R5RS Scheme (Dr.Racket)
;; Towards a Scheme Interpreter for the Lambda Calculus — Part 1: Syntax
;; 5 points
;; , and pre-requisite for all subsequent parts of the project
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;; All programming is to be carried out using the pure functional sublanguage of R5RS Scheme.
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;; You might want to have a look at
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;; 1. The lambda calculus is a particularly simple programming language consisting only of
;; variable references, lambda expressions with a single formal parameter, and function
;; applications. A BNF definition of lambda calculus expressions is
;; <expr> ::= <variable> | (lambda ( <variable> ) <expr> ) | ( <expr> <expr> )
;; Design a data type for the lambda calculus, with constructors, selectors, and classifiers.
;; For concrete representation, use Scheme, as follows: an identifier should be represented as
;; a quoted Scheme variable, a lambda expression (lambda (x) E) as the quoted 3-element list
;; ‘(lambda (x) [list representing E]), and an application (E1 E2) as the quoted 2-element list
;; ‘([list representing E1] [list representing E2])
;; 2. In (lambda (<variable>) <expr>), we say that <variable> is a binder that
;; binds all occurrences of that variable in the body, <expr>, unless some intervening
;; binder of the same variable occurs. Thus in (lambda (x) (x (lambda (x) x))),
;; the first occurrence of x binds the second occurrence of x, but not
;; the fourth. The third occurrence of x binds the fourth occurrence of x.
;; A variable x occurs free in an expression E if there is some occurrence of x which is not
;; bound by any binder of x in E. A variable x occurs bound in an expression E if it is
;; not free in E. Thus x occurs free in (lambda (y) x), bound in (lambda (x) x), and both
;; free and bound in (lambda (y) (x (lambda (x) x))).
;; As a consequence of this definition, we can say that a variable x occurs free in a
;; lambda calculus expression E iff one of the following holds:
;; (i) E = x
;; (ii) E = (lambda (y) E’), where x is distinct from y and x occurs free in E’
;; (iii) E = (E’ E”) and x occurs free in E’ or x occurs free in E”
;; Observe that this is an inductive definition, exploiting the structure of lambda calculus
;; expressions.
;; Similarly, a variable x occurs bound in a lambda calculus expression E iff one of the
;; following holds:
;; (i) E = (lambda (x) E’) and x occurs free in E’
;; (ii) E = (lambda (y) E’), and x occurs bound in E’: here, y may be x, or distinct from x
;; (iii) E = (E1 E2) and x occurs bound in either E1 or E2
;; Develop and prove correct a procedure free-vars that inputs a list representing a lambda calculus
;; expression E and outputs a list without repetitions (that is, a set) of the variables occurring
;; free in E.
;; Develop and prove correct a procedure bound-vars that inputs a list representing a lambda calculus
;; expression E and outputs the set of variables which occur bound in E.
;; 3. Define a function all-ids which returns the set of all symbols — free or bound variables,
;; as well as the lambda identifiers for which there are no bound occurrences — which occur in
;; a lambda calculus expression E.
