]                 
    sum(s(x)) = [2 1 0]x + [1] >= [1 1 0]x + [1] = +(sum(x),s(x))
                [2 0 1]    [0]    [0 0 0]    [0]                 
    
                 [3 1 1]    [0]    [3 1 1]    [0]                       
    sum1(s(x)) = [3 1 1]x + [1] >= [3 1 1]x + [0] = s(+(sum1(x),+(x,x)))
                 [3 1 1]    [1]    [3 1 1]    [1]                       
   problem:
    sum(s(x)) -> +(sum(x),s(x))
    sum1(s(x)) -> s(+(sum1(x),+(x,x)))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                  [1 1 1]  
     [sum1](x0) = [1 1 1]x0
                  [1 1 1]  ,
     
                 [1 1 0]     [0]
     [sum](x0) = [1 1 0]x0 + [1]
                 [0 0 0]     [1],
     
                   [1 0 0]     [1 0 0]  
     [+](x0, x1) = [0 0 0]x0 + [0 0 1]x1
                   [0 0 0]     [0 0 0]  ,
     
               [1 1 0]     [0]
     [s](x0) = [1 1 1]x0 + [0]
               [1 0 0]     [1]
    orientation:
                 [2 2 1]    [0]    [2 2 0]    [0]                 
     sum(s(x)) = [2 2 1]x + [1] >= [1 0 0]x + [1] = +(sum(x),s(x))
                 [0 0 0]    [1]    [0 0 0]    [0]                 
     
                  [3 2 1]    [1]    [3 1 1]    [0]                       
     sum1(s(x)) = [3 2 1]x + [1] >= [3 1 1]x + [0] = s(+(sum1(x),+(x,x)))
                  [3 2 1]    [1]    [3 1 1]    [1]                       
    problem:
     sum(s(x)) -> +(sum(x),s(x))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [sum](x0) = 4x0 + 3,
      
      [+](x0, x1) = 3x0 + x1,
      
      [s](x0) = 4x0 + 3
     orientation:
      sum(s(x)) = 16x + 15 >= 16x + 12 = +(sum(x),s(x))
     problem:
      
     Qed