1. Chapter 1 Review: BNF Grammar for lists:
<list> ::= ({<datum>}*)
<dotted-datum> ::= ({<datum>}+ . <datum>)
<vector> ::= #({<datum>}*)
<datum> ::= <number> | <symbol> | <boolean> | <string>
::= <list> | <dotted-datum> | <vector>
Let's derive (a “mixed” # (bag (of . data))) from this grammar....

2. Chapter 1 Review: Free and Bound Variables
A variable x occurs free in a l-calculus expression E iff:
1. E is a variable reference and E = x; or
2. E is of the form (l (y) E') where y ≠ x and x occurs free
in E'; or
3. E is of the form (E1 E2) and x occurs free in E1 or E2
Let's prove that b occurs free in (l (a) (a b))

3. Chapter 1 Review: Free and Bound Variables
A variable x occurs bound in a l-calculus expression E iff:
1. E is of the form (l (y) E') where x occurs bound in E'
or x = y and y occurs free in E'; or
2. E is of the form (E1 E2) and x occurs bound in E1 or E2
Let's prove that a occurs bound in (l (a) (a b))

4. Coding It Up: Follow the Definition
(define occurs-free?
(lambda (var exp)
(cond
1. E is a variable reference and E = x; or
((symbol? exp) (eqv? var exp))
2. E is of the form (l (y) E') where y ≠ x and x occurs free in E'; or
((eqv? (car exp) 'lambda)
(and (not (eqv? (caadr exp) var))
(occurs-free? var (caddr exp))))
3. E is of the form (E1 E2) and x occurs free in E1 or E2
((else (or (occurs-free? var (car exp))
(occurs-free? var (cadr exp)))))))

5. (define occurs-bound?
(lambda (var exp)
(cond
0. symbol by itself isn't bound
((symbol? exp) #f)
1. E is of the form (l (y) E') where x occurs bound in E'
((eqv? (car exp) 'lambda)
(or (occurs-bound? var (caddr exp))
or x = y and y occurs free in E'; or
(and (eqv? (caadr exp) var)
(occurs-free? var (caddr exp)))))
2. E is of the form (E1 E2) and x occurs bound in E1 or E2
(else (or (occurs-bound? var (car exp))
(occurs-bound? var (cadr exp)))))))

6. Use helper functions! (define (lambda-exp? exp)
(eqv? (car exp?) 'lambda))
(define (lambda-arg exp)
(caadr exp))
(define (lambda-body exp)
(caddr exp))
(define (app-fun exp)
(car exp))
(define (app-arg exp)
(cadr exp))

7. (define occurs-free?
(lambda (var exp)
(cond
1. E is a variable reference and E = x; or
((symbol? exp) (eqv? var exp))
2. E is of the form (l (y) E') where y ≠ x and x occurs free in E'; or
((lambda-exp? exp)
(and (not (eqv? (lambda-arg exp) var)
(occurs-free? var (lambda-body exp))))
3. E is of the form (E1 E2) and x occurs free in E1 or E2
((else (or (occurs-free? var (app-fun exp))
(occurs-free? var (app-arg exp)))))))