Proof that Alternating Step Generator and modifed ASG' have equivalent security?
Okay, I had a look at your source [12], e.g. ON EDIT DISTANCE ATTACK TO ALTERNATING STEP GENERATOR.

The original ASG defines the key stream Z = {z_i} from three shift registers X = {x_i}, Y = {y_i} and C = {c_i} via these steps:


  
  Initially, l = 0, t = 1.
  If ct = 1, then l = l + 1;
  Output z_t = x_l ⊕ y_(t−l) , where ⊕ is XOR operator;
  t = t + 1 and go to step 2.
  


E.g. we step one them depending on c_t, and use the latest output of both.



For the variant, we define w_i = z_i ⊕ z_(i-1), u_i = x_i ⊕ x_(i-1), v_i = y_i ⊕ y_(i-1), e.g simply the "difference sequences" of the original ones.

U = {u_i} and V = {v_i} can easily be created by LFSRs of the same length as these for X and Y, simply by adding or removing one feedback link. (And if we have the registers for U and V, we can get the registers for X and Y back.)

It now shows that we can derive W = {w_i} not only directly from Z, but also from U, V and C, using the modified construction:


  
  Initially l = 0, t = 1;
  If c_t = 1, then l = l + 1 and output w_t = u_l ;
  If c_t = 0, output w_t = v_(t−l).
  t = t + 1, go to step 2.
  


Why? Simply put the formulas:

If c_t = 1, then

w_t =      z_t        ⊕        z_(t-1)
    = (x_l ⊕ y_(t−l)) ⊕ (x_(l-1) ⊕ y_((t-1)−(l-1)))
    = (x_l ⊕ x_(l-1)) ⊕ (y_((t-1)−(l-1)) ⊕ y_(t−l))
    =      u_l        ⊕ (y_(t−l) ⊕ y_(t−l))
    =      u_l        ⊕         0
    =      u_l.


If c_t = 0, then

w_t =     z_t         ⊕     z_(t-1)
    = (x_l ⊕ y_(t−l)) ⊕ (x_l ⊕ y_((t-1)−l))
    = (x_l ⊕ x_l) ⊕ (y_(t−l) ⊕ y_((t-1)−l)
    =       0     ⊕ (y_(t−l) ⊕ y_((t-l)−1)
    =       0     ⊕      v_(t-l)
    =                    v_(t-l).


It is a bit confusing because of 1 and l looking quite similar here.

So, how would we do our attack on Z, given an attack on W?


Transform the observed sequence of Z into a sequence for W, by calculating differences of consecutive elements.
Break W, get formulas for C, U and V.
Transform the formulas for U and V back into these for X and Y by adding/removing
one feedback link.