 Solution

On two finite total lists, the equations agree. If either list is empty they return the other; if both have heads, expanding twiddle twice gives x : y : twiddle xs ys, exactly the third equation for twiddle'. The same finite-prefix reasoning proves agreement on infinite inputs with defined spines.

They differ on partial lists. Take the first argument [1] and the second argument $\perp$. The original definition can emit its first head immediately:After recognizing the nonempty first argument, twiddle' must inspect the second argument to distinguish its second and third equations. That inspection diverges, soFor example, demanding only the head distinguishes them: the original returns 1, while the primed version diverges. Equality of functions under lazy evaluation requires agreement on these partial inputs too.