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Hypothetical “Implementation” of pairs
(define cons (lambda (a b) (lambda (x) (if x a b))))
(define (car p) (p #t))
Do type analysis on the above. p is located inside a single open paren and is followed by a Boolean. It must therefore be a procedure which takes in a Boolean and returns something (we don’t know what).
(define (cdr p) (p #f))
(car (cons 1 2)) => 1 (cdr (cons 1 2)) => 2
((lambda (p) (p #t)) ((lambda (a b) (lambda (x) (if x a b)))) 1 (2))
((lambda (p) (p #t)) (lambda (x) (if x 1 2)))
((lambda (x) (if x 1 2)) #t)
(if #t 1 2) => 1
Three arguments of cons
(define cons (lambda (a b c) (lambda (x) (if (= x 0) a (if (= x 4 2) b c)))))
(define (car p) (p 0))
(define (cdr p) (p 42))
(define (cgr p) (p 7))
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The three above definitions are nonsensical, created to make a point that we can define functions any way we wish, with any number or value we choose.
Sorting…in ascending order.
(define (min x y) (if (> x y) y x))
(define (min-list lst) (if (null? (cdr lst)) (car lst) (min (car lst) (min-list (cdr lst)))))
(define (without-n lst n) (if (null? lst) nil (if (= (car lst) n) (cdr lst) (cons (car lst) (without-n (cdr lst) n)))))
(define (sort-list lst) (if (or (null? lst) (null? (cdr lst))) lst (let ((least (min-list lst))) (cons least (sort-list (without-n lst least))))))
Create a function insert, to insert an element into a list (unsorted), using only one helper function.
(define (insert elem lst) (if (null? lst) (list elem) (if (< elem (car lst)) (cons elem lst) (cons (car lst) (insert elem (cdr lst))))))
(define (insert-sort lst) (if (or (null? lst) (null? cdr lst)) lst (insert (car lst) (insert-sort (cdr lst)))))
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