$3$ Evaluation

For the tasks in this section, consider the following program, which we'll call $e.$

$e(fst(x$: int $$ int.$fstx,x$: int $$ int.$sndx))(snd((\frac{?}{\text{b}\text{a}\text{r}}(1)\frac{,}{b}ar(2)),(\frac{?}{\text{b}\text{a}\text{r}}(3)\frac{,}{b}ar(4))))$

We'll use $v$ to refer to the value that results from evaluating $e.$

Task $3\mathrm{.}1(7$ points$).$

Write a full derivation $($proof tree$)$ for a big$-$step evaluation of $e,$ that is$,$ a derivation of the conclusion $ev.$

Task $3\mathrm{.}2(7$ points$).$

Evaluate $e$ fully using control stacks. That is$,$ fill out all of the steps in the following outline:

$e$

$|...$

$|v$

where $e$ and $v$ are the program and return value above.

Task $3\mathrm{.}3(3$ points$).$

Decompose $e($from above$)$ into an evaluation context $E$ and expression $e_0$ such that $e=E[e_0]$ and $e_0|e_0^{\text{'}}$ for some $e_0^{\text{'}}.$ You don't need to define $e_0^{\text{'}}($just $E$ and $e_0)$ or justify this step. And you only need to do it for the first step of the evaluation, you don't need to step it all the way to a value.