YES

Problem:
 strict:
  top(left(car(x,y),car(old(),z))) -> top(right(y,car(old(),z)))
  top(left(car(x,car(old(),y)),z)) -> top(right(car(old(),y),z))
  top(right(x,car(y,car(old(),z)))) -> top(left(x,car(old(),z)))
  top(right(car(old(),x),car(y,z))) -> top(left(car(old(),x),z))
  top(left(bot(),car(old(),x))) -> top(right(bot(),car(old(),x)))
  top(right(car(old(),x),bot())) -> top(left(car(old(),x),bot()))
 weak:
  top(left(car(x,y),z)) -> top(left(y,z))
  top(right(x,car(y,z))) -> top(right(x,z))
  bot() -> car(new(),bot())

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
                     [1 1]     [1 0]     [1]
   [right](x0, x1) = [1 0]x0 + [0 0]x1 + [0],
   
           [0]
   [old] = [1],
   
               [1 0]  
   [top](x0) = [1 0]x0,
   
           [0]
   [bot] = [0],
   
                   [2 0]     [1 2]  
   [car](x0, x1) = [1 1]x0 + [1 0]x1,
   
           [0]
   [new] = [0],
   
                         [1 2]  
   [left](x0, x1) = x0 + [0 0]x1
  orientation:
                                      [2 0]    [1 2]    [3 2]    [2]    [1 1]    [1 2]    [1]                             
   top(left(car(x,y),car(old(),z))) = [2 0]x + [1 2]y + [3 2]z + [2] >= [1 1]y + [1 2]z + [1] = top(right(y,car(old(),z)))
   
                                      [2 0]    [3 2]    [1 2]    [2]    [2 2]    [1 0]    [2]                             
   top(left(car(x,car(old(),y)),z)) = [2 0]x + [3 2]y + [1 2]z + [2] >= [2 2]y + [1 0]z + [2] = top(right(car(old(),y),z))
   
                                       [1 1]    [2 0]    [3 2]    [3]    [1 0]    [3 2]    [2]                            
   top(right(x,car(y,car(old(),z)))) = [1 1]x + [2 0]y + [3 2]z + [3] >= [1 0]x + [3 2]z + [2] = top(left(x,car(old(),z)))
   
                                       [2 2]    [2 0]    [1 2]    [2]    [1 2]    [1 2]                             
   top(right(car(old(),x),car(y,z))) = [2 2]x + [2 0]y + [1 2]z + [2] >= [1 2]x + [1 2]z = top(left(car(old(),x),z))
   
                                   [3 2]    [2]    [1 2]    [1]                                 
   top(left(bot(),car(old(),x))) = [3 2]x + [2] >= [1 2]x + [1] = top(right(bot(),car(old(),x)))
   
                                    [2 2]    [2]    [1 2]                                 
   top(right(car(old(),x),bot())) = [2 2]x + [2] >= [1 2]x = top(left(car(old(),x),bot()))
   
                           [2 0]    [1 2]    [1 2]     [1 0]    [1 2]                  
   top(left(car(x,y),z)) = [2 0]x + [1 2]y + [1 2]z >= [1 0]y + [1 2]z = top(left(y,z))
   
                            [1 1]    [2 0]    [1 2]    [1]    [1 1]    [1 0]    [1]                  
   top(right(x,car(y,z))) = [1 1]x + [2 0]y + [1 2]z + [1] >= [1 1]x + [1 0]z + [1] = top(right(x,z))
   
           [0]    [0]                   
   bot() = [0] >= [0] = car(new(),bot())
  problem:
   strict:
    top(left(car(x,car(old(),y)),z)) -> top(right(car(old(),y),z))
   weak:
    top(left(car(x,y),z)) -> top(left(y,z))
    top(right(x,car(y,z))) -> top(right(x,z))
    bot() -> car(new(),bot())
  Matrix Interpretation Processor: dim=2
   
   interpretation:
                      [2 0]     [1 1]  
    [right](x0, x1) = [0 0]x0 + [0 2]x1,
    
            [1]
    [old] = [2],
    
                [1 1]     [0]
    [top](x0) = [0 0]x0 + [3],
    
            [0]
    [bot] = [0],
    
                           
    [car](x0, x1) = x0 + x1,
    
            [0]
    [new] = [0],
    
                     [2 0]     [2 0]  
    [left](x0, x1) = [1 0]x0 + [0 3]x1
   orientation:
                                       [3 0]    [3 0]    [2 3]    [3]    [2 0]    [1 3]    [2]                             
    top(left(car(x,car(old(),y)),z)) = [0 0]x + [0 0]y + [0 0]z + [3] >= [0 0]y + [0 0]z + [3] = top(right(car(old(),y),z))
    
                            [3 0]    [3 0]    [2 3]    [0]    [3 0]    [2 3]    [0]                 
    top(left(car(x,y),z)) = [0 0]x + [0 0]y + [0 0]z + [3] >= [0 0]y + [0 0]z + [3] = top(left(y,z))
    
                             [2 0]    [1 3]    [1 3]    [0]    [2 0]    [1 3]    [0]                  
    top(right(x,car(y,z))) = [0 0]x + [0 0]y + [0 0]z + [3] >= [0 0]x + [0 0]z + [3] = top(right(x,z))
    
            [0]    [0]                   
    bot() = [0] >= [0] = car(new(),bot())
   problem:
    strict:
     
    weak:
     top(left(car(x,y),z)) -> top(left(y,z))
     top(right(x,car(y,z))) -> top(right(x,z))
     bot() -> car(new(),bot())
   Qed