$1$ Functions with Multiple Arguments $($with Deferred Substitutions$)$

Start with the f$1$WAE interpreter for deferred substitution, and extend the implementation to support any number of arguments to a function $($including zero$),$ and any number of arguments $($including zero$)$ in a function application:As with homework $1,$ you must change the f$1$ WAE datatype, and you must thus provide a parse function that produces values of your modified f$1$WAE datatype. It must accept a quoted expression and produce an $$ nWAE value. Similarly, you must provide a parse$-$defn function to parse definitions.

See homework $1$ for details on these functions.

You must also provide an interp$-$expr function with signature:

fnWAE $($listof FunDef$)->$ Number

This function should be a very simple wrapper for your interp function, which our tests will call on the results of your parsers. If yours is more than a few lines $($one line is possible$),$ you may be overthinking it$.$

$2$ Errors

Your interpreter and parser must obey the formats and precedence rules described in homework $1.$

Otherwise, assume that the input to your parser is a well$-$formed program.

$3$ Conditionals

Add if$0,$ a conditional expression. It has three subexpressions:$2$ Errors

Your interpreter and parser must obey the formats and precedence rules described in homework $1.$ Otherwise, assume that the input to your parser is a well$-$formed program.

$3$ Conditionals

Add if$0,$ a conditional expression. It has three subexpressions:

$$ fnWAE$>=$cdots

$|if0||$

$1$

Evaluating an if$0$ expression evaluates the first subexpression; if it produces $0,$ then the result of the entire expression is the result of the second subexpression. Otherwise, the result is the result of the third subexpression.

Examples:

$($test $($interp$-$expr $($parse $\text{'}\{$if$0012\})\text{'}())1)$

$($test $($interp$-$expr $($parse $\text{'}\{$if$0123\})\text{'}())3)4$ Negative predicate

Implement, in the fnWAE language $($without any extensions, i$.$e$.,$ you cannot add new kinds of expressions to the language or to your interpreter$),$ a predicate neg? that determines if an integer is negative. That is$,$ write a defun like this one that can be passed to your parse function and then used with your interp function $($with a suitable expression that calls it$)$:

$\{$deffun neg? x$\}...\}$

It must return either $0($if the input is negative$),$ or $1($if not$).$ The number $0$ is not itself considered negative.

$5$ Multiplication on integers

As you did in the previous problem, implement $($in the fnWAE language without any extensions$),$ a function mult that computes the product of two integers.

$\{$deffun mult x y$\}...\}$

$6$ Handin instructions

Provide definitions for parse, parse$-$defn, and interp$-$expr, as above.

Provide a PLAI$-$level definition of mult$-$and$-$neg$-$deffuns that is bound to a list of $($unparsed$)$ deffuns that contains both neg? and mult as well as any helper functions you need: