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:
   [and](x0, x1) = 4x0 + x1 + 5,
   
   [implies](x0, x1) = 6x0 + 4x1 + 2,
   
   [false] = 0,
   
   [not](x0) = 5x0 + 2
  orientation:
   and(x,false()) = 4x + 5 >= 0 = false()
   
   and(x,not(false())) = 4x + 7 >= x = x
   
   not(not(x)) = 25x + 12 >= x = x
   
   implies(false(),y) = 4y + 2 >= 2 = not(false())
   
   implies(x,false()) = 6x + 2 >= 5x + 2 = not(x)
   
   implies(not(x),not(y)) = 30x + 20y + 22 >= 16x + 10y + 22 = implies(y,and(x,y))
  problem:
   implies(false(),y) -> not(false())
   implies(x,false()) -> not(x)
   implies(not(x),not(y)) -> implies(y,and(x,y))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [and](x0, x1) = 4x0 + x1 + 2,
    
    [implies](x0, x1) = 5x0 + 4x1 + 6,
    
    [false] = 2,
    
    [not](x0) = 5x0 + 1
   orientation:
    implies(false(),y) = 4y + 16 >= 11 = not(false())
    
    implies(x,false()) = 5x + 14 >= 5x + 1 = not(x)
    
    implies(not(x),not(y)) = 25x + 20y + 15 >= 16x + 9y + 14 = implies(y,and(x,y))
   problem:
    
   Qed