As you can see, Language Four extends Language Three with three new constructs: the $<$ operator for comparison, the $-$ operator for subtraction, and the conditional $($if$-$then$-$else$)$ expression. The sample program above defines a recursive facto$-$rial function and uses it to compute the factorial of $5.$

a$.$ Define the syntax of Language Four by extending the syntax of Lan$-$guage Three with the three additional constructs required. Show your new BNF$.$ Make sure it is unambiguous.

b$.$ Define the three new kinds of AST nodes you need to match your extended syntax. Extend the Prolog implementation of Language Three to handle them. $($You will need to start with the dynamic$-$scoping implementation, since the static$-$scoping one cannot handle recursive definitions. For this reason, the sample program above is not legal in ML$.)$ Verify that your imple$-$mentation evaluates the sample program correctly$$the factorial of $5$ is $120.$

c$.$ Give a natural semantics for Language Four.