YES

Problem:
 or(x,x) -> x
 and(x,x) -> x
 not(not(x)) -> x
 not(and(x,y)) -> or(not(x),not(y))
 not(or(x,y)) -> and(not(x),not(y))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [not](x0) = 4x0 + 2,
   
   [and](x0, x1) = x0 + 5x1 + 4,
   
   [or](x0, x1) = x0 + 5x1 + 6
  orientation:
   or(x,x) = 6x + 6 >= x = x
   
   and(x,x) = 6x + 4 >= x = x
   
   not(not(x)) = 16x + 10 >= x = x
   
   not(and(x,y)) = 4x + 20y + 18 >= 4x + 20y + 18 = or(not(x),not(y))
   
   not(or(x,y)) = 4x + 20y + 26 >= 4x + 20y + 16 = and(not(x),not(y))
  problem:
   not(and(x,y)) -> or(not(x),not(y))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [not](x0) = x0 + 1,
    
    [and](x0, x1) = x0 + x1 + 2,
    
    [or](x0, x1) = x0 + x1
   orientation:
    not(and(x,y)) = x + y + 3 >= x + y + 2 = or(not(x),not(y))
   problem:
    
   Qed