YES

Problem:
 and(not(not(x)),y,not(z)) -> and(y,band(x,z),x)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [band](x0, x1) = 3x0 + x1 + 3,
   
   [and](x0, x1, x2) = 6x0 + 6x1 + 5x2,
   
   [not](x0) = 2x0 + 1
  orientation:
   and(not(not(x)),y,not(z)) = 24x + 6y + 10z + 23 >= 23x + 6y + 6z + 18 = and(y,band(x,z),x)
  problem:
   
  Qed