YES

Problem:
 and(x,false()) -> false()
 and(x,not(false())) -> x
 not(not(x)) -> x
 implies(false(),y) -> not(false())
 implies(x,false()) -> not(x)
 implies(not(x),not(y)) -> implies(y,and(x,y))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [implies](x0, x1) = 2x0 + 2x1 + 6,
   
   [not](x0) = 2x0 + 5,
   
   [and](x0, x1) = x0 + x1,
   
   [false] = 0
  orientation:
   and(x,false()) = x >= 0 = false()
   
   and(x,not(false())) = x + 5 >= x = x
   
   not(not(x)) = 4x + 15 >= x = x
   
   implies(false(),y) = 2y + 6 >= 5 = not(false())
   
   implies(x,false()) = 2x + 6 >= 2x + 5 = not(x)
   
   implies(not(x),not(y)) = 4x + 4y + 26 >= 2x + 4y + 6 = implies(y,and(x,y))
  problem:
   and(x,false()) -> false()
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                    [1 0 0]     [1 0 0]     [1]
    [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                    [0 0 0]     [0 0 0]     [0],
    
              [0]
    [false] = [0]
              [0]
   orientation:
                     [1 0 0]    [1]    [0]          
    and(x,false()) = [0 0 0]x + [0] >= [0] = false()
                     [0 0 0]    [0]    [0]          
   problem:
    
   Qed