YES

Problem:
 norm(nil()) -> 0()
 norm(g(x,y)) -> s(norm(x))
 f(x,nil()) -> g(nil(),x)
 f(x,g(y,z)) -> g(f(x,y),z)
 rem(nil(),y) -> nil()
 rem(g(x,y),0()) -> g(x,y)
 rem(g(x,y),s(z)) -> rem(x,z)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [rem](x0, x1) = 4x0 + x1,
   
   [f](x0, x1) = 4x0 + 2x1,
   
   [s](x0) = x0,
   
   [g](x0, x1) = x0 + 4x1,
   
   [0] = 0,
   
   [norm](x0) = 4x0 + 6,
   
   [nil] = 0
  orientation:
   norm(nil()) = 6 >= 0 = 0()
   
   norm(g(x,y)) = 4x + 16y + 6 >= 4x + 6 = s(norm(x))
   
   f(x,nil()) = 4x >= 4x = g(nil(),x)
   
   f(x,g(y,z)) = 4x + 2y + 8z >= 4x + 2y + 4z = g(f(x,y),z)
   
   rem(nil(),y) = y >= 0 = nil()
   
   rem(g(x,y),0()) = 4x + 16y >= x + 4y = g(x,y)
   
   rem(g(x,y),s(z)) = 4x + 16y + z >= 4x + z = rem(x,z)
  problem:
   norm(g(x,y)) -> s(norm(x))
   f(x,nil()) -> g(nil(),x)
   f(x,g(y,z)) -> g(f(x,y),z)
   rem(nil(),y) -> nil()
   rem(g(x,y),0()) -> g(x,y)
   rem(g(x,y),s(z)) -> rem(x,z)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                    [1 0 1]     [1 0 0]  
    [rem](x0, x1) = [0 1 0]x0 + [0 0 0]x1
                    [0 0 1]     [0 0 1]  ,
    
                  [1 0 1]     [1 1 0]  
    [f](x0, x1) = [1 0 0]x0 + [1 1 1]x1
                  [0 1 0]     [0 0 1]  ,
    
              [1 0 0]     [0]
    [s](x0) = [0 0 0]x0 + [0]
              [0 0 1]     [1],
    
                       [1 0 0]     [0]
    [g](x0, x1) = x0 + [0 0 0]x1 + [1]
                       [0 1 0]     [0],
    
          [0]
    [0] = [0]
          [0],
    
                 [1 1 0]  
    [norm](x0) = [0 0 0]x0
                 [0 1 0]  ,
    
            [0]
    [nil] = [0]
            [1]
   orientation:
                   [1 1 0]    [1 0 0]    [1]    [1 1 0]    [0]             
    norm(g(x,y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0] = s(norm(x))
                   [0 1 0]    [0 0 0]    [1]    [0 1 0]    [1]             
    
                 [1 0 1]    [0]    [1 0 0]    [0]             
    f(x,nil()) = [1 0 0]x + [1] >= [0 0 0]x + [1] = g(nil(),x)
                 [0 1 0]    [1]    [0 1 0]    [1]             
    
                  [1 0 1]    [1 1 0]    [1 0 0]    [1]    [1 0 1]    [1 1 0]    [1 0 0]    [0]              
    f(x,g(y,z)) = [1 0 0]x + [1 1 1]y + [1 1 0]z + [1] >= [1 0 0]x + [1 1 1]y + [0 0 0]z + [1] = g(f(x,y),z)
                  [0 1 0]    [0 0 1]    [0 1 0]    [0]    [0 1 0]    [0 0 1]    [0 1 0]    [0]              
    
                   [1 0 0]    [1]    [0]        
    rem(nil(),y) = [0 0 0]y + [0] >= [0] = nil()
                   [0 0 1]    [1]    [1]        
    
                      [1 0 1]    [1 1 0]    [0]        [1 0 0]    [0]         
    rem(g(x,y),0()) = [0 1 0]x + [0 0 0]y + [1] >= x + [0 0 0]y + [1] = g(x,y)
                      [0 0 1]    [0 1 0]    [0]        [0 1 0]    [0]         
    
                       [1 0 1]    [1 1 0]    [1 0 0]    [0]    [1 0 1]    [1 0 0]            
    rem(g(x,y),s(z)) = [0 1 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 1 0]x + [0 0 0]z = rem(x,z)
                       [0 0 1]    [0 1 0]    [0 0 1]    [1]    [0 0 1]    [0 0 1]            
   problem:
    f(x,nil()) -> g(nil(),x)
    rem(g(x,y),0()) -> g(x,y)
    rem(g(x,y),s(z)) -> rem(x,z)
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                     [1 0 0]     [1 0 1]  
     [rem](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                     [0 0 0]     [1 0 0]  ,
     
                   [1 0 0]     [1 0 1]     [0]
     [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                   [0 0 0]     [0 0 0]     [1],
     
               [1 0 0]     [0]
     [s](x0) = [0 0 0]x0 + [0]
               [0 0 1]     [1],
     
                   [1 0 0]     [1 0 0]     [0]
     [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                   [0 0 0]     [0 0 0]     [1],
     
           [1]
     [0] = [0]
           [0],
     
             [0]
     [nil] = [0]
             [1]
    orientation:
                  [1 0 0]    [1]    [1 0 0]    [0]             
     f(x,nil()) = [0 0 0]x + [0] >= [0 0 0]x + [0] = g(nil(),x)
                  [0 0 0]    [1]    [0 0 0]    [1]             
     
                       [1 0 0]    [1 0 0]    [1]    [1 0 0]    [1 0 0]    [0]         
     rem(g(x,y),0()) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = g(x,y)
                       [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]         
     
                        [1 0 0]    [1 0 0]    [1 0 1]    [1]    [1 0 0]    [1 0 1]            
     rem(g(x,y),s(z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z + [0] >= [0 0 0]x + [0 0 0]z = rem(x,z)
                        [0 0 0]    [0 0 0]    [1 0 0]    [0]    [0 0 0]    [1 0 0]            
    problem:
     
    Qed