YES

Problem:
 implies(not(x),y) -> or(x,y)
 implies(not(x),or(y,z)) -> implies(y,or(x,z))
 implies(x,or(y,z)) -> or(y,implies(x,z))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [or](x0, x1) = x0 + x1 + 1,
   
   [implies](x0, x1) = 2x0 + 4x1 + 4,
   
   [not](x0) = 2x0 + 4
  orientation:
   implies(not(x),y) = 4x + 4y + 12 >= x + y + 1 = or(x,y)
   
   implies(not(x),or(y,z)) = 4x + 4y + 4z + 16 >= 4x + 2y + 4z + 8 = implies(y,or(x,z))
   
   implies(x,or(y,z)) = 2x + 4y + 4z + 8 >= 2x + y + 4z + 5 = or(y,implies(x,z))
  problem:
   
  Qed