 Solution

Keep the neighboring Fibonacci number in a private closure. Before calling the transition with $x=F_n$, let its captured following equal $F_{n+1}$.The transition returns $F_{n+1}$ and stores $F_n+F_{n+1}=F_{n+2}$, preserving the paired Fibonacci recurrence. Initialization gives $F_0=0$ and $F_1=1$, soUsing BigInt allows all Fibonacci numbers rather than a finite prefix before integer overflow. Replacing BigInt by Int gives the source's suggested machine-integer interface with that limitation. With the FunIter implementation from part (c), each transition runs once per generated cell, and copied iterators can advance independently while sharing the privately owned generator. A simpler alternative uses a pure function on pairs, $(a,b)\mapsto (b,a+b)$, and projects the first component; the closure above follows the source's suggested single-value approach directly.