7. Numbers Games¶
Contents
本章节概要：
主要引入了数字，以及数字列表tup，
围绕着上面两个概念，定义了相关的一系列操作，并用递归实现之。
7.1. 基本概念¶
数字也是一个atom
number?: 判断一个S表达式是否为数字
add1: 对数字加1
(define add1 1+)
sub1: 对数字减1
(define sub1 1-)
zero?: 判断一个数字是否为0
tup: 列表中包含的每个s表达式都是数字的列表。
注解
书中为了简化概念，不考虑负数的情况。
7.2. +¶
将两个数字相加
; +; number number -> number
(define +
(lambda (a b)
(cond
((zero? b) a)
(else (add1 (+ a (sub1 b)))))))
; 另外一个版本
(define +
(lambda (a b)
(cond
((zero? b) a)
(else (+ (add1 a) (sub1 b))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define +&co
(lambda (a b col)
(cond
((zero? b) (col a))
(else (+&co a
(sub1 b)
(lambda (num)
(col (add1 num))))))))
; 另外一个版本
(define +&co
(lambda (a b col)
(cond
((zero? b) (col a))
(else (+&co (add1 a)
(sub1 b)
col)))))
7.3. -¶
将两个数字相减
注解
不考虑相减的结果为负数的情况
; -: number number -> number
(define -
(lambda (a b)
(cond
((zero? b) a)
(else (sub1 (- a (sub1 b)))))))
; 另一个版本
(define -
(lambda (a b)
(cond
((zero? b) a)
(else (- (sub1 a) (sub1 b))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define -&co
(lambda (a b col)
(cond
((zero? b) (col a))
(else (-&co a
(sub1 b)
(lambda (num)
(col (sub1 num))))))))
; 另一个版本
(define -&co
(lambda (a b col)
(cond
((zero? b) (col a))
(else (-&co (sub1 a) (sub1 b) col)))))
7.4. addtup¶
将一个tup中的所有数字相加
; addtup: tup -> number
(define addtup
(lambda (tup)
(cond
((null? tup) 0)
(else (+ (car tup) (addtup (cdr tup)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define addtup&co
(lambda (tup col)
(cond
((null? tup) (col 0))
(else
(addtup&co (cdr tup)
(lambda (sum)
(col (+ sum (car tup)))))))))
7.5. *¶
将两个数字相乘
; *: number number -> number
(define *
(lambda (n m)
(cond
((zero? m) 0)
(else (+ n (* n (sub1 m)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define *&co
(lambda (n m col)
(cond
((zero? m) (col 0))
(else (*&co n
(sub1 m)
(lambda (num)
(col (+ num n))))))))
7.6. tup+¶
将两个tup中相对的数字相加,然后返回相加后的tup
; tup1+: tup tup -> tup
; 该函数允许两个参数的长度不一样
(define tup1+
(lambda (tup1 tup2)
(cond
((null? tup2) tup1)
((null? tup1) tup2)
(else (cons (+ (car tup1) (car tup2))
(tup+ (cdr tup1) (cdr tup2)))))))
; tup2+: tup tup -> tup
; 该函数的两个参数长度必须一样
(define tup2+
(lambda (tup1 tup2)
(cond
((and (null? tup1) (null? tup2)) '())
(else (cons (+ (car tup1) (car tup2))
(tup+ (cdr tup1) (cdr tup2)))))))
(define col
(lambda (newtup)
(display newtup)
(newline)
newtup))
(define tup+&co
(lambda (tup1 tup2 col)
(cond
((null? tup1) (col tup2))
((null? tup2) (col tup1))
(else (tup+&co (cdr tup1)
(cdr tup2)
(lambda (newtup)
(col (cons (+ (car tup1)
(car tup2))
newtup))))))))
7.7. <¶
比较两个数字的大小
注解
< 函数未考虑两个数字都为0的情况
; >: number number -> bool
(define >
(lambda (n m)
(cond
((zero? n) #f)
((zero? m) #t)
(else (> (sub1 n) (sub1 m))))))
(define col
(lambda (bflag)
(display bflag)
(newline)
bflag))
(define >&co
(lambda (n m col)
(cond
((zero? n) (col #f))
((zero? m) (col #t))
(else (>&co (- n 1) (- m 1) col)))))
7.8. >¶
比较两个数字的大小
注解
< 函数未考虑两个数字都为0的情况
; <: number number -> bool
(define <
(lambda (n m)
(cond
((zero? m) #f)
((zero? n) #t)
(else (< (sub1 n) (sub1 m))))))
(define col
(lambda (bflag)
(display bflag)
(newline)
bflag))
(define <&co
(lambda (n m col)
(cond
((zero? m) (col #f))
((zero? n) (col #t))
(else (<&co (sub1 n) (sub1 m) col)))))
7.9. =¶
比较两个数字是否相等
; =: number number -> bool
(define =
(lambda (n m)
(cond
((zero? m) (zero? n))
((zero? n) #f)
(else (= (sub1 n) (sub1 m))))))
; 另一版本
(define =
(lambda (n m)
(cond
((or (> n m) (< n m)) #f)
(else #t))))
(define col
(lambda (bflag)
(display bflag)
(newline)
bflag))
(define =&co
(lambda (n m)
(cond
((zero? m) (col (zero? n)))
((zero? n) (col #f))
(else (=&co (sub1 n) (sub1 m) col)))))
7.10. ^¶
阶乘
; ^: number number -> number
(define ^
(lambda (n m)
(cond
((zero? n) 0)
((zero? m) 1)
(else (* n (^ n (sub1 m)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define ^&co
(lambda (n m col)
(cond
((zero? n) (col 0))
((zero? m) (col 1))
(else (^&co n
(sub1 m)
(lambda (num)
(col (* n num))))))))
7.11. /¶
除
; /: number number -> number
(define /
(lambda (n m)
(cond
((< n m) 0)
(else (add1 (/ (- n m) m))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define /&co
(lambda (n m col)
(cond
((< n m) (col 0))
(else (/&co (- n m)
m
(lambda (num)
(col (add1 num))))))))
7.12. length¶
返回一个lat的长度
; length: lat -> number
(define length
(lambda (lat)
(cond
((null? lat) 0)
(else (add1 (length (cdr lat)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define length&co
(lambda (lat col)
(cond
((null? lat) (col 0))
(else (length&co (cdr lat)
(lambda (num)
(col (add1 num))))))))
7.13. pick¶
根据传入的数字,获取其对应在lat中位置的S表达式
; pick: number lat -> symbol
(define pick
(lambda (n lat)
(cond
((zero? (sub1 n)) (car lat))
(else (pick (sub1 n) (cdr lat))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define pick&co
(lambda (n lat col)
(cond
((null? lat) (col '()))
((zero? (sub1 n)) (col (car lat)))
(else (pick&co (sub1 n) (cdr lat) col)))))
7.14. rempick¶
根据传入的数字, 删除其对应在lat中位置的S表达式,并返回剩余列表
; rempick: number lat -> lat
(define rempick
(lambda (n lat)
(cond
((null? lat) '())
((zero? (sub1 n)) (cdr lat))
(else (cons (car lat)
(rempick (sub1 n) (cdr lat)))))))
(define col
(lambda (lat)
(display lat)
(newline)
lat))
(define rempick&co
(lambda (n lat col)
(cond
((null? lat) (col '()))
((zero? (sub1 n)) (col (cdr lat)))
(else (rempick&co (sub1 n)
(cdr lat)
(lambda (newlat)
(col (cons (car lat)
newlat))))))))
7.15. no-nums¶
选出列表中的非数字S表达式, 并以列表形式返回
; no-nums: lat -> lat
(define no-nums
(lambda (lat)
(cond
((null? lat) '())
((number? (car lat)) (no-nums (cdr lat)))
(else (cons (car lat)
(no-nums (cdr lat)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define no-nums&co
(lambda (lat col)
(cond
((null? lat) (col '()))
((number? (car lat))
(no-nums&co (cdr lat) col))
(else
(no-nums&co (cdr lat)
(lambda (newlat)
(col (cons (car lat) newlat))))))))
7.16. all-nums¶
跟上面相反, 只返回数字列表
; all-nums: lat -> lat
(define all-nums
(lambda (lat)
(cond
((null? lat) '())
((not (number? (car lat))) (all-nums (cdr lat)))
(else (cons (car lat) (all-nums (cdr lat)))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define all-nums&co
(lambda (lat col)
(cond
((null? lat) (col '()))
((not (number? (car lat)))
(all-nums&co (cdr lat) col))
(else
(all-nums&co (cdr lat)
(lambda (newlat)
(col (cons (car lat) newlat))))))))
小技巧
no-nums 和 all-nums 的逻辑基本一样，
其实可以将它们的共同点抽出来。
(define col
(lambda (no-nums all-nums)
(dispaly no-nums)
(newline)
(display all-nums)))
(define split-nums&co
(lambda (lat col)
(cond
((null? lat) (col '() '()))
((number? (car lat))
(split-nums&co (cdr lat)
(lambda (no all)
(col no (cons (car lat) all)))))
(else
(split-nums&co (cdr lat)
(lambda (no all)
(col (cons (car lat) no) all)))))))
7.17. eqan?¶
比较两个S表达式是否是相等
小技巧
实现该函数是为了扩展 eq? 的功能，后面会用到该函数
; eqan? symbol symbol -> bool
(define eqan?
(lambda (a1 a2)
(cond
((and (number? a1) (number? a2))
(= a1 a2))
((or (number? a1) (number? a2))
#f)
(else (eq? a1 a2)))))
7.18. occur?¶
检查列表中有几个指定的S表达式
; occur: symbol lat -> number
(define occur
(lambda (a lat)
(cond
((null? lat) 0)
((eqan? a (car lat))
(add1 (occur a (cdr lat))))
(else (occur a (cdr lat))))))
(define col
(lambda (num)
(display num)
(newline)
num))
(define occur&co
(lambda (n lat col)
(cond
((null? lat) (col 0))
((eq? n (car lat))
(occur&co n
(cdr lat)
(lambda (num)
(col (add1 num)))))
(else
(occur&co n
(cdr lat)
col)))))
7.19. one?¶
判断一个数字是否为1
小技巧
写这个函数，应该是稍微提一下Scheme中的第八戒。
; one?: number -> bool
(define one?
(lambda (n)
(cond
((zero? n) #f)
(else (zero? (sub1 n))))))
; 另一版本
(define one?
(lambda (n)
(cond
(else (= n 1)))))
通过 one? 咱们就可以简写 rempick 函数了。
; rempick: number lat -> lat
(define rempick
(lambda (n lat)
(cond
((null? lat) '())
((one? n) (cdr lat))
(else (cons (car lat)
(rempick (sub1 n)
(cdr lat)))))))