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=3
  
  interpretation:
                  [1 1 0]     [1 0 0]  
   [or](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                  [0 0 0]     [0 0 0]  ,
   
                       [1 1 0]     [1 0 0]  
   [implies](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                       [0 0 0]     [0 0 0]  ,
   
               [1 0 0]     [0]
   [not](x0) = [0 1 0]x0 + [1]
               [0 0 0]     [0]
  orientation:
                       [1 1 0]    [1 0 0]    [1]    [1 1 0]    [1 0 0]           
   implies(not(x),y) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y = or(x,y)
                       [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]           
   
                             [1 1 0]    [1 1 0]    [1 0 0]    [1]    [1 1 0]    [1 1 0]    [1 0 0]                      
   implies(not(x),or(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z + [0] >= [0 0 0]x + [0 0 0]y + [0 0 0]z = implies(y,or(x,z))
                             [0 0 0]    [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]    [0 0 0]                      
   
                        [1 1 0]    [1 1 0]    [1 0 0]     [1 1 0]    [1 1 0]    [1 0 0]                      
   implies(x,or(y,z)) = [0 0 0]x + [0 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = or(y,implies(x,z))
                        [0 0 0]    [0 0 0]    [0 0 0]     [0 0 0]    [0 0 0]    [0 0 0]                      
  problem:
   implies(x,or(y,z)) -> or(y,implies(x,z))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [or](x0, x1) = 4x0 + x1 + 4,
    
    [implies](x0, x1) = x0 + 6x1
   orientation:
    implies(x,or(y,z)) = x + 24y + 6z + 24 >= x + 4y + 6z + 4 = or(y,implies(x,z))
   problem:
    
   Qed